Harmonic Implications of the Overtone Series, Part III
In the first two installments of this subject (None of this will make sense to you unless you read and comprehend them first), I demonstrated how the implications of the harmonic overtone series lead to tonality, and how the secondary dominant system (Extended with French-derived secondary dominants, fully-diminished seventh secondary dominants, and German-derived secondary dominants) was simply a result of these implications: Every degree with a target degree a fifth below can carry an overtone chord, or an altered overtone chord. I realize that there are other altered overtone chords, and that some of these require the assumed-root treatment, but the demonstration thus far will equip the student desiring to integrate those sonorities with what is needed. I wish to move on.
Today I want to demonstrate how the integrated tonality found in Arnold Schoenberg's "Structural Functions" and Gene Cho's "Harmonic Convergence" actually has a simpler solution than even Dr. Cho's slim monograph posits. Not only that, but several centuries of experimentation and anguished doubt about temperament schemes were also mostly a waste of time: The series tells what is required through its implications.
The second example in this series is reproduced as the first example here. It demonstrates the technologically correct direct crosswise transformation between two overtone chords a perfect fifth apart. But, as I finally brought up in the second installment, it is the tri-tone that is the nuclear energy in music: The root and fifth of the overtone chord are "free" with respect to where they go, but the major third and minor seventh are not; they have leading-tone and leaning-tone energies, respectively, and desire to resolve.
Diatonic chords of the major tonal system can be directly connected by these crosswise transformations, but once a secondary dominant function sonority is assigned to a degree, the more proper implied resolution is in the second example: There is a momentary triad with a doubled root (in the upper strata), and the crosswise resolution's completion is delayed.
*****
There are various rationalizations for the minor modality, and all have some merit or other, but the traditional "pure minor" derivation with minor triads on all of the cardinal degrees (Aolean mode), is really not very sound. For one thing, the Aolean mode is almost never heard in harmony: The minor mode is almost always simply a minor tonic with overtone chords on the fourth and fifth degrees, and perhaps a diminished minor seventh chord on the second degree. In any case, the overtone series makes the minor triad conspicuous by it's absense. Whether you prefer the arithmetic mean versus geometric mean explaination, or the cosine series explaination (My personal prefference), the one thing that the series itself implies is that a minor triad can be targeted. There really isn't any more to understand.
In the third example on this page, I have interjected minor triads as targets, which first become major triads, and then acquire a minor seventh to become overtone chords. This is it in a nutshell: Every tone desires to acquire a perfect fifth and become a tonic, even if it is a minor tonic, but a major tonic is more perfectly in tune with the series (And so, all of those Picardy Thirds throughout history ending minor mode pieces). Then, after its part on the stage is over, the major tonic desires to acquire a minor seventh and be absorbed into a new tonic a perfect fifth below: The previous root is demoted to become the new root's perfect fifth.
If we extend this sequence through twelve roots, we get the example at the bottom of the page. If the fifths were absolutely perfect 3:2 ratio Pythagorean fifths, we would end up at a point not exactly where we departed from. The so-called Pythagorean Comma of 23.46+ cents sharp. Well, since the octave is the first interval (And most important) in the series, it can't be adjusted. But the fifth can. If you adjust each fifth by 1/12 of a Pythagorean comma, you get... Twelve Tone Equal Temperament. In other words,
THE SERIES IMPLIES THAT AN EQUALLY TEMPERED TWELVE TONE SYSTEM IS THE MOST PERFECT ONE!
Not meaning to yell, but just so I'm clear on that point: The series implies equal temperament (So don't get your panties in a bunch about discrepancies between TTET and the ratios in the series, because they are not only red herring issues and straw man arguements, but they are a result of the desires of the series!
I don't know what about this is difficult to comprehend, but some people do seem to have issues with it.
