Pianotech

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074

Original Message:
Sent: 05-15-2024 15:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 12:47
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Tim,

Thanks for reposting, as yours was simply taken down by the 'system' when it was following one that was taken down for not being on point.

That being said, I encourage you to post all UT preference opinions on the new thread by S. Rosenthal, which he humorously and appropriately entitled "A Tempermental Journey." This is a great new place for the UT guys, since it is a very small minority who want UT on a modern concert grand piano. To each his own.

There is room at the table for all here! I respect that, but also adhere to the 150-year-old tradition of the move toward atonality that Liszt and Wagner started and matured around the turn of 1900 with the New Viennese School of composers. The German tuners had, by historic accounts, already embraced ET by 1850 and some say the English tuners even before that. I love the sound of Rubinstein, Horowitz, Bronfman, Argerich, Hamlin, and all the 'greats' of prior and recent who play the full range of literature on great-sounding ET tunings on fabulous Steinway D's. So, ET is here to stay, and I don't see serious pianists turning back the clock to pre-1850. 

On a personal 'note', my Fazioli sounds to the ears of everyone who has been in the room, 'the best sounding tuning they have ever heard' and that came from 4 tuners each with over 35 years of high-level tuning history. It simply came from Pure12th being spot on using the 'new technology'. I can play literature from Bach to Bartok and it all comes off well. Everything is unbiased and consonant sounding. It sounds 'mainstream.'

Respectfully, 

Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101
------------------------------

Original Message:
Sent: 05-15-2024 19:49
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074

Original Message:
Sent: 05-15-2024 15:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 12:47
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Kent,

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Wikipedia

I intentionally wrote the article from the perspective of a professional musician who grew up in that culture.

I knew in advance I would catch flack from the piano tuners. So be it.

Best,

Steve

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Kent,

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Wikipedia

I intentionally wrote the article from the perspective of a professional musician who grew up in that culture.

I knew in advance I would catch flack from the piano tuners. So be it.

Best,

Steve

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Kent,

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Wikipedia

I intentionally wrote the article from the perspective of a professional musician who grew up in that culture.

I knew in advance I would catch flack from the piano tuners. So be it.

Best,

Steve

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

While it is true that in music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, this definition is not sufficient for pianos because having a fundamental frequency ratio of 3:2 does not guarantee a beatless fifth, which is the way a listener recognizes a perfect interval.  The only reason the precise mathematical ratio has any significance to the human ear is that harmonics are exact whole-number multiples of the fundamental.  In a perfect fifth the 3rd harmonic of the lower note and the 2nd harmonic of the upper note are identical in frequency and therefore will be beatless.  But pianos do not have harmonics.  They have partials that are slightly sharp from what the harmonic would be.  Therefore a perfect fifth on a piano (the one that would sound beatless) would require that the fundamental have a ratio that is slightly more than 3:2.  But it gets even more complicated when we consider that a fifth has other coincident partials besides 3:2, such as 6:4 and 9:6.  And unlike instruments like wind instruments with pure harmonics, the 6:4 perfect fifth will not occur at the same tuning as the perfect 3:2 perfect fifth.  Therefore there are several different kinds of "perfect fifths" on a piano, and we need to specify which one we mean.  I suspect that if you tuned C1/G1 for a perfect fifth would be listening for the 6:4 fifth, but when tuning C5/G5 as a perfect fifth you would be listening for the 3:2 fifth because that would be the most prominent coincident partials.  For instruments without inharmonicity, the 3:2, 6:4, 9:6, etc. would all be perfect at the same time. 

------------------------------
Robert Scott
Real-Time Specialties (TuneLab)
fixthatpiano@yahoo.com
Hopkins MN

Original Message:
Sent: 05-16-2024 13:16
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Kent,

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Wikipedia

I intentionally wrote the article from the perspective of a professional musician who grew up in that culture.

I knew in advance I would catch flack from the piano tuners. So be it.

Best,

Steve

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-16-2024 10:56
From: Kent Swafford
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/15/2024 7:50:00 PM
From: Tim Foster
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074

Original Message:
Sent: 05-15-2024 15:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 12:47
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Robert,

All wind and brass instruments have inharmonicity in their natural 'positions'. For example, the exponential shape of a brass instrument bell is a mathematical approximation to put the overtone series as close as possible to harmonic integers but it is still 'imperfect' and the instrumentalist has to make on-the-fly adjustments of embouchure to compensate. Of course the pianist cannot make those adjustments during a performance. Steven Rosenthal's post of the Debussy Clair de Lune is very revealing. The human listener has 'adapted' to ET on the piano as a compromise to the full range of keys and repertoire. Does anyone think the example of Evgeny Kissin's Clair de Lune is anything other than 'aesthetically lovely?' You can bet his Hamburg D was tuned with ET of some sort. 

