Schoenberg: Theory Of Harmony (1)

A month or so ago I started reading from Schoenberg’s Theory of Harmony. The Preface to the first edition is signed “Vienna, July 1911”, conspicuously just two months after Mahler died and was buried – in Vienna. Mahler’s death had devastated Schoenberg; he had been one of Mahler’s “top students” — although clearly already masterful by this time. In the first edition there is a profuse dedication to Mahler about how Schoenberg didn’t believe his contemporaries had sufficiently praised Mahler, and then in a subsequent edition, published some ten years later, he had decided that people had, by that point, come to praise Mahler enough; in the edition I have, the dedication reads simply “To the hallowed memory of GUSTAV MAHLER”.

It is not just any book about harmony or some esoteric technical subject — Schoenberg was a smart dude. He writes holistically, philosophically. He writes about the folly of studying theory without actually doing the art — and also he writes “against comfort” —

I hope my pupils will commit themselves to searching! Because they will know that one searches for the sake of searching. That finding, which is indeed the goal, can easily put an end to striving.

and

those who so love comfort will never seek where there is not definitely something to find.

Mahler revisited

As writes Schoenberg,

It is indeed our duty to reflect over and over again upon the mysterious origins of the powers of art. And again and again to begin at the beginning; again and again to examine anew for ourselves and attempt to organize anew for ourselves. Regarding nothing as given but the phenomena. These we may more rightly regard as eternal than the laws we believe we have found.

And so, in such a spirit, I begin by paying respect to Mahler, once more.

As Bernstein taught in his Harvard Lectures, Mahler had operated at the limit of tonality. He was essentially as “atonal” as one can get while still working in the “system of Western tonality”. He was the last statement, a final recapitulation of a long arc of tradition. Let us also remember that the cataclysm of the Great War was the abyss into which the culture would fall a mere 3 years after his death, that all the arts would lose their grasp on narrative, that Mahler’s music was always spiritually seeking, that the ending of the 9th symphony was, in Bernstein’s words an expression of the act of dying. All of these fateful themes — the end of tonality, the end of his life, the end of narrative, the end of hope, the immense shock to the culture — it all came in the span of just a few short years, and I, along with Bernstein (who called Mahler a prophet, and was later buried with the score of Mahler’s 5th symphony), do not believe it to be mere coincidence.

To directly quote Bernstein from “The Unanswered Question” Harvard Lectures, on the final movement of the 9th symphony:

… the fourth and last movement, the Adagio, the final farewell. And it takes the form of a prayer. Mahler’s last chorale. His closing hymn. A super-prayer for the restoration of life, of tonality, of faith. This is tonality unashamed, presented in all its aspects, ranging from the diatonic simplicity of the hymn tune that opens it, through every possible chromatic ambiguity.

the final incredible page — this page, I think, is the closest we have ever come, in any work of art, to experiencing the very act of dying … the slowness of this page is terrifying

And so but Schoenberg — one of Mahler’s protégés — was one of the two composers, along with Stravinsky, chiefly responsible for, indeed, leaving tonality completely behind. Or, I don’t know if it’s fair to say that he was responsible. Maybe … tonality left us. But he was involved.

My mission is not to follow Schoenberg beyond tonality (although if that is where he will lead me, then so be it, but somehow I rather doubt that even such an energetic pedagogue as he will be able to). My notion and hope is that in this book is the clearest formulation of the methods used by Mahler.

John Coltrane remarked about his early experiences playing in Charlie Parker’s bands (Coltrane on Coltrane, p. 24):

Charlie [Parker], he swayed me so much, you know. But after I had come out from under it a little bit, after I had been under it a little bit, under his spell, then I could listen to other people too.

I went through something like this with Mahler in the mid-2010s. In 2011 I had the glorious opportunity to record the 9th at New World Symphony. By the final performance, I had heard its message, had come under its spell, and it turned out to be an inevitability that over the next several years I would explore all Mahler’s symphonies. Now granted, I never knew Mahler nor played in Mahler’s “band”, but it was like this. In those years, music began and ended with his symphonies.

The expressions ‘consonance’ and ‘dissonance’ are false

Before you leave behind tonality, you should know what you are leaving behind. In the early part of Schoenberg’s book is quite simply the clearest formulation I have ever read about consonance.

The expressions ‘consonance’ and ‘dissonance’, which signify an antithesis, are false. It all simply depends on the growing ability of the analyzing ear to familiarize itself with the remote overtones, thereby expanding the conception of what is euphonious, suitable for art, so that it embraces the whole natural phenomenon.

