A process that enumerates its own form

Most musical forms are decisions. How long the exposition runs, how many times the refrain comes back, whether there is a coda — all of it chosen, and choosable otherwise.

There is a small family of pieces where the form is not a decision at all. A rule is stated, the rule is applied until it exhausts itself, and the length of the piece is whatever the arithmetic gives. The composer picks the rule and the material; the shape is a consequence.

Phase music is the clearest instance, and the arithmetic is short enough to do in a paragraph.

The rule, and the number it produces #

Take a rhythm of $n$ steps. Have two players play it in unison, then move one player forward by one step, then by another, and so on. After $n$ shifts that player has moved a whole cycle and the two are in unison again.

So the piece has exactly $n$ stages, and $n$ is a property of the pattern rather than of anybody’s plan — with one condition that is worth stating because most patterns satisfy it and some do not. If the pattern maps onto itself under a rotation smaller than $n$, the piece is over sooner: a twelve-step pattern with period three has three stages, not twelve, and the fourth shift is the first repeated. Fifteen of the 495 eight-in-twelve patterns are like that. So the length is the pattern’s own period rather than its length, and the two agree for everything a composer would plausibly choose.

The material is twelve steps with eight struck, and the first shift already fills the cycle completely. The densest stage is the first one, which is not what a listener expects of a process that is supposed to build — and it is a consequence of the pattern’s own gaps rather than of anything the composer arranged.

Twelve steps, so twelve stages. That is the whole of the form.

What is heard at each stage #

The interesting part is not the count but the content, because at every stage the two parts together produce a pattern that neither of them is.

The hero figure lists all twelve. Each row is the union of the original and the shifted copy: a step is struck if either player strikes it, and the darker cells are the steps where both do.

Read down the third column and the numbers move. The unison stage has eight onsets, because both players are hitting the same eight. Every other stage has ten, eleven or twelve, and three of them — the shifts by one, by six and by eleven — produce a completely saturated resultant, all twelve steps struck, a continuous stream from a pattern that is two-thirds full.

Read the coincidence count and something else appears. Among the shifted stages it runs 4, 5, 6, 5, 6, 4, 6, 5, 6, 5, 4 — and that sequence is a palindrome. It has to be: the number of strokes two copies share at a shift of $k$ is the same as at a shift of $n-k$, because shifting one copy forward by $k$ and the other forward by $n-k$ are the same relative displacement. The second half of the piece mirrors the first in coincidence, without mirroring it in content.

None of that was composed. It is the autocorrelation of an eight-in-twelve pattern, and it would be the same for anybody who chose the same pattern.

Why this is not a polyrhythm #

The site’s rhythm field already has an essay about two clocks running at once, and phasing is not that, which is worth making explicit because the two get confused.

A polyrhythm is two rates. Three against two means one part divides the bar in three and the other in two, and the combined pattern repeats after the least common multiple of the two — six steps, in that case, permanently. It is a fixed object.

Phasing is two copies at the same rate, displaced. The combined pattern is different at every stage and the displacement is what changes. Nothing about it repeats until the whole orbit is complete.

Compare stage three, which is also six coincidences and ten struck. The orbit is symmetric about its middle — shifting by k and by twelve minus k give the same combined pattern — so a phase piece that runs the whole way round plays every distinct stage twice, and the second half is the first half backwards.

There is a real relation between them, though, and it is worth stating because it explains where the audible interest in a phase piece comes from. Each stage of a phase piece is a canon at a fixed time interval, and a canon at a time interval is what a polyrhythm becomes when the two parts play the same material. The stages are twelve canons on one subject, enumerated.

The stages are a rhythm of their own #

There is a level above the stages that the hero figure makes visible and that no description of the process mentions, and it is the reason a phase piece has a shape rather than merely a length.

Take the resultant onset count at each stage — 8, 12, 11, 10, 11, 10, 12, 10, 11, 10, 11, 12 — and read it as a curve. It starts low, jumps to its maximum, and then oscillates between ten and twelve for the rest of the piece, touching twelve at three points. That is a density profile of exactly the kind a cyclic piece uses instead of a cadence, except that here nobody arranged it.

