Why Music Works: Chapter Fourteen
The Pythagorean Comma and Equal Temperament
PREFACE to All Posts:
This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.
Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.
Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.
Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.
Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.
INTRODUCTION to Chapter Fourteen:
In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.
Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.
After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. Now it's time to look at the series generated intervals, the Pythagorean Comma, and the inevitability of equal temperament for fixed pitch instruments.
*****
CHAPTER FOURTEEN:
EXAMPLE 72:
Listen to Example 69
The simplest way to wrap your brain around the Pythagorean comma and equal temperament is the illustration in example seventy-two above. If we stack seven perfect 2:1 octaves next to twelve perfect 3:2 fifths - both starting on F-natural - we should end up on the same pitch, which is the TTET-equivalent F-natural/E-sharp. The problem is that with twelve pure 3:2 perfect fifths, this doesn't happen: The E-sharp is 23.46 cents above the F-natural. In TTET, a cent is 1/100 - or 1% - of an equally tempered semitone, so the E-sharp is slightly less than eighth-tone sharp. Since average ears can detect five to seven cents of difference and well trained ears two or three cents, this is obviously out of tune.
Since the octave is inviolable - pitch recognition depends on octave equivalence (Though piano tuners mess with them all the time) - the simplest and most elegant solution is to simply reduce the size of the perfect fifths by one twelfth of the comma, which comes out to 1.955 cents, or less than 2% of a semitone. I can hear this with two simultaneous notes a fifth apart in TTET (So long as there are no phase, flange, chorus, or reverb effects present) as a slow cross-modulation - or beating - but I understand that the average listener doesn't perceive that level of detail. This trivial adjustment would seem to completely cure the "problem," but it doesn't really, since the other ratios that the harmonic series generates do not equally fill the perfect fifth (A just major and minor third, for example).
EXAMPLE 72B:
There is a useful chart demonstrating the differences between just and equally tempered intervals at the Wikipedia article about equal temperament.
As you can see, while the deviations from pure for the perfect fifth and perfect fourth are indeed trivial, that is not the case for some of the other intervals. What is true, however, is that all of the deviations from pure in equal temperament are trivial for the perfect consonant intervals. This really is the most important thing, because those are the lowest - and therefore most prominent - overtones.
For the imperfect consonances, there are some seeming problems, but perceptually some of those seeming problems are not as bad as they look. Almost no average listener has any problems with minor thirds and major sixths in TTET, for example, but I and some others do perceive them as slightly "off," but still not bothersome. In contrast to that, the major thirds and minor sixths do bother some people, including me, even though their deviations are smaller than those for the minor third/major sixth pair. Again, that is because the major third is lower in the series, more prominent as an overtone, and the overtone chord is based on it as a lowest gender-defining interval. As a result of the strident nature of equally tempered thirds, I and some other guitarists detune our G strings because in guitar music so many major thirds occur between the D and G strings. Luthiers often compensate saddles there too.
As for the dissonances, there is so much interference in them already because of the complexities involved in their relationships that it really doesn't matter much, with one exception: The tritone. In just tuning an augmented fourth and a diminished fifth are not the same interval: The diminished fifth is sharp compared to an equally tempered tritone, and an augmented fourth is flat compared to an equally tempered tritone (I use tritone to describe both augmented fourths and diminished fifths in TTET because they are the same 600 cent interval). That means when passive tones are present to define those intervals in their respective ways, the pure diminished fifth does not have as strong of a leading tone/leaning tone impetus to contract, and the pure augmented fourth doesn't have as strong of a leaning tone/leading tone impetus to expand.
This means that, as far as I am concerned, an equally tempered tritone is superior to the two just versions. The naturally just overtone sonority already wishes to resolve, and TTET only adds to that desire.
*****
I found an interesting YouTube video that purports to demonstrate the superiority of root-calculated just harmony, but which to me in fact proves just the opposite: The superiority of equal temperament. The two things that make this example so excellent are that the sound is generated as sine waves, and that there is a wrapped oscilloscope for visual reenforcement.
What I think is one of the most humorous aspects to that video is the fact that any critical thinker will immediately notice that you can see a lot more difference than you can hear.
*****
Another way to view the naturalness of the equal temperament solution is by noting that the harmonic series is made up of simple ratios generated beyond the octave while equal temperament is made up primarily of simple ratios plotted within the octave. For example, in equal temperament the octave is 1, the tritone is 1/2, the major third is 1/3, the minor third is 1/4, a whole step is 1/6, and a semitone is 1/12. The perfect fourth and fifth, which are 5/12 and 7/12 respectively, are really the only complex intervals in the system, and those are the ones closest to just. Noting TTET variations from just isn't exactly a making a red herring argument, but it's close.
A final thing to think about is what kinds of effects we apply to music today. I previously mentioned phasing and flanging, which are short time delay effects with small pitch modulations that increase disturbances to solid perception of pitch but it is really the ubiquitous use of pitch-shift and delay/pitch mod chorus that puts the lie to the notion that people want to hear just harmony: Just the opposite is true, and it has been since people first noticed how nice a section of doubled or trebled voices or strings sounded. And voices and strings can theoretically reproduce just intervals and harmony.
I enjoy hearing historical temperaments for period music, but I want all twenty-four major and minor keys to sound the same relative to each other, and TTET is the only way to get that with fixed pitch instruments, so it is the only viable solution as far as I am concerned.
That leads us to the final chapter in the current version of WMW on harmony.
PREFACE to All Posts:
This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.
Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.
Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.
Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.
Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.
INTRODUCTION to Chapter Fourteen:
In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.
Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.
After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. Now it's time to look at the series generated intervals, the Pythagorean Comma, and the inevitability of equal temperament for fixed pitch instruments.