You missed my point that the 12th or 3rd overtone IS PERFECT with Pure12th. That is a very common interval for spread chords on a piano. So, 'no', the 5th (3:2) is not Pure/Perfect but close (-1.23¢) but the 3:1 is Pure/Perfect with proper Pure12th ET using the best new tech app and device.

Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101
------------------------------

Original Message:
Sent: 05-16-2024 15:39
From: Robert Scott
Subject: It's All About the Perfect 5th! A Musicological Perspective

While it is true that in music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, this definition is not sufficient for pianos because having a fundamental frequency ratio of 3:2 does not guarantee a beatless fifth, which is the way a listener recognizes a perfect interval.  The only reason the precise mathematical ratio has any significance to the human ear is that harmonics are exact whole-number multiples of the fundamental.  In a perfect fifth the 3rd harmonic of the lower note and the 2nd harmonic of the upper note are identical in frequency and therefore will be beatless.  But pianos do not have harmonics.  They have partials that are slightly sharp from what the harmonic would be.  Therefore a perfect fifth on a piano (the one that would sound beatless) would require that the fundamental have a ratio that is slightly more than 3:2.  But it gets even more complicated when we consider that a fifth has other coincident partials besides 3:2, such as 6:4 and 9:6.  And unlike instruments like wind instruments with pure harmonics, the 6:4 perfect fifth will not occur at the same tuning as the perfect 3:2 perfect fifth.  Therefore there are several different kinds of "perfect fifths" on a piano, and we need to specify which one we mean.  I suspect that if you tuned C1/G1 for a perfect fifth would be listening for the 6:4 fifth, but when tuning C5/G5 as a perfect fifth you would be listening for the 3:2 fifth because that would be the most prominent coincident partials.  For instruments without inharmonicity, the 3:2, 6:4, 9:6, etc. would all be perfect at the same time. 

------------------------------
Robert Scott
Real-Time Specialties (TuneLab)
fixthatpiano@yahoo.com
Hopkins MN

Original Message:
Sent: 05-16-2024 13:16
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Kent,

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Wikipedia

I intentionally wrote the article from the perspective of a professional musician who grew up in that culture.

I knew in advance I would catch flack from the piano tuners. So be it.

Best,

Steve

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-16-2024 10:56
From: Kent Swafford
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/15/2024 7:50:00 PM
From: Tim Foster
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074

Original Message:
Sent: 05-15-2024 15:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 12:47
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Robert,

All wind and brass instruments have inharmonicity in their natural 'positions'. For example, the exponential shape of a brass instrument bell is a mathematical approximation to put the overtone series as close as possible to harmonic integers but it is still 'imperfect' and the instrumentalist has to make on-the-fly adjustments of embouchure to compensate. Of course the pianist cannot make those adjustments during a performance. Steven Rosenthal's post of the Debussy Clair de Lune is very revealing. The human listener has 'adapted' to ET on the piano as a compromise to the full range of keys and repertoire. Does anyone think the example of Evgeny Kissin's Clair de Lune is anything other than 'aesthetically lovely?' You can bet his Hamburg D was tuned with ET of some sort. 

You missed my point that the 12th or 3rd overtone IS PERFECT with Pure12th. That is a very common interval for spread chords on a piano. So, 'no', the 5th (3:2) is not Pure/Perfect but close (-1.23¢) but the 3:1 is Pure/Perfect with proper Pure12th ET using the best new tech app and device.

Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-16-2024 15:39
From: Robert Scott
Subject: It's All About the Perfect 5th! A Musicological Perspective