So the piece has a form at two levels. Inside a stage, a repeating rhythm. Across the stages, a density curve. Neither was composed, both are computable from the pattern before a note is played, and the second is a fair description of the large shape of the piece.

That is worth putting beside the other forms in this field. A sonata movement’s shape is a key plan and a composer’s decision at every point; a phase piece’s shape is a density curve and a consequence of one decision at the start. The two are not different amounts of planning. They are planning applied at different places.

The pattern decides how good the piece is #

If the form is arithmetic, everything a composer controls is in the choice of pattern — and the choice matters enormously, in ways the arithmetic makes visible.

A pattern of $n$ steps gives $n$ stages, so a longer pattern gives a longer piece. A pattern with more onsets saturates sooner, so a very full pattern produces stages that are all indistinguishable streams. A pattern with very few onsets produces stages that are all sparse and similar. What is wanted looks like a pattern whose rotations against itself produce different amounts of coincidence — which is a property of its autocorrelation.

Scoring all 495 eight-in-twelve patterns by the variance of that autocorrelation says two things, and the first is that the criterion as stated defeats itself. The three patterns it ranks highest are the ones whose coincidence counts run 4, 4, 8, 4, 4, 8, 4, 4, 8 — an eight at every third shift, five times the bell pattern’s variance. An eight means every onset has landed on an onset, which means the rotation has mapped the pattern onto itself, which means the pattern has period three and the piece has three stages rather than twelve.

So maximising the spread of coincidence selects patterns with internal periodicity, and internal periodicity collapses the form. The criterion has to be applied subject to the orbit being full — no rotation mapping the pattern onto itself — and 480 of the 495 satisfy that.

The second thing is where the bell pattern sits, and it is not at the top. Its variance is 0.63 against a best of 1.54 among the full-orbit patterns, and 96 of the 480 beat it: it is at about the eightieth percentile, comfortably above the median and a long way from the maximum. Its coincidence counts run 4, 5, 6, 5, 6, 4, 6, 5, 6, 5, 4 — a spread of two — where the best available runs 4, 7, 4, 6, 4, 6, 4, 6, 4, 7, 4, a spread of three and a much sharper alternation.

Which is a fair verdict on the criterion rather than on the pattern. A pattern chosen for a phase piece has to be playable by two people in strict unison at speed, has to be memorable enough that a listener can hear which copy is which, and has to be a real timeline rather than an arrangement of eight dots — and the variance measures none of that. The arithmetic says the choice was not the extremal one, and the reason it was not is that the arithmetic is not the whole brief.

The bell pattern is a good choice on exactly that test. Its coincidence counts run from four to six out of eight, so the stages differ audibly; its resultants run from eight onsets to twelve, so the density varies; and three of the twelve saturate completely, which gives the piece landmarks.

Six patterns generated by Bjorklund’s algorithm rather than chosen show that any of them could be the material for a phase piece, and that the form would differ: the orbit’s shape is a function of where the onsets are, so the sequence of densities a process enumerates is decided entirely by the pattern it starts from.

A maximally even pattern is not obviously the best material for this, and there is a reason to think it is close to the worst. Bjorklund’s construction spreads onsets as evenly as possible, which means a maximally even pattern resembles its own rotations more than an uneven one does — its autocorrelation is flat. Flat autocorrelation means every stage sounds much like every other, which is the one thing a piece whose form is its stages cannot afford.

That is a testable claim about a compositional choice, arrived at from a property of a pattern, and it is the kind of thing this whole field is for.

The pattern’s length is the piece’s length #

One consequence of the arithmetic deserves separating out, because it is the strongest form of the claim that the form is not chosen.

A four-step pattern gives four stages. A nine-step pattern gives nine. A sixteen-step pattern gives sixteen. If each stage is held for a fixed number of repetitions, the piece’s duration in seconds is the pattern length squared, times the repetition count, times the step duration — so choosing a sixteen-step pattern instead of a twelve-step one does not make the piece a third longer, it makes it nearly twice as long.

That quadratic is the practical constraint on the whole idea. At twelve steps and a comfortable tempo the piece runs a few minutes; at twenty-four steps it would run four times as long with stages that differ half as much, and nobody has written that piece. The form is arithmetic and the arithmetic is not gentle, which is why the repertoire clusters at short pattern lengths.

