Musical Implications of the Harmonic Overtone Series: Chapter IV
The Series' Prediction of Integrated Modality and Equal Temperament
In Example I I have arranged a twelve bar phrase containing twelve progressive root motions. Each target triad is minor, which then becomes major, and finally, it acquires a minor seventh to become an overtone chord before proceeding to the next target. Not only does this progress through all twelve chromatic degrees, but all twenty-four major and minor tonics and all twelve overtone sonorities as well. This is the series working out for itself the concept of integrated modality in the most direct way possible.
This example also represents, as a musical example, the desire of each and every one of the twelve tones: A tone first desires to rise to the rank of a tonic, regardless of whether or not it is major or minor, by acquiring for itself a perfect fifth. Then, if it is minor, it desires to become a more perfect tonic with a major third. Finally, after the tone has had its time on the stage of the piece, it wishes to acquire a minor seventh, become an overtone sonority, and then it is absorbed into a new tonic a perfect fifth below, where the process starts anew.
Note that during this chromatic cycle, the voices in the upper stratum descend an octave. In Schenker-speak this would be an 8, 7, 6, 5, 4, 3, 2, 1 line, and that points out what a joke Schenker's theories actually are. The lines Schenker describes, in breathless terms which suggest that they are some big, important discovery, are nothing more than natural artifacts which result from any functional harmonic continuity which has a preponderance of progressive root movements and a primarily four-voice texture. And, since progressive root movements are statistically the most common type in all forms of traditional music, and since overtone chords resolving to targets will always imply four voices, that means almost all functional harmonic continuities create, as artifacts, decending lines, whether they be 3,2,1's; 3, 2, 3, 2, 1's; 5, 4, 3, 2, 1's; 5, 4, 3, 2, 5, 4, 3, 2, 1's; or 8, 7, 6, 5, 4, 3, 2, 1's.
I spent a lot of time on Schenker when I was working on my Master of Music degree, and I "Schenked" my share of pieces during that time, so when I finally realized that these descending lines are present with or without the composer's being aware of them, and are in fact a natural side effect of the implications of the overtone series in action - simple musical gravity - I about laughed myself silly. I can't think of any music theory that is a bigger waste of time than Schenker's. Well, there is Paul Hindemith I guess. To be valuable as a music theory, that theory must be of use to those who wish to compose music. Schenkerian analysis does not help the prospective composer one whit.
Finally, note that from a technical standpoint the bass part of Example I is a single-interval twelve tone row (Counting octave displacement inversions as equivalent). Not particularly impressive as a tone row, but it does not have to consist of only a single interval, as you can see in Example II. Here, there are two intervals in the bass part: a half-progressive descending minor third followed by a fully progressive rising fourth. The chords which are exited via half-progression start out as major triads and become overtone sonorities (Which are not functional in nature), and their targets start out as minor sevenths, become overtone chords, and then aquire a diminished fifth to become French-derived secondary dominants. When you create intervallically precise root progression patterns such as these, and make all of the transformations according to hoyle, an interesting entity is created: The harmonic canon, which we will look more closely at in chapter six.
For now I want to point out the brief fifth chapter example. It's quite an irony that the example is so simple, because it took an entire millennium of Western music history to work out the solution to it, which I find positively flabbergasting, but I guess cultural hindsight is 20:20.
The problem here is what is called the Pythagorean Comma. If you take twelve absolutely pure perfect fifths with the overtone series ratio of 3:2, you get the derivation of the chromatic scale as implied by the series (But the series actually implies a series of descending fifths, which we just saw harmonically in Example I above). If you compare that to the same span in perfectly pure 2:1 octaves, it takes seven octaves to match the span of the twelve pure perfect fifths. The problem arises between the e-sharp and the f-natural enharmonics at the top of each series: They are not the same pitch. The e-sharp that the pure perfect fifths generated is circa 23.46 cents sharp compared to the f-natural. This caused ten centuries worth of headaches.
It goes without saying - or, at least it should - that the pureness of the octave is inviolable, so stretching it should not be an option (But, this is done by piano tuners all the time). The fifth, on the other hand, is clearly subservient to the octave in rank within the series. So, the only logical solution would be to pinch the fifths each by 1/12th of the Pythagorean comma, which comes out to a measly 1.95 cents. Why this obviously implied solution took a thousand years to be implimented is simply beyond me, but it does make for fascinating reading. Sure, I like to hear Baroque music in well tempered harpsichord tunings and at philosophical pitch (And, I tune my guitars to a philosophical pitch based on an A0 of 27.0 Hz versus the typical 27.5 Hz), but fixed pitch instruments, especially fretted string instruments, really have no other viable option than equal temperament. To be fair, at least one ancient Greek musician-philosopher suggested it, so the idea was around, but even as late as Beethoven's day the matter wasn't completely settled. What finally settled the matter was the realization - in the actual practice of composers - of the series' implied integrated modality: After the Romantics, it was all over but the shouting of the period instrument and ancient music crowds.
The series itself is unstable, so the idea that the ideal tuning system should be stable is not only an impossible goal, but the series simply does not imply that it is required (Or, that it is even particualrly desirable). What the series does imply is that any one of the twenty-four possible tonics can be the locus of a composition, and for them to all sound the same relative to each other, twelve tone equal temperament is required.
In the electronic realm, octave divisions up to over 170 divisions can be used, and they sound to most people the same as just intonation (Since at circa 7 cents, the divisions border on the smallest differences that the average person can detect), but such ponderous systems will never offer any potential solution for the human performer with a fixed-pitch instrument in hand. As with every ultimate theoretical solution, twelve tone equal temperament is elegantly simple, and since the series specifies twelve pitches in an octave, it is the only real solution the series implies.
I've had a lot of very good days fishing, but never one that good.