A twenty-five is a nine until its last unit

A bar of nine in Balkan practice is four beats of unequal length, the ratio between them is fixed at three to two by arithmetic rather than by hearing, and the length of the bar is bounded by the psychological present — a bar of sixteen units just fits inside three and a half seconds at the subdivision that suits both beats best, and a bar of seventeen does not.

That bound arrived with a caveat attached and the caveat was taken on trust. Every description of the long metres — the Macedonian and Bulgarian bars of eighteen, twenty-two, twenty-five — says that they are not single bars at all but combinations: a twenty-five is a seven and a seven and an eleven, and what a listener holds is three short bars in a row rather than one long one. Constantin Brăiloiu’s classification of 1951, which gave these metres their name, says so; the dance teachers say so; and nothing in this collection had ever asked whether it is true.

It is a claim about an inference, and there is machinery here for exactly that. The preference-rule model — Lerdahl and Jackendoff’s, in the form this collection uses — scores a candidate metre against a stream of onsets and returