CAUT

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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

------------------------------
Steven Norsworthy
PianoSens
Cardiff By The Sea CA
(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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Dennis Johnson, Ed Foote, and others of the UT school of tuning, I have some important questions. First a statement:

"TUNING IS MATH" and I think that is indisputable.

So, "Show me the math" you are using, full chromatic outcomes. What happens to your 5ths and 12ths when you tune your UT in the key of C, and then play in the key of Db or the key of F#? What happens when you play a Scriabin or Bartok or Schoenberg, which has no tonality per se?

Let's even harken back to Chopin. Listen to Arthur Rubinstein's Chopin recordings, universally acclaimed as one of the greatest performers of Chopin of all time. Do you not like the sound of the piano overall on those recordings? You know that Franz Mohr probably was the tuner on most of the Rubinstein and Horowitz recordings. You know Franz was not only a ET tuner, but specifically was focused on getting the 5ths / 12ths nearly pure. That was a stated objective of the Steinway company regarding 5ths from what I can gather. 

I want to see all your ratios for full chromatic offsets, so please show me your math. I seriously am skeptical that it will result in an unbiased tonal result that fits the full spectrum of tonal and atonal repertoire without screwing it all up.

Also, if you really have something, then after showing us the math, also show us the recording with good mics and good piano and play all the interval relationships of your UT and let is listen carefully. Then get a highly regarded concert pianist and have them play the full range of repertoire. Let's compare that with Rubinstein or Horowitz recordings which were ET.

Best regards,

Steve N

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Dennis Johnson, Ed Foote, and others of the UT school of tuning, I have some important questions. First a statement:

"TUNING IS MATH" and I think that is indisputable.

So, "Show me the math" you are using, full chromatic outcomes. What happens to your 5ths and 12ths when you tune your UT in the key of C, and then play in the key of Db or the key of F#? What happens when you play a Scriabin or Bartok or Schoenberg, which has no tonality per se?

Let's even harken back to Chopin. Listen to Arthur Rubinstein's Chopin recordings, universally acclaimed as one of the greatest performers of Chopin of all time. Do you not like the sound of the piano overall on those recordings? You know that Franz Mohr probably was the tuner on most of the Rubinstein and Horowitz recordings. You know Franz was not only a ET tuner, but specifically was focused on getting the 5ths / 12ths nearly pure. That was a stated objective of the Steinway company regarding 5ths from what I can gather. 

I want to see all your ratios for full chromatic offsets, so please show me your math. I seriously am skeptical that it will result in an unbiased tonal result that fits the full spectrum of tonal and atonal repertoire without screwing it all up.

Also, if you really have something, then after showing us the math, also show us the recording with good mics and good piano and play all the interval relationships of your UT and let is listen carefully. Then get a highly regarded concert pianist and have them play the full range of repertoire. Let's compare that with Rubinstein or Horowitz recordings which were ET.

Best regards,

Steve N

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

------------------------------
Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

So, unless I can clone one of you (UT proponents, that is) and transmit your intuitive brain processing to others, we can never appreciate the aesthetic superiority of the intuitive approach which is totally individualistic and subjective, and has no fundamental mathematical foundation. -- I see... We all wish that we could take one of you back in a Time Machine and show Rubinstein and Horowitz what 'could have been' (Please forgive the humor, I mean no harm, but playfully put)! Smiles!  --- Steve 

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

Original Message:
Sent: 05-11-2024 14:13
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

------------------------------
Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

So, unless I can clone one of you (UT proponents, that is) and transmit your intuitive brain processing to others, we can never appreciate the aesthetic superiority of the intuitive approach which is totally individualistic and subjective, and has no fundamental mathematical foundation. -- I see... We all wish that we could take one of you back in a Time Machine and show Rubinstein and Horowitz what 'could have been' (Please forgive the humor, I mean no harm, but playfully put)! Smiles!  --- Steve 

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

Original Message:
Sent: 05-11-2024 14:13
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

------------------------------
Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Ed and Dennis,

ANY temperament can be described as a weighted linear equation of various ratios of partials.

