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
------------------------------

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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
------------------------------

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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
------------------------------

Steven and others. There are a lot of temperaments that are candidates for use in tuning an acoustic piano, particularly with the assistance of an ETD. The excellent creativity in this area includes what we have recently seen in this listserver.

There is Steven's special P12 ET as an example, as well as Kent Swafford's extensive work with P12 ET. Ron Koval has his KV1.3 and KV2.1 and more. David Pinnegar has several historic temperaments that he commonly uses, such as Kellner and Werkmeister, among the hundreds that are available. And all the variants of Equal Beating Victorian Temperament, and more. And don't forget the industry standard octave-based ET that most of us use. All with potential variations such as stretching octaves and adjusting fifths or other intervals. Not to mention variation of pitch (versus A440) and other refinements.

My point: Some tuners like myself can use some assistance, particularly when we talk to a customer and try to explain what the current tuning options are in our industry (versus the industry standard). Please consider the following for a start:

Documentation is needed. Modern digital-based is preferred, like the Scala freeware. https://huygens-fokker.org/scala/ It is made for documenting keyboard temperaments, and can be used to, in turn, interface to MIDI keyboards and other digital instruments.

Demonstrations are essential. I know that an all-digital piano capability (including the hybrid piano), or a MIDI keyboard with a controller like Pianoteq, is a preferred option and that some of our customers are already equipped for such (including music schools and university music departments). It should only take a few minutes to do various recordings. But let's think initial/simple at the start. Some A-B audio recordings from the same music played on different acoustic piano tunings for comparison use would be a great start. Regards, Norman.

------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321

Original Message:
Sent: 05-11-2024 01:14
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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
------------------------------

Norman,

You previously heard these but some did not.

Here are the intervals played and recorded after I tuned Pure12ths: Playing 5ths, Octaves, 12ths, consecutively and chromatically.

https://soundcloud.com/snorsworthy/piano-tuning-check-with-5ths-8ths-12ths

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

Original Message:
Sent: 05-11-2024 08:11
From: Norman Brickman
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steven and others. There are a lot of temperaments that are candidates for use in tuning an acoustic piano, particularly with the assistance of an ETD. The excellent creativity in this area includes what we have recently seen in this listserver.

There is Steven's special P12 ET as an example, as well as Kent Swafford's extensive work with P12 ET. Ron Koval has his KV1.3 and KV2.1 and more. David Pinnegar has several historic temperaments that he commonly uses, such as Kellner and Werkmeister, among the hundreds that are available. And all the variants of Equal Beating Victorian Temperament, and more. And don't forget the industry standard octave-based ET that most of us use. All with potential variations such as stretching octaves and adjusting fifths or other intervals. Not to mention variation of pitch (versus A440) and other refinements.

My point: Some tuners like myself can use some assistance, particularly when we talk to a customer and try to explain what the current tuning options are in our industry (versus the industry standard). Please consider the following for a start:

Documentation is needed. Modern digital-based is preferred, like the Scala freeware. https://huygens-fokker.org/scala/ It is made for documenting keyboard temperaments, and can be used to, in turn, interface to MIDI keyboards and other digital instruments.

Demonstrations are essential. I know that an all-digital piano capability (including the hybrid piano), or a MIDI keyboard with a controller like Pianoteq, is a preferred option and that some of our customers are already equipped for such (including music schools and university music departments). It should only take a few minutes to do various recordings. But let's think initial/simple at the start. Some A-B audio recordings from the same music played on different acoustic piano tunings for comparison use would be a great start. Regards, Norman.

------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321

Original Message:
Sent: 05-11-2024 01:14
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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
------------------------------

Norman, here are 3 Pianoteq generated samples:

https://soundcloud.com/stevelehmannmusic/sets/alternative-tuning-systems

Norman,

You previously heard these but some did not.

Here are the intervals played and recorded after I tuned Pure12ths: Playing 5ths, Octaves, 12ths, consecutively and chromatically.

https://soundcloud.com/snorsworthy/piano-tuning-check-with-5ths-8ths-12ths

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

Original Message:
Sent: 05-11-2024 08:11
From: Norman Brickman
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steven and others. There are a lot of temperaments that are candidates for use in tuning an acoustic piano, particularly with the assistance of an ETD. The excellent creativity in this area includes what we have recently seen in this listserver.

There is Steven's special P12 ET as an example, as well as Kent Swafford's extensive work with P12 ET. Ron Koval has his KV1.3 and KV2.1 and more. David Pinnegar has several historic temperaments that he commonly uses, such as Kellner and Werkmeister, among the hundreds that are available. And all the variants of Equal Beating Victorian Temperament, and more. And don't forget the industry standard octave-based ET that most of us use. All with potential variations such as stretching octaves and adjusting fifths or other intervals. Not to mention variation of pitch (versus A440) and other refinements.