The listener does not hear the process #

There is a well-known gap between what such a piece is and what it sounds like, and it is worth being precise about, because the composers of this music were explicit about wanting the process audible.

What a listener actually attends to, at any moment, is the resultant — the bottom row of the stage figure. It is a rhythm, it repeats, it is heard as a pattern in its own right. The two parts producing it are usually not separable by ear once the shift is more than a step or two, because the ear builds objects out of what fuses and two identical timbres striking near-simultaneously fuse readily.

So the piece is heard as a sequence of rhythms, each arriving suddenly, each stable for a while. The process is audible in the sense that a listener knows something systematic is happening; it is not audible in the sense of being followable step by step.

That is not a criticism and it is close to the point. A form determined entirely by a rule produces musical events the composer did not select and could not easily have invented — the saturated stages, the near-palindrome, the particular resultant at shift seven — and a listener meets those as music rather than as arithmetic.

Where else the shape turns up #

Phase music is a small repertoire and the shape is not.

A change-ringing peal is the same idea with permutations rather than rotations: a set of bells, a rule for exchanging adjacent pairs, and the piece is over when every ordering has been rung once. The number of changes is a factorial, the ringers execute a rule, and the length is arithmetic. That practice is four centuries older than phase music and nobody involved would call it composition.

A canon with a stated rule — by inversion, by augmentation, by a fixed delay — is a partial version: the rule determines the second voice and the composer still chooses when to stop.

And any rotation of a cyclic pattern is available as a device wherever cyclic music is played, which is most places. What is unusual about the phase piece is only that it commits to enumerating all of them.

Two shifts that produce the same coincidence do not produce the same music #

The palindrome in the coincidence counts invites a wrong conclusion, and heading it off makes the structure clearer.

Shift two and shift ten both have five coinciding strokes out of eight. They are not the same stage. Their resultants are different patterns — one has its rest in the ninth position and the other in the seventh — and they sound different, because a rhythm is where the onsets are and not how many there are.

Counting distinct resultants across the whole orbit gives ten, not twelve: two pairs of shifts happen to produce the same combined pattern, which is a property of this particular pattern’s symmetry and not a general fact. So even the count of musically distinct stages is something the arithmetic has to be asked for rather than assumed.

That is the useful discipline in this whole field. The number of stages is $n$; the number of distinct coincidence counts is smaller; the number of distinct resultants is smaller still. Three different questions about one orbit, three different answers, and only the first is obvious.

Taken as spans rather than as patterns the density is flat once both parts are playing, and that is the honest picture of what a phase piece does: the amount of sound does not change and the pattern does, so the whole of the form is carried by which steps coincide rather than by how many.

What this cannot show #

The figures treat a stage as an instant, and it is not. Reich’s process moves gradually — one player accelerates very slightly until they arrive at the next position — and the transitional passages, where the two parts are between stages, are a substantial part of the listening experience and are not in any figure here. What is drawn is the sequence of stable states, and the transitions between them are where a great deal of the difficulty and most of the risk lies for the performers.

The figures also have no dynamics, no timbre and no accent. The two clapping parts in a real performance are two pairs of hands with different sounds, and a listener uses that difference to separate them for longer than the fusion argument above suggests.

And the resultant treats a coincidence as one onset. Two hands striking together are louder than one, which is a real accent pattern laid over the resultant and is arguably the most audible thing in the piece — the coincidence count in the figures is exactly that pattern’s onset count, and nothing here draws it as the accent it is. Since the same onsets support several metres depending on where the accents fall, that missing accent pattern is not a detail: it is what decides how each stage is heard, and a listener’s reading of a stage may change during it.

There is also nothing here about pitch. The process was first applied to tape loops of speech and then to a twelve-note piano figure, where the resultant is a melody rather than a rhythm and the two parts produce intervals as well as coincidences. The rhythmic version is used here because an encoding of a rhythm is unambiguous in a way an encoding of a melody is not, and every claim above would need a second set of arguments about intervals if pitch were in it.

Where the ladder goes #

A phase piece is a canon whose delay changes. A canon whose delay is fixed raises a different question — not what shape the piece takes, but whether the tune works against itself at all — and that turns out to be decidable in advance by scanning every delay, with a result that separates two nursery tunes completely.