While it is true that in music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, this definition is not sufficient for pianos because having a fundamental frequency ratio of 3:2 does not guarantee a beatless fifth, which is the way a listener recognizes a perfect interval.  The only reason the precise mathematical ratio has any significance to the human ear is that harmonics are exact whole-number multiples of the fundamental.  In a perfect fifth the 3rd harmonic of the lower note and the 2nd harmonic of the upper note are identical in frequency and therefore will be beatless.  But pianos do not have harmonics.  They have partials that are slightly sharp from what the harmonic would be.  Therefore a perfect fifth on a piano (the one that would sound beatless) would require that the fundamental have a ratio that is slightly more than 3:2.  But it gets even more complicated when we consider that a fifth has other coincident partials besides 3:2, such as 6:4 and 9:6.  And unlike instruments like wind instruments with pure harmonics, the 6:4 perfect fifth will not occur at the same tuning as the perfect 3:2 perfect fifth.  Therefore there are several different kinds of "perfect fifths" on a piano, and we need to specify which one we mean.  I suspect that if you tuned C1/G1 for a perfect fifth would be listening for the 6:4 fifth, but when tuning C5/G5 as a perfect fifth you would be listening for the 3:2 fifth because that would be the most prominent coincident partials.  For instruments without inharmonicity, the 3:2, 6:4, 9:6, etc. would all be perfect at the same time. 

------------------------------
Robert Scott
Real-Time Specialties (TuneLab)
fixthatpiano@yahoo.com
Hopkins MN

Original Message:
Sent: 05-16-2024 13:16
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Kent,

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Wikipedia

I intentionally wrote the article from the perspective of a professional musician who grew up in that culture.

I knew in advance I would catch flack from the piano tuners. So be it.

Best,

Steve

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-16-2024 10:56
From: Kent Swafford
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/15/2024 7:50:00 PM
From: Tim Foster
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074

Original Message:
Sent: 05-15-2024 15:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 12:47
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Steve, you are right about the natural resonant frequencies of a wind instrument being inharmonic.  But I was referring to the overtones of a single note.  In a wind instrument those overtones will be locked together.  The waveform of a wind instrument is periodic.  Every periodic waveform of frequency F can be decomposed into a series of sine waves of frequencies nF for n = 1,2,3....  In other words, harmonics.  The waveform of a piano note cannot be represented in this manner because the waveform is not periodic.

As for which intervals are perfect, any interval can be made perfect by a corresponding tuning.  You can have a perfect 3:2 fifth tuning, a perfect 6:2 twelfth tuning, or a perfect 4:2 octave tuning, etc.  As for which of these tunings sound the best, that is entirely another matter and not one I am qualified to comment on.  I will leave it up to musicologists like you to decide that.                      

Robert,

All wind and brass instruments have inharmonicity in their natural 'positions'. For example, the exponential shape of a brass instrument bell is a mathematical approximation to put the overtone series as close as possible to harmonic integers but it is still 'imperfect' and the instrumentalist has to make on-the-fly adjustments of embouchure to compensate. Of course the pianist cannot make those adjustments during a performance. Steven Rosenthal's post of the Debussy Clair de Lune is very revealing. The human listener has 'adapted' to ET on the piano as a compromise to the full range of keys and repertoire. Does anyone think the example of Evgeny Kissin's Clair de Lune is anything other than 'aesthetically lovely?' You can bet his Hamburg D was tuned with ET of some sort. 

You missed my point that the 12th or 3rd overtone IS PERFECT with Pure12th. That is a very common interval for spread chords on a piano. So, 'no', the 5th (3:2) is not Pure/Perfect but close (-1.23¢) but the 3:1 is Pure/Perfect with proper Pure12th ET using the best new tech app and device.

Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-16-2024 15:39
From: Robert Scott
Subject: It's All About the Perfect 5th! A Musicological Perspective

While it is true that in music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, this definition is not sufficient for pianos because having a fundamental frequency ratio of 3:2 does not guarantee a beatless fifth, which is the way a listener recognizes a perfect interval.  The only reason the precise mathematical ratio has any significance to the human ear is that harmonics are exact whole-number multiples of the fundamental.  In a perfect fifth the 3rd harmonic of the lower note and the 2nd harmonic of the upper note are identical in frequency and therefore will be beatless.  But pianos do not have harmonics.  They have partials that are slightly sharp from what the harmonic would be.  Therefore a perfect fifth on a piano (the one that would sound beatless) would require that the fundamental have a ratio that is slightly more than 3:2.  But it gets even more complicated when we consider that a fifth has other coincident partials besides 3:2, such as 6:4 and 9:6.  And unlike instruments like wind instruments with pure harmonics, the 6:4 perfect fifth will not occur at the same tuning as the perfect 3:2 perfect fifth.  Therefore there are several different kinds of "perfect fifths" on a piano, and we need to specify which one we mean.  I suspect that if you tuned C1/G1 for a perfect fifth would be listening for the 6:4 fifth, but when tuning C5/G5 as a perfect fifth you would be listening for the 3:2 fifth because that would be the most prominent coincident partials.  For instruments without inharmonicity, the 3:2, 6:4, 9:6, etc. would all be perfect at the same time. 