My point: Some tuners like myself can use some assistance, particularly when we talk to a customer and try to explain what the current tuning options are in our industry (versus the industry standard). Please consider the following for a start:

Documentation is needed. Modern digital-based is preferred, like the Scala freeware. https://huygens-fokker.org/scala/ It is made for documenting keyboard temperaments, and can be used to, in turn, interface to MIDI keyboards and other digital instruments.

Demonstrations are essential. I know that an all-digital piano capability (including the hybrid piano), or a MIDI keyboard with a controller like Pianoteq, is a preferred option and that some of our customers are already equipped for such (including music schools and university music departments). It should only take a few minutes to do various recordings. But let's think initial/simple at the start. Some A-B audio recordings from the same music played on different acoustic piano tunings for comparison use would be a great start. Regards, Norman.

------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321

Original Message:
Sent: 05-11-2024 01:14
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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
------------------------------

Steven, thank you for the samples. I'm afraid, however, that I probably failed – I liked all three recordings: Zarlino, Pythagorean, and Equal. Others might be more "atuned" to different temperaments than myself and will hopefully also respond here. And of course the particular piece being played will make a difference as to which temperament is preferred.

Unfortunately, your type of demonstration in turn illustrates a conundrum I have, and why I think that discussion of multiple tuning temperaments on this PTG forum is important but is also counter-productive to all here who enjoy working on acoustic pianos. We all understand why in the early 1900's octave-based ET was the only appropriate choice to tune pianos. But in 2024, if I introduce alternative tunings to my customers, and since they play music of multiple composers written for multiple musical keys, I worry that I become a walking advertisement for (a) their purchase and use of all-digital pianos (including hybrid pianos), or (b) their use of a keyboard + MIDI environment. Music schools are already following this path (this is the digital age!). Regards, Norman.

Norman, here are 3 Pianoteq generated samples:

https://soundcloud.com/stevelehmannmusic/sets/alternative-tuning-systems

Norman,

You previously heard these but some did not.

Here are the intervals played and recorded after I tuned Pure12ths: Playing 5ths, Octaves, 12ths, consecutively and chromatically.

https://soundcloud.com/snorsworthy/piano-tuning-check-with-5ths-8ths-12ths

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

Original Message:
Sent: 05-11-2024 08:11
From: Norman Brickman
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steven and others. There are a lot of temperaments that are candidates for use in tuning an acoustic piano, particularly with the assistance of an ETD. The excellent creativity in this area includes what we have recently seen in this listserver.

There is Steven's special P12 ET as an example, as well as Kent Swafford's extensive work with P12 ET. Ron Koval has his KV1.3 and KV2.1 and more. David Pinnegar has several historic temperaments that he commonly uses, such as Kellner and Werkmeister, among the hundreds that are available. And all the variants of Equal Beating Victorian Temperament, and more. And don't forget the industry standard octave-based ET that most of us use. All with potential variations such as stretching octaves and adjusting fifths or other intervals. Not to mention variation of pitch (versus A440) and other refinements.

My point: Some tuners like myself can use some assistance, particularly when we talk to a customer and try to explain what the current tuning options are in our industry (versus the industry standard). Please consider the following for a start:

Documentation is needed. Modern digital-based is preferred, like the Scala freeware. https://huygens-fokker.org/scala/ It is made for documenting keyboard temperaments, and can be used to, in turn, interface to MIDI keyboards and other digital instruments.

Demonstrations are essential. I know that an all-digital piano capability (including the hybrid piano), or a MIDI keyboard with a controller like Pianoteq, is a preferred option and that some of our customers are already equipped for such (including music schools and university music departments). It should only take a few minutes to do various recordings. But let's think initial/simple at the start. Some A-B audio recordings from the same music played on different acoustic piano tunings for comparison use would be a great start. Regards, Norman.

------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321

Original Message:
Sent: 05-11-2024 01:14
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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
------------------------------

Steven, thank you for the samples. I'm afraid, however, that I probably failed – I liked all three recordings: Zarlino, Pythagorean, and Equal. Others might be more "atuned" to different temperaments than myself and will hopefully also respond here. And of course the particular piece being played will make a difference as to which temperament is preferred.

Unfortunately, your type of demonstration in turn illustrates a conundrum I have, and why I think that discussion of multiple tuning temperaments on this PTG forum is important but is also counter-productive to all here who enjoy working on acoustic pianos. We all understand why in the early 1900's octave-based ET was the only appropriate choice to tune pianos. But in 2024, if I introduce alternative tunings to my customers, and since they play music of multiple composers written for multiple musical keys, I worry that I become a walking advertisement for (a) their purchase and use of all-digital pianos (including hybrid pianos), or (b) their use of a keyboard + MIDI environment. Music schools are already following this path (this is the digital age!). Regards, Norman.