Steve

So, unless I can clone one of you (UT proponents, that is) and transmit your intuitive brain processing to others, we can never appreciate the aesthetic superiority of the intuitive approach which is totally individualistic and subjective, and has no fundamental mathematical foundation. -- I see... We all wish that we could take one of you back in a Time Machine and show Rubinstein and Horowitz what 'could have been' (Please forgive the humor, I mean no harm, but playfully put)! Smiles!  --- Steve 

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

Original Message:
Sent: 05-11-2024 14:13
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

------------------------------
Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

Original Message:
Sent: 5/11/2024 4:07:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Ed and Dennis,

ANY temperament can be described as a weighted linear equation of various ratios of partials.

Steve

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

Original Message:
Sent: 05-11-2024 15:35
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 2:38:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

So, unless I can clone one of you (UT proponents, that is) and transmit your intuitive brain processing to others, we can never appreciate the aesthetic superiority of the intuitive approach which is totally individualistic and subjective, and has no fundamental mathematical foundation. -- I see... We all wish that we could take one of you back in a Time Machine and show Rubinstein and Horowitz what 'could have been' (Please forgive the humor, I mean no harm, but playfully put)! Smiles!  --- Steve 

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

Original Message:
Sent: 05-11-2024 14:13
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

------------------------------
Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

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: 05-12-2024 00:24
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 4:07:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Ed and Dennis,

ANY temperament can be described as a weighted linear equation of various ratios of partials.

Steve

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

Original Message:
Sent: 05-11-2024 15:35
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 2:38:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

So, unless I can clone one of you (UT proponents, that is) and transmit your intuitive brain processing to others, we can never appreciate the aesthetic superiority of the intuitive approach which is totally individualistic and subjective, and has no fundamental mathematical foundation. -- I see... We all wish that we could take one of you back in a Time Machine and show Rubinstein and Horowitz what 'could have been' (Please forgive the humor, I mean no harm, but playfully put)! Smiles!  --- Steve 

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

Original Message:
Sent: 05-11-2024 14:13
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

------------------------------
Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

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: 05-12-2024 00:24
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 4:07:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Ed and Dennis,

ANY temperament can be described as a weighted linear equation of various ratios of partials.

Steve

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

Original Message:
Sent: 05-11-2024 15:35
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 2:38:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

So, unless I can clone one of you (UT proponents, that is) and transmit your intuitive brain processing to others, we can never appreciate the aesthetic superiority of the intuitive approach which is totally individualistic and subjective, and has no fundamental mathematical foundation. -- I see... We all wish that we could take one of you back in a Time Machine and show Rubinstein and Horowitz what 'could have been' (Please forgive the humor, I mean no harm, but playfully put)! Smiles!  --- Steve 

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

Original Message:
Sent: 05-11-2024 14:13
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

------------------------------
Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

------------------------------
Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 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!

Steven Norsworthy

May 8, 2024

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

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: 05-12-2024 00:24
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 4:07:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Ed and Dennis,

ANY temperament can be described as a weighted linear equation of various ratios of partials.

Steve

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

Original Message:
Sent: 05-11-2024 15:35
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 2:38:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

So, unless I can clone one of you (UT proponents, that is) and transmit your intuitive brain processing to others, we can never appreciate the aesthetic superiority of the intuitive approach which is totally individualistic and subjective, and has no fundamental mathematical foundation. -- I see... We all wish that we could take one of you back in a Time Machine and show Rubinstein and Horowitz what 'could have been' (Please forgive the humor, I mean no harm, but playfully put)! Smiles!  --- Steve 

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

Original Message:
Sent: 05-11-2024 14:13
From: Edward Foote
Subject: It's All About the Perfect 5th! A Musicological Perspective

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Ed Foote RPT

Original Message:
Sent: 05-11-2024 09:50
From: Robert Bussell
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/11/2024 8:42:00 AM
From: Edward Foote
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/10/2024 12:41:00 PM
From: Steven Norsworthy
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Hi Dennis,

If we simply use the Pure12th method as taught by Kent Swafford, then we get the perfect balance between the 5ths at -1.2347¢ and the octaves at +1.2347¢ and then the 3rds take care of themselves automatically. We cannot independently adjust the 5ths, octaves, and 3rds. They are all interdependent. That simply does not work mathematically. However, I am 'yet' to hear anyone say that Pure12th tuning is anything but musically spectacular from a subjective point of view. It provides a special clarity or bell-like sound from being so consonant.