------------------------------
Robert Scott
Real-Time Specialties (TuneLab)
fixthatpiano@yahoo.com
Hopkins MN

Original Message:
Sent: 05-16-2024 13:16
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Kent,

In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. Wikipedia

I intentionally wrote the article from the perspective of a professional musician who grew up in that culture.

I knew in advance I would catch flack from the piano tuners. So be it.

Best,

Steve

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-16-2024 10:56
From: Kent Swafford
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/15/2024 7:50:00 PM
From: Tim Foster
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

------------------------------
Tim Foster RPT
New Oxford PA
(470) 231-6074

Original Message:
Sent: 05-15-2024 15:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 12:47
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Kent writes: 

>>Tim cites the Pythagorean M3rd as 22 cents wide, and the pure octave ET M3rd as 14 cents wide. These are rounded figures. Please note that the pure 12th ET M3rd _also_ rounds to 14 cents wide.(!) Where are these wide M3rds of pure 12th ET so often complained about? They don't actually exist! To be a bit more precise, Pythagorean M3rds are 21.5 cents expanded.<< 

    Actually, the Pythagorean comma is 23.4 cents wide.   The 21.5 cents being referenced is the Syntonic comma.  These are arrived at by different means, and the syntonic comma is the near-universally accepted limit for a M3 in the WT's. I also submit that the word "Pure" in tempering discussions is best used to describe an untempered interval.  

Regards,  

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Kent writes: 

>>Tim cites the Pythagorean M3rd as 22 cents wide, and the pure octave ET M3rd as 14 cents wide. These are rounded figures. Please note that the pure 12th ET M3rd _also_ rounds to 14 cents wide.(!) Where are these wide M3rds of pure 12th ET so often complained about? They don't actually exist! To be a bit more precise, Pythagorean M3rds are 21.5 cents expanded.<< 

    Actually, the Pythagorean comma is 23.4 cents wide.   The 21.5 cents being referenced is the Syntonic comma.  These are arrived at by different means, and the syntonic comma is the near-universally accepted limit for a M3 in the WT's. I also submit that the word "Pure" in tempering discussions is best used to describe an untempered interval.  

Regards,  

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

------------------------------
Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
------------------------------

Kent writes: 

>>Tim cites the Pythagorean M3rd as 22 cents wide, and the pure octave ET M3rd as 14 cents wide. These are rounded figures. Please note that the pure 12th ET M3rd _also_ rounds to 14 cents wide.(!) Where are these wide M3rds of pure 12th ET so often complained about? They don't actually exist! To be a bit more precise, Pythagorean M3rds are 21.5 cents expanded.<< 

    Actually, the Pythagorean comma is 23.4 cents wide.   The 21.5 cents being referenced is the Syntonic comma.  These are arrived at by different means, and the syntonic comma is the near-universally accepted limit for a M3 in the WT's. I also submit that the word "Pure" in tempering discussions is best used to describe an untempered interval.  

Regards,  

I am a little confused why my critique of the OP was removed, presumably marked "inappropriate." It was a direct reply to the content of the OP from a musicological perspective. I would like to repost since it was in fact on topic.

___________

Steven,

Thank you for sharing this perspective. I agree that the fifth is incredibly important for many of the reasons you state, but I would disagree with your conclusion that tuners should tune P12 for the reasons you gave. This is not to say that I don't think there is value in P12 tuning and if you or others like it, by all means use it. 

Certainly, parallel organum of the Middle Ages utilizes the P5, but during the Middle Ages the M3 was considered dissonant. In Pythagorean tuning, M3s are 22% wide from perfect if memory serves. In ET they are around 14%. As music developed in western society, 3rds became a function of consonance in music in contrast to earlier times. Tempering at least some 5ths not only allowed avoiding the wolf but also allowed for more pleasing (slower beating) 3rds. 

If there is any doubt about the importance of 3rds in western music, consider the 5ths that were compromised almost universally in well and Victorian temperaments, following the circle of fifths: C-G, G-D, D-A, A-E. This was very intentional, since this sequence placed E closer to C which allows the M3 to beat more slowly, in some cases making a nearly P3. Well temperaments usually prioritize nearly P3rds in the keys closer to C in the circle of 5ths, such as G and F, all of which keys were considered the most consonant, not because of P5s but because of the 3rds. 3rds were in fact foundational to western music as we know it, and the most consonant keys had the most tempered fifths.

