Sociological Analogies to Pitch Pathologies
A passage in *The Dawn of Everything* by Graeber and Wengrow grabbed me. It describes the *seka* system of Bali, where a rigidly hierarchical Hindu caste society — in which every individual occupies a unique and cosmically fixed position, with no two people truly equal — nevertheless organizes its practical communal life on fully egalitarian terms. Governance, temple management, agriculture: all decided by consensus, everyone participating equally.
I had been turning over two sociological analogies for what I think of as pitch pathologies — pathologies, I should say, only from the perspective of a committed 12EDO partisan, which I am. The Balinese image suddenly clarified both analogies at once, and suggested a third.
A composer I knew, sometime after the fall of the Berlin Wall, said with evident relief: *finally, the twelve-tone regime, where all the pitches are equal, can end.* I loved that. The communist analogy for equal-weight atonality had been sitting in plain sight.
The Boulez diktat — his famous demand for *equal weight* among the pitch classes — produced some extraordinary music. We revere so much of Boulez's work. But as a compositional imperative it encouraged what I'll call **symmetry death**. When all pitches carry equal weight, phrases dissolve. Phrases are middle-ground rhythms; they require weighting and unweighting.
The symmetric collections — octatonic, whole-tone, augmented — move without hierarchy, like a society where no one outranks anyone. The atonal equivalent of pan-diatonicism is the result: a wash. Boulez' 12-tone practice aimed for equal weight, inevitably more bumpy than the through symmetries, but still very far from the hierarchies of tonal music. Flatness as a positive stylistic value led to composers staying in octotonic scale for entire movments. I think of much of Meyer Kupferman's music, to which I was utterly devoted when I was in my 20ths and 30s.
The communist analogy describes this well enough. Total equality of pitches, like total equality of citizens, produces a particular kind of flatness.
It is important to say clearly that Babbitt and his school never subscribed to equal weight — not ideologically, not compositionally. I rely here on Frank Brickle's testimony; Frank studied with Babbitt and the transmission is direct. The confusion is understandable: the density and complexity of Babbitt's arrays produces, for most listeners, the perceptual sensation of a wash, of symmetry death. But tI learned to hear it on its own terms. It was not easy. Underneath, the hierarchical structure is ferociously present. Roger Sessions, after the war, sent Babbitt to Europe specifically to find living Schenkerians. Babbitt told me himself that Schenker fascinated him deeply. This is not musicological gossip; it is the record of a lineage. Schenker's beautiful metaphors — the foreground, the middleground, the Urlinie descending to rest — are baked into the logic of Babbitt's time-points. The time-points are not a workaround for the phrase; they are a synthesis: Schenkerian middleground hierarchy smuggled into the heart of serial practice.
Nevertheless, Babbitt will likely be for connoisseurs. It operates at a level of complexity that most ears cannot track in real time, even trained ears. This is what makes him, for me, the late Beethoven of the serial world: a peak that is also a precipice. The late quartets are not a dead end, but no one could follow Beethoven *there* without pastiche. Same with Babbitt's arrays. The system is so internally consistent, so perfectly itself, that it becomes a gilded cage — what Lewin might have called a superposition, holding phrase and non-phrase, hierarchy and its suspension, simultaneously in solution.
Even Babbitt seems to have felt it. *Swan Song No. 1* suggests a composer loosening, on the way toward something more open.
I am aware that dwelling on these distinctions — Babbitt versus Boulez, Schenker in the arrays, the difference between perceived and actual equal weight — marks me as a man of the Cold War. I am that man, partly. But the distinctions matter. Letting Babbitt blur into Boulez because both sound difficult is exactly the kind of historical flattening that makes the defense of 12EDO harder than it needs to be. And I am not only a man of the Cold War. The path I've chosen — a pull back toward quasi-tonal phraseology, toward hierarchy made audible rather than private — is not regression. It is the only honest move available after that particular precipice. In the twenty-first century, a return to the phrase can only be a return of our own time. Think of Borges' Pierre Menard rewriting *Don Quixote* word for word: unless we resort to pastiche, our phrases can only be of now.
---
The second pathology runs in the opposite direction. Just intonation and Pythagorean tuning — and the contemporary enthusiasm for spectral music -- insist on the absolute uniqueness of each pitch and each interval. In JI there are many minor thirds, each one slightly different; many fifths, each irreducibly itself. This has a genuine beauty. Within a single mode, committed to monody, these unique pitches carry a lovely hierarchy, the tonic sitting at the bottom with minimum potential energy.
But our composers of the last 1000 years were ambitious. They wanted to *pun*.
What I mean by punning is what Georg Joseph Vogler called *Mehrdeutigkeit* — multiple meaning, ambiguity. The tritone that is both a diminished fifth and an augmented fourth. The enharmonic pivot that shifts the tonal center. The dim7 chord that belongs to four keys simultaneously. Look at the Diabelli Variations, No. 20, where each chord trumps the next through exactly this mechanism. Look at the appogiaturas in Bach's BWV 998. The whole history of Western harmonic adventurism, from early modal ambiguity through chromatic voice-leading to the tritone substitution, is a history of composers exploiting Mehrdeutigkeit.