What today is remote can tomorrow be close at hand; it is all a matter of whether one can get closer. And the evolution of music has followed this course: it has drawn into the stock of artistic resources more and more of the harmonic possibilities inherent in the tone.

Consonance and dissonance are on a continuum; the closer to the fundamental, the more consonant, the further, the more dissonant.

The consonances are accordingly the first overtones, and they are the more nearly perfect the closer they are to the fundamental. That means, the closer they lie to the fundamental, the more easily we can grasp their similarity to it, the more easily the ear can fit them into the total sound and assimilate them, and the more easily we can determine that the sound of these overtones together with the fundamental is ‘restful’ and euphonious, needing no resolution.

Efforts to make use of the more remote consonances (today called ‘dissonances’) as artistic means… led necessarily to many an error, to many a detour. The way of history, as we can see it in that which has actually been selected by practice from the practicable dissonances, hardly leads here to a correct judgment of the real relations. That assertion is proved by the incomplete or unusual scales of many other peoples, who have, nevertheless, as much right as we to explain them by appeal to nature. Perhaps their tones are often even more natural than ours (that is, more exact, more correct, better); for the tempered system, which is only an expedient for overcoming the difficulties of the material, has indeed only a limited similarity to nature. That is perhaps an advantage, but hardly a mark of superiority.

The harmonic series (in equal temperament approximation)

The harmonics are actually just integer multiples of the fundamental frequency. Not all of these actually appear in our equal temperament tuning system.

Gravity and constructing the Major scale

1. A musical sound is a composite, made up of a series of tones sounding together, the overtones; hence it forms a chord.
2. In [the series built on C] c is the strongest sound, because it occurs the greatest number of times, and because it is actually played or sung itself, as a fundamental.
3. After c the next strongest tone is g, because it occurs earlier in the series, therefore more often than the other tones. If we think of this g as a real tone, it then has overtones itself (as a tone actually played); and at the same time this g, together with its overtones, presupposes the C. The overtones of the overtones also contribute to the sound.
4. Consequently: an actual tone (the g) appears as dependent upon a tone a fifth below.

Unfortunately I cannot follow that “consequently”. But Schoenberg uses this “consequently” to introduce the concept of “gravity”. The notion is (shown for 2 notes, but applies to any 2 notes a fifth apart):

• G “depends” on C
• C exerts “gravity” on G

From here we see that the first 5 harmonics (each of which account for 3 individual notes) of each of the subdominant (e.g., F), the fundamental (C), and the dominant (G)… together form the major scale.

“Tonal gravity” would go on to play a bigger role in 20th century music theory. It is attributed to George Russell, so I was surprised to find mention of it far earlier. Schoenberg promises to “speak often of this characteristic” and to “draw a number of conclusions from it”.

The power of consonance

My interest in this theory comes from the immediate “power” of playing and experiencing this sound in all 12 keys.

A hand-wavy argument: even on my Kawai electronic keyboard — which doesn’t emit all the real harmonics of a piano, but the samples are presumably of high enough quality that there is some approximation — I can hear that if I play all the notes in the harmonic series, there is “more” “consonance” than if I add some other note to the series — for example even adding the fifth between the first and second harmonics “muddies” the sound.

Other observations:

• It’s kind of eerie that the 11th harmonic sounds the ♯11 and the 13th harmonic sounds the 13th.
• The melodic minor scale based on the dominant is hidden inside the harmonic series of the fundamental. This luxurious, rich, restful sound I would associate with Debussy more than with any other composer. It stands to reason, in accord with Schoenberg’s argument about dissonance, that Debussy (died in 1918, only a few years after Mahler), was operating in the remote regions of the harmonic series.
  • The dissonances around the (nearly complete instance of the) melodic minor scale, in various chord voicings, form very compelling harmonic counterpoint to a pedal formed with the 1st and 2nd harmonics. The focus is the dominant 7 chord with the upper ♯11 and 13 extensions. The consonance is uncannily powerful — it almost feels like “cheating”.
• I note that Rachmaninoff’s left hand (my most ready example comes from the middle section of Prelude in G minor Op. 23, No. 5 — the first arpeggiated D major chord is built using the harmonics numbers 1 + 2, 3, 4, 5, 7, 8, 10), was clearly governed by the harmonic series (and by extension Chopin, and others).
• The major 7 interval between the dominant and the ♯11 occurs earlier in the series than the major 7 interval between the tonic and the major 7. According to Schoenberg’s theory of consonance, it means the ♯11 is less dissonant than the major 7. And that’s how I hear it. It’s how I’ve always heard it; that is, for years and years I have felt the “gravity” of the ♯11.