Norman, here are 3 Pianoteq generated samples:

https://soundcloud.com/stevelehmannmusic/sets/alternative-tuning-systems

Norman,

You previously heard these but some did not.

Here are the intervals played and recorded after I tuned Pure12ths: Playing 5ths, Octaves, 12ths, consecutively and chromatically.

https://soundcloud.com/snorsworthy/piano-tuning-check-with-5ths-8ths-12ths

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

Original Message:
Sent: 05-11-2024 08:11
From: Norman Brickman
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steven and others. There are a lot of temperaments that are candidates for use in tuning an acoustic piano, particularly with the assistance of an ETD. The excellent creativity in this area includes what we have recently seen in this listserver.

There is Steven's special P12 ET as an example, as well as Kent Swafford's extensive work with P12 ET. Ron Koval has his KV1.3 and KV2.1 and more. David Pinnegar has several historic temperaments that he commonly uses, such as Kellner and Werkmeister, among the hundreds that are available. And all the variants of Equal Beating Victorian Temperament, and more. And don't forget the industry standard octave-based ET that most of us use. All with potential variations such as stretching octaves and adjusting fifths or other intervals. Not to mention variation of pitch (versus A440) and other refinements.

My point: Some tuners like myself can use some assistance, particularly when we talk to a customer and try to explain what the current tuning options are in our industry (versus the industry standard). Please consider the following for a start:

Documentation is needed. Modern digital-based is preferred, like the Scala freeware. https://huygens-fokker.org/scala/ It is made for documenting keyboard temperaments, and can be used to, in turn, interface to MIDI keyboards and other digital instruments.

Demonstrations are essential. I know that an all-digital piano capability (including the hybrid piano), or a MIDI keyboard with a controller like Pianoteq, is a preferred option and that some of our customers are already equipped for such (including music schools and university music departments). It should only take a few minutes to do various recordings. But let's think initial/simple at the start. Some A-B audio recordings from the same music played on different acoustic piano tunings for comparison use would be a great start. Regards, Norman.

------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321

Original Message:
Sent: 05-11-2024 01:14
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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 have been repeatedly amazed at seeing different pianists relish extremely fast RBI's that typically make me cringe a little. Everyone hears things a little differently. David's tuning style has both extremes of  quietude and "aggression". Yet musicians hear these in a different context than piano technicians (I think). It's an interesting paradox. 

Peter Grey Piano Doctor 

Steven, thank you for the samples. I'm afraid, however, that I probably failed – I liked all three recordings: Zarlino, Pythagorean, and Equal. Others might be more "atuned" to different temperaments than myself and will hopefully also respond here. And of course the particular piece being played will make a difference as to which temperament is preferred.

Unfortunately, your type of demonstration in turn illustrates a conundrum I have, and why I think that discussion of multiple tuning temperaments on this PTG forum is important but is also counter-productive to all here who enjoy working on acoustic pianos. We all understand why in the early 1900's octave-based ET was the only appropriate choice to tune pianos. But in 2024, if I introduce alternative tunings to my customers, and since they play music of multiple composers written for multiple musical keys, I worry that I become a walking advertisement for (a) their purchase and use of all-digital pianos (including hybrid pianos), or (b) their use of a keyboard + MIDI environment. Music schools are already following this path (this is the digital age!). Regards, Norman.

Norman, here are 3 Pianoteq generated samples:

https://soundcloud.com/stevelehmannmusic/sets/alternative-tuning-systems

Norman,

You previously heard these but some did not.

Here are the intervals played and recorded after I tuned Pure12ths: Playing 5ths, Octaves, 12ths, consecutively and chromatically.

https://soundcloud.com/snorsworthy/piano-tuning-check-with-5ths-8ths-12ths

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

Original Message:
Sent: 05-11-2024 08:11
From: Norman Brickman
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steven and others. There are a lot of temperaments that are candidates for use in tuning an acoustic piano, particularly with the assistance of an ETD. The excellent creativity in this area includes what we have recently seen in this listserver.

There is Steven's special P12 ET as an example, as well as Kent Swafford's extensive work with P12 ET. Ron Koval has his KV1.3 and KV2.1 and more. David Pinnegar has several historic temperaments that he commonly uses, such as Kellner and Werkmeister, among the hundreds that are available. And all the variants of Equal Beating Victorian Temperament, and more. And don't forget the industry standard octave-based ET that most of us use. All with potential variations such as stretching octaves and adjusting fifths or other intervals. Not to mention variation of pitch (versus A440) and other refinements.