The statement "perfect 5ths are essential for building chords" is not correct. While 5ths are essential, P5s are not as I have demonstrated above. Even the relatively rapidly beating G-D in Kirnberger III is partially masked by the 3rd in a G major triad.

In functional harmony, it is correct to see significance in the Tonic/Dominant (I/V) relationship, but again this is not a necessarily a P5 relationship. Further, it is not so much the fifth comprising the root of the dominant chord that creates tension, but the tritone which is created from the 5th and 7th partials of the dominant root. This translates primarily to the 3rd and 7th (i.e. 3rd from the root and a 3rd from the 5th) of the chord being the primarily creators of tension that "want" to resolve back to the tonic (I).

For these reasons, I do not recommend P12 tuning universally, especially for classical music. P12 tuning by necessity makes faster beating 3rds which I believe I have given a sufficient argument as to their historic importance in the development of western music. Functional harmony depends on 5ths for certain, but the P5s as you stated in the OP are not necessary for functional harmony at all. Since they are not necessary for functional harmony, P12 tuning is not necessarily the best tuning for western music.

Colleagues, in compliance with Patrick's request, and then seeing thereafter extremely repetitive long replies from one in particular that were an afront after Patrick made the polite request on the rules, I marked those particular long replies that were off-topic as inappropriate. For those who made comments after, I enjoyed reading your on-topic replies and invite you to repost them. If we keep venturing into UT, I request those interested in UT to start a whole new thread on that subject. Let's keep the subject on the original subject matter. Kindly, and professionally, Steve. 

Peter, from what I can gather, you will need 'not just any app', but an app that has accurate full IH (some do and some don't) in addition to an accurate tuning curve generation for Pure 12th. Given those conditions, the result will be fine on any scale piano. I'd recommend a consultation with a Pure-12th expert who has evaluated the apps that provide this. I think we both know someone who can opine privately on this from experience! --- Steve N.

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(619) 964-0101

Original Message:
Sent: 05-09-2024 10:27
From: Peter Grey
Subject: It's All About the Perfect 5th! A Musicological Perspective

Another (but not necessarily the smartest, best,  or most authoritative) answer is: On a well scaled (and generally reasonably large) instrument...no argument whatsoever from an ET perspective. However,  on a PSO (of which there are MANY in existence and often comprise a high % of the typical piano tuner's clientele ownership and thus a proportionately higher % of exposure to requiring 'best compromise' theory and practice for optimal musical usage) it simply doesn't always "work as advertised". Therefore we need other "tools" at our disposal to effect a reasonably acceptable result. 

Additionally, in the case of PSO's that are used almost universally for beginners or "elementary" playing, there exist highly satisfactory compromises that will enhance the musicality of the most used key signatures at the expense of the least (or never) used key signatures, and if mutually agreed upon can produce a superior result under such limited playing capacity. 

Each temperament scheme has its place and since piano tuners are the ones doing it (in most cases) it's up to us to determine (in consultation with whoever is playing (and/or paying) what is the best application under the circumstances. 

Others may disagree... 🤔 

Peter Grey Piano Doctor 

------------------------------
Peter Grey
Stratham NH
(603) 686-2395
pianodoctor57@gmail.com

Original Message:
Sent: 05-09-2024 08:46
From: Rick Butler
Subject: It's All About the Perfect 5th! A Musicological Perspective

Question: Why would a piano tuner, from a musicological perspective, consider using thirds, tenths, and seventeenths, given the fundamental importance of the Perfect 5th to tune a piano? 

Answer: The use of interval comparisons using thirds, tenths, seventeenths, and, in particular, minor thirds can be used to accurately measure and adjust the temper of octaves and fifths. They allow one to form and execute a tuning strategy that produces the best musical result for each piano they tune. 

------------------------------
Rick Butler RPT
The Butler School of Piano Technology
Bowie MD
240 396 7480
RickRickRickRickRick

Original Message:
Sent: 05-09-2024 01:22
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

IT'S ALL ABOUT THE PERFECT 5^th: A Musicological Perspective 

I am writing this little essay 'for piano tuners' but 'from the perspective of a musicologist.' Let's talk about what is actually important in music regarding the intervals we use and how we tune.

Overall, perfect fifths are fundamental to the structure and sound of Western music, playing a key role in harmony, chord construction, tuning systems, tonal relationships, and overall musical expression.