In JI and strict Pythagorean tuning, punning is impossible. Every pitch is what it is. The intervals are unique phenomena and cannot be made to mean two things at once. This is **particularity death** — the Hindu extreme, where every individual occupies an absolutely fixed and irreducible cosmic position. The hierarchy is real, even beautiful, but the system cannot modulate. It cannot pun its way to a new tonal center.
---
The third pathology is subtler and, to me, the most poignant, because it is the most ambitious.
Composers working in X≠12EDO systems — 18-tone, 31-tone, microtonal experiments of various kinds — are often motivated by exactly the right instinct: they want *more* Mehrdeutigkeit, not less. A composer I know who trained himself rigorously in 18-tone music made the mathematically defensible claim that 18EDO is richer in ambiguity than 12EDO, from a purely combinatorial standpoint.
He is not wrong. But there is a problem that the mathematics cannot see.
18EDO contains three whole-tone scales a third-tone apart. The whole-tone scale is something we know — it rings our bells, it lives in us from a century of music. But the harmonies *between* those familiar islands are ones we do not have in us. They don't ring anything. And so 18EDO creates a peculiar situation: the familiar collections are there but recontextualized into a harmonic neighborhood that is, to our experienced ears, impossibly cluttered. In an odd way, 18EDO's own ambitions push it toward 36EDO — which would contain 12EDO itself, shifted by a third-tone, plus all the intervening material. The cognitive result is not enrichment but overload.
The bells don't ring. That is the problem.
---
Here is what I want to defend, and why the Balinese *seka* image struck me so forcefully.
12EDO is not natural in any simple sense. Its intervals are irrational numbers — no perfect fifth, no pure octave harmonic. When critics of 12EDO call it unnatural, they are not wrong by their own lights. But the same mathematics that generates 12EDO generates quasicrystals, which occur in nature, rarely but really. And 12EDO's behavior — the way it superconducts harmonic motion, the way it allows energy to flow without resistance from one tonal center to another — emerged not from a conspiracy against just intonation but from compositional practice itself. Composers wanted to pun. Equal temperament is what punning demands.
More importantly: 12EDO is the *only* division of the octave that preserves Mehrdeutigkeit at the level of *experienced* harmony. It carries within it the harmonies that a millennium of notated music has put in our bodies. The thirds, the fifths, the dominant seventh, the dim7, the augmented sixth — these are not just intervals, they are memories. They ring us. 12EDO enables those memories to be redirected, punned upon, made to mean something new.
Perhaps this is the *seka* paradox. In principle, 12EDO presents equal pitches — the communist surface. But in practice, in the hands of composers who understand weighting and unweighting, phrase rhythm and hierarchical counterpoint, it is anything but equal. The hierarchy lives in us, not in the tuning system. The tuning system merely makes it possible to exploit that hierarchy fluidly, to shift it, to trick it, to send it somewhere unexpected.
Bali managed total cosmic hierarchy and practical egalitarianism simultaneously. 12EDO manages irrational equality and deeply felt harmonic hierarchy simultaneously. Both systems work because the important structure is carried by the participants — by the Balinese who know their position, by the listeners who carry the harmonies in their bones.
---
I can be among those who get a bit thrown off by composers make ideological commitments that exile them from what is *in us*, but who am I to say what's *in us*? The rebellion against Boulez's equal-weight diktat is understandable — symmetry death is real. The attraction of JI's pure intervals is real. The mathematical ambitions of X≠12EDO are real and not to be dismissed.
But I want to make the case, before the argument moves on without me, that what 12EDO related 12 keys to one another and we have 12EDO subsets in us through our exposure to a thousand years of notated music.
We are a rich, diverse, pluralistic musical universe. Let things play out. We'll see what new harmonies can get into us. But let this defense be on record to empower those who see the 12EDO space as something barely recovering from its 20th C. caprices, overreach, and often hubristic historicism.
William Anderson
I compose, play plucked instruments, run ensembles, produce concerts, and advocate for new music, I soaked up contemporary American music for 30 years, working directly with composers from all over the map. In the last 20 years I began composing works that draw on that experience, with a particular concern for finding a coherent way to make reconciling musics that were sealed off from one another. How to make embarrassingly simple and charming music work with the most complex musical modalities? Tipping in this direction has made me aware of the dangers of grasping too firmly onto any one notion.
I've come to Macherey's vision that our shadow (what we're *not*) is radiantly present in our work. So it's perfectly reasonable to *love everything*. We're imbricated. --
**The true necessity of the text manifests itself in the fact that not a word of it can be changed and nothing can be added, even though it appears at each moment as though a new topic could be chosen, an alternative narrative selected. But it is precisely this ceaseless shadowy presence of other possible phrases which could be pronounced, this ineradicable sense that things could have been other than they are, which enforces the constraining necessity of the text we actually have before us.**--Macherey