'Equal' means that there is absolutely no bias for any key. 'Tuning Is Math', one of my slogans that says that anything we do to tune an instrument can be described by a mathematical relationship between the notes. If we tune intuitively, it is not duplicatable and requires us to be cloned as human beings! A little humor to make the point. 

Furthermore, the 'high tech' tools allow us to have a nearly 'perfect' implementation that we humans cannot quite achieve, so we don't have to clone ourselves to have others achieve the same great result. That is the breakthrough of high-tech tuning that was not previously available. One of the top aural tuners in the country is a friend of mine, and I told him that we no longer need to clone him (LOL) because we can apply modern high-tech apps and sensors and make a piano sound better in tune than we could even dream of in the past aurally. He did not particularly 'like' that but could not refute it either, especially when I demonstrated it with the most up-to-date advanced app and sensor.

The turn of the 20th Century led to the end of traditional 'tonality' as we knew it in Western Classical Musical Traditions, requiring us to be more 'equal' in our treatment of the notes, ending the notion of key signatures. Subjectively speaking now, a concert pianist has to play a wide variety of material on a program and needs this 'equality' to pull it off. Some may disagree and wish we harkened back to the late 18th century way of tuning unequal temperament. It becomes an emotional argument with them and all I can say is they have the right to like this but it is not the main stream for today.

Best and kind regards,

Steve N.

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

Original Message:
Sent: 05-10-2024 08:46
From: Dennis Johnson
Subject: It's All About the Perfect 5th! A Musicological Perspective

Steve-

I appreciate your enthusiasm and agree that the P12 tuning style works nicely, especially for ET.  No argument, however I think we can agree it is the 3rd that provide color, and that is also an important part of our musicological history.  It doesn't have to be either/or.  One thing that wasn't mentioned is that P5ths also imply equal beating triads, which is part of the improved clarity.  However, if we blend that with a subtle variety of color, which is consistent with compositional styles of the majority of music people play on the piano, then in my opinion it takes the whole experience to a even higher musical experience.  It's not good or bad though, it's all good.  The reason I got away from P12 tuning is because it can stretch the 10ths in black keys a little more than my comfort level. 

thanks,

Dennis.  

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Dennis Johnson, R.P.T.
Johnson Piano Service
djpianoservice@gmail.com
djpianoservice.com
612-599-6437

Original Message:
Sent: 05-09-2024 17:08
From: Steven Norsworthy
Subject: It's All About the Perfect 5th! A Musicological Perspective

Fred,

I would call myself a 'disciple' of Kent Swafford. We talk nearly every week. He 'converted' me to the Pure 12th. I can really hear the difference using it. Kent listened to my recordings after tuning and really approved of what it achieved. Pianoscope's IH and tuning curves are what I use, along with the sensor of course. That combination nails the 12th so well that it sounds 'hollow' and totally beat-less. The rest of the piano sings with a bell-like clarity that it never did before. Kent and I looked at my tuning curves in Pianoscope. We jointly determined that due to the extremely low IH of this Fazioli, that it was the ideal candidate for the 3:1 only, and set the curve to 3:1, and turned off the 6:2 and 9:3, and it was a wise choice. My 3:1 is nearly perfectly flat.

Thanks for the references. 

I think some on the PTG who read my essay will appreciate the motive of offering a musicology perspective. Most tuners don't have formal music education and degrees. I have gotten to know now about 100 tuners in the past 6 months. Only one has a Masters Degree in Keyboard and is an active performer. I love to teach, it was my 'calling' from a young age. I do not mean to 'talk down' to anyone with the essay. I hope it added perspective to those who have not thought as much about the musicology of pitch perception in music history.

Best and kind regards,

Steve

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

Original Message:
Sent: 05-09-2024 16:22
From: Fred Sturm
Subject: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:39:00 PM
From: Richard West
Subject: RE: It's All About the Perfect 5th! A Musicological Perspective

Original Message:
Sent: 5/9/2024 1:22:00 AM
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 name, 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.' Violina, 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 trombone section's importance 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 is 1.955 ¢ flat, since 

1200*log2(1.4983 / 1.5) = -1.955 ¢ or nearly 2 cents flat.

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 a Perfect 12^th, so that 5^th above the octave IS A PERFECT 5^th!