The intervals of the octave and the perfect fifth have remained the two most important building blocks in Western music; the octave providing the framework of stability and repetition; and the fifth, generating the primary harmonic driving force, the dominant to tonic cadence.

The 'perfect fifth' is more consonant (consonance vs. dissonance), or stable, than any other interval except the unison and the octave. In some sense, it is even more important than the octave, which I will explain. It occurs above the root of all major (and minor) chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth, hence, the 'Pure-12^th Piano Tuning' achieves the musical goal of a 'Perfect 5th.' Violins, violas, and cellos  are tuned in 5ths. The 'Circle of 5ths' is a fundamental concept. So, it wasn't by accident that the first steps towards harmony in Western music started with the first intervals of the overtone series. The early polyphony of the middle ages consisted in one or more voice parts accompanying the cantus firmus, often in parallel motion using the interval of the octave, fifth or fourth (the first intervals found in the overtone series).    

Harmony: Perfect fifths are considered one of the most consonant intervals in music. When two notes are played a perfect fifth apart, they sound pleasing and stable to the ear. This stability forms the foundation of harmony in Western music.

Chord Building: Perfect fifths are essential for building chords. In traditional Western music theory, chords are often constructed by stacking notes in intervals of a third (major or minor). The perfect fifth is a common interval found in many chords, such as the power chord (used in rock music) and the perfect fifth interval itself forms the basis of the dominant triad in tonal music.

Tuning Systems: Perfect fifths are crucial in tuning systems. In the context of equal temperament tuning (the tuning system commonly used in Western music), the perfect fifth is the interval that is tempered or adjusted to allow all keys to be played equally well. This compromise is necessary to ensure that music can be played in all keys without sounding out of tune.

Functional Harmony: In tonal music, the perfect fifth plays a significant role in establishing tonal centers and creating harmonic tension and resolution. For example, the perfect fifth relationship between the tonic and dominant is fundamental in establishing tonality and creating a sense of resolution when moving from the dominant back to the tonic.

Melodic and Harmonic Context: Perfect fifths are prevalent in melodies and harmonies across various musical genres. They provide a sense of stability, direction, and resolution, contributing to the overall structure and coherence of musical compositions.

On a personal note, in my early years as an orchestral trombonist, it was obvious that an important role of the trombone section included providing underlying support chords. Three trombones can provide a triad chord. The section member who had the 5^th had to be 'perfect', even more so than the member who had the 3^rd, since half the literature is in a minor key.

Now, let's look at the math. An octave-based equal temperament puts the 5^th  at 

2^(7/12) = 1.4983, 

where the perfect 5^th is 1.5. 

Therefore, the 5^th nearly 2 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢.

Now let's examine the 5^th using the 'Pure 12th System'

3^(7/19) = 1.4989

1200*log2(1.4989 / 1.5) = -1.2347 ¢ 

Therefore, 1.955 – 1.2347  = 0.7203 ¢, or in other words, the Pure 12^th System produced a 5^th that is ¾ ¢ better to being a perfect 5^th. But, more importantly, it produces an absolutely Perfect 12^th, therefore, the 5^th above the octave IS A PERFECT 5^th!

Now ask yourself, why would a piano tuner, from a musicological perspective, want to even consider tuning using 3rds, 10ths, 17ths and counting beats from these 3rds? Yes, historically, it is easier to 'count beats' of progressive 3rds, but we now have 'new technology' that eliminates the need for this. The musicologist is telling us about the fundamental importance of the Perfect 5^th! We now can achieve it with the 12th if we use new technology.

Steven Norsworthy

May 8, 2024

http://rf2bits.com/

http://PianoSens.com

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Steven Norsworthy
Cardiff By The Sea CA

steven@rf2bits.com
(619) 964-0101
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Kent writes: 

>>Tim cites the Pythagorean M3rd as 22 cents wide, and the pure octave ET M3rd as 14 cents wide. These are rounded figures. Please note that the pure 12th ET M3rd _also_ rounds to 14 cents wide.(!) Where are these wide M3rds of pure 12th ET so often complained about? They don't actually exist! To be a bit more precise, Pythagorean M3rds are 21.5 cents expanded.<< 

    Actually, the Pythagorean comma is 23.4 cents wide.   The 21.5 cents being referenced is the Syntonic comma.  These are arrived at by different means, and the syntonic comma is the near-universally accepted limit for a M3 in the WT's. I also submit that the word "Pure" in tempering discussions is best used to describe an untempered interval.  

Regards,