CHAPTER FOURTEEN:
EXAMPLE 72:
Listen to Example 69
The simplest way to wrap your brain around the Pythagorean comma and equal temperament is the illustration in example seventy-two above. If we stack seven perfect 2:1 octaves next to twelve perfect 3:2 fifths - both starting on F-natural - we should end up on the same pitch, which is the TTET-equivalent F-natural/E-sharp. The problem is that with twelve pure 3:2 perfect fifths, this doesn't happen: The E-sharp is 23.46 cents above the F-natural. In TTET, a cent is 1/100 - or 1% - of an equally tempered semitone, so the E-sharp is slightly less than eighth-tone sharp. Since average ears can detect five to seven cents of difference and well trained ears two or three cents, this is obviously out of tune.
Since the octave is inviolable - pitch recognition depends on octave equivalence (Though piano tuners mess with them all the time) - the simplest and most elegant solution is to simply reduce the size of the perfect fifths by one twelfth of the comma, which comes out to 1.955 cents, or less than 2% of a semitone. I can hear this with two simultaneous notes a fifth apart in TTET (So long as there are no phase, flange, chorus, or reverb effects present) as a slow cross-modulation - or beating - but I understand that the average listener doesn't perceive that level of detail. This trivial adjustment would seem to completely cure the "problem," but it doesn't really, since the other ratios that the harmonic series generates do not equally fill the perfect fifth (A just major and minor third, for example).
EXAMPLE 72B:
There is a useful chart demonstrating the differences between just and equally tempered intervals at the Wikipedia article about equal temperament.
As you can see, while the deviations from pure for the perfect fifth and perfect fourth are indeed trivial, that is not the case for some of the other intervals. What is true, however, is that all of the deviations from pure in equal temperament are trivial for the perfect consonant intervals. This really is the most important thing, because those are the lowest - and therefore most prominent - overtones.
For the imperfect consonances, there are some seeming problems, but perceptually some of those seeming problems are not as bad as they look. Almost no average listener has any problems with minor thirds and major sixths in TTET, for example, but I and some others do perceive them as slightly "off," but still not bothersome. In contrast to that, the major thirds and minor sixths do bother some people, including me, even though their deviations are smaller than those for the minor third/major sixth pair. Again, that is because the major third is lower in the series, more prominent as an overtone, and the overtone chord is based on it as a lowest gender-defining interval. As a result of the strident nature of equally tempered thirds, I and some other guitarists detune our G strings because in guitar music so many major thirds occur between the D and G strings. Luthiers often compensate saddles there too.
As for the dissonances, there is so much interference in them already because of the complexities involved in their relationships that it really doesn't matter much, with one exception: The tritone. In just tuning an augmented fourth and a diminished fifth are not the same interval: The diminished fifth is sharp compared to an equally tempered tritone, and an augmented fourth is flat compared to an equally tempered tritone (I use tritone to describe both augmented fourths and diminished fifths in TTET because they are the same 600 cent interval). That means when passive tones are present to define those intervals in their respective ways, the pure diminished fifth does not have as strong of a leading tone/leaning tone impetus to contract, and the pure augmented fourth doesn't have as strong of a leaning tone/leading tone impetus to expand.
This means that, as far as I am concerned, an equally tempered tritone is superior to the two just versions. The naturally just overtone sonority already wishes to resolve, and TTET only adds to that desire.
I found an interesting YouTube video that purports to demonstrate the superiority of root-calculated just harmony, but which to me in fact proves just the opposite: The superiority of equal temperament. The two things that make this example so excellent are that the sound is generated as sine waves, and that there is a wrapped oscilloscope for visual reenforcement.
What I think is one of the most humorous aspects to that video is the fact that any critical thinker will immediately notice that you can see a lot more difference than you can hear.
Another way to view the naturalness of the equal temperament solution is by noting that the harmonic series is made up of simple ratios generated beyond the octave while equal temperament is made up primarily of simple ratios plotted within the octave. For example, in equal temperament the octave is 1, the tritone is 1/2, the major third is 1/3, the minor third is 1/4, a whole step is 1/6, and a semitone is 1/12. The perfect fourth and fifth, which are 5/12 and 7/12 respectively, are really the only complex intervals in the system, and those are the ones closest to just. Noting TTET variations from just isn't exactly a making a red herring argument, but it's close.
A final thing to think about is what kinds of effects we apply to music today. I previously mentioned phasing and flanging, which are short time delay effects with small pitch modulations that increase disturbances to solid perception of pitch but it is really the ubiquitous use of pitch-shift and delay/pitch mod chorus that puts the lie to the notion that people want to hear just harmony: Just the opposite is true, and it has been since people first noticed how nice a section of doubled or trebled voices or strings sounded. And voices and strings can theoretically reproduce just intervals and harmony.
I enjoy hearing historical temperaments for period music, but I want all twenty-four major and minor keys to sound the same relative to each other, and TTET is the only way to get that with fixed pitch instruments, so it is the only viable solution as far as I am concerned.
That leads us to the final chapter in the current version of WMW on harmony.
posted by Hucbald at 3:27 PM
1 Comments:
Wow, that video is terrible argument for JI. With those particular timbres and music, the difference is barely audible. And I can totally understand someone having preference for either version anyway. The video's implication is nothing more than dogmatic mathematical bias similar to the nonsense of Pythagoras or (initially) Kepler insisting that the 5 perfect solids must be the foundation of the universe.
The truth about harmonic tuning is that JI does achieve beatlessness with harmonic timbres but beatlessness isn't necessarily the fundamental absolute definition of "in-tune" which is a psychological experience.
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