Finally, note that the upper four voices of the final example make a harmonic double canon. A simple one, to be sure, but it would still be a perfect canon if all the targets were major (But, all of the targets would have to be major: This does not work in a diatonic system). So, it is not really the physical phenomenon of echo which is responsible for canon, as Schillinger argued, but rather canon is one of the things harmonically implied by the series.
*****
Redhead, red dress, and goldfish. Just shoot me.
Today I want to demonstrate how the integrated tonality found in Arnold Schoenberg's "Structural Functions" and Gene Cho's "Harmonic Convergence" actually has a simpler solution than even Dr. Cho's slim monograph posits. Not only that, but several centuries of experimentation and anguished doubt about temperament schemes were also mostly a waste of time: The series tells what is required through its implications.
The second example in this series is reproduced as the first example here. It demonstrates the technologically correct direct crosswise transformation between two overtone chords a perfect fifth apart. But, as I finally brought up in the second installment, it is the tri-tone that is the nuclear energy in music: The root and fifth of the overtone chord are "free" with respect to where they go, but the major third and minor seventh are not; they have leading-tone and leaning-tone energies, respectively, and desire to resolve.
Diatonic chords of the major tonal system can be directly connected by these crosswise transformations, but once a secondary dominant function sonority is assigned to a degree, the more proper implied resolution is in the second example: There is a momentary triad with a doubled root (in the upper strata), and the crosswise resolution's completion is delayed.
*****
There are various rationalizations for the minor modality, and all have some merit or other, but the traditional "pure minor" derivation with minor triads on all of the cardinal degrees (Aolean mode), is really not very sound. For one thing, the Aolean mode is almost never heard in harmony: The minor mode is almost always simply a minor tonic with overtone chords on the fourth and fifth degrees, and perhaps a diminished minor seventh chord on the second degree. In any case, the overtone series makes the minor triad conspicuous by it's absense. Whether you prefer the arithmetic mean versus geometric mean explaination, or the cosine series explaination (My personal prefference), the one thing that the series itself implies is that a minor triad can be targeted. There really isn't any more to understand.
In the third example on this page, I have interjected minor triads as targets, which first become major triads, and then acquire a minor seventh to become overtone chords. This is it in a nutshell: Every tone desires to acquire a perfect fifth and become a tonic, even if it is a minor tonic, but a major tonic is more perfectly in tune with the series (And so, all of those Picardy Thirds throughout history ending minor mode pieces). Then, after its part on the stage is over, the major tonic desires to acquire a minor seventh and be absorbed into a new tonic a perfect fifth below: The previous root is demoted to become the new root's perfect fifth.
If we extend this sequence through twelve roots, we get the example at the bottom of the page. If the fifths were absolutely perfect 3:2 ratio Pythagorean fifths, we would end up at a point not exactly where we departed from. The so-called Pythagorean Comma of 23.46+ cents sharp. Well, since the octave is the first interval (And most important) in the series, it can't be adjusted. But the fifth can. If you adjust each fifth by 1/12 of a Pythagorean comma, you get... Twelve Tone Equal Temperament. In other words,
THE SERIES IMPLIES THAT AN EQUALLY TEMPERED TWELVE TONE SYSTEM IS THE MOST PERFECT ONE!
Not meaning to yell, but just so I'm clear on that point: The series implies equal temperament (So don't get your panties in a bunch about discrepancies between TTET and the ratios in the series, because they are not only red herring issues and straw man arguements, but they are a result of the desires of the series!
I don't know what about this is difficult to comprehend, but some people do seem to have issues with it.
Finally, note that the upper four voices of the final example make a harmonic double canon. A simple one, to be sure, but it would still be a perfect canon if all the targets were major (But, all of the targets would have to be major: This does not work in a diatonic system). So, it is not really the physical phenomenon of echo which is responsible for canon, as Schillinger argued, but rather canon is one of the things harmonically implied by the series.
*****
Redhead, red dress, and goldfish. Just shoot me.
posted by Hucbald at 8:27 PM
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