My point: Some tuners like myself can use some assistance, particularly when we talk to a customer and try to explain what the current tuning options are in our industry (versus the industry standard). Please consider the following for a start:

Documentation is needed. Modern digital-based is preferred, like the Scala freeware. https://huygens-fokker.org/scala/ It is made for documenting keyboard temperaments, and can be used to, in turn, interface to MIDI keyboards and other digital instruments.

Demonstrations are essential. I know that an all-digital piano capability (including the hybrid piano), or a MIDI keyboard with a controller like Pianoteq, is a preferred option and that some of our customers are already equipped for such (including music schools and university music departments). It should only take a few minutes to do various recordings. But let's think initial/simple at the start. Some A-B audio recordings from the same music played on different acoustic piano tunings for comparison use would be a great start. Regards, Norman.

------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321

Original Message:
Sent: 05-11-2024 01:14
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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 have been repeatedly amazed at seeing different pianists relish extremely fast RBI's that typically make me cringe a little. Everyone hears things a little differently. David's tuning style has both extremes of  quietude and "aggression". Yet musicians hear these in a different context than piano technicians (I think). It's an interesting paradox. 

Peter Grey Piano Doctor 

Steven, thank you for the samples. I'm afraid, however, that I probably failed – I liked all three recordings: Zarlino, Pythagorean, and Equal. Others might be more "atuned" to different temperaments than myself and will hopefully also respond here. And of course the particular piece being played will make a difference as to which temperament is preferred.

Unfortunately, your type of demonstration in turn illustrates a conundrum I have, and why I think that discussion of multiple tuning temperaments on this PTG forum is important but is also counter-productive to all here who enjoy working on acoustic pianos. We all understand why in the early 1900's octave-based ET was the only appropriate choice to tune pianos. But in 2024, if I introduce alternative tunings to my customers, and since they play music of multiple composers written for multiple musical keys, I worry that I become a walking advertisement for (a) their purchase and use of all-digital pianos (including hybrid pianos), or (b) their use of a keyboard + MIDI environment. Music schools are already following this path (this is the digital age!). Regards, Norman.

Norman, here are 3 Pianoteq generated samples:

https://soundcloud.com/stevelehmannmusic/sets/alternative-tuning-systems

Norman,

You previously heard these but some did not.

Here are the intervals played and recorded after I tuned Pure12ths: Playing 5ths, Octaves, 12ths, consecutively and chromatically.

https://soundcloud.com/snorsworthy/piano-tuning-check-with-5ths-8ths-12ths

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

Original Message:
Sent: 05-11-2024 08:11
From: Norman Brickman
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steven and others. There are a lot of temperaments that are candidates for use in tuning an acoustic piano, particularly with the assistance of an ETD. The excellent creativity in this area includes what we have recently seen in this listserver.

There is Steven's special P12 ET as an example, as well as Kent Swafford's extensive work with P12 ET. Ron Koval has his KV1.3 and KV2.1 and more. David Pinnegar has several historic temperaments that he commonly uses, such as Kellner and Werkmeister, among the hundreds that are available. And all the variants of Equal Beating Victorian Temperament, and more. And don't forget the industry standard octave-based ET that most of us use. All with potential variations such as stretching octaves and adjusting fifths or other intervals. Not to mention variation of pitch (versus A440) and other refinements.

My point: Some tuners like myself can use some assistance, particularly when we talk to a customer and try to explain what the current tuning options are in our industry (versus the industry standard). Please consider the following for a start:

Documentation is needed. Modern digital-based is preferred, like the Scala freeware. https://huygens-fokker.org/scala/ It is made for documenting keyboard temperaments, and can be used to, in turn, interface to MIDI keyboards and other digital instruments.

Demonstrations are essential. I know that an all-digital piano capability (including the hybrid piano), or a MIDI keyboard with a controller like Pianoteq, is a preferred option and that some of our customers are already equipped for such (including music schools and university music departments). It should only take a few minutes to do various recordings. But let's think initial/simple at the start. Some A-B audio recordings from the same music played on different acoustic piano tunings for comparison use would be a great start. Regards, Norman.

------------------------------
Norman Brickman
Potomac Piano Service
Potomac, Maryland
potomacpiano@verizon.net
https://potomacpiano.com
(301) 983.9321

Original Message:
Sent: 05-11-2024 01:14
From: Tim Foster
Subject: It's All About the Perfect 5th! A Musicological Perspective

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, 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 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 of 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: 05-09-2024 14:49
From: David Pinnegar
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 12:47:00 PM
From: Steven Norsworthy
Subject: RE: 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
------------------------------

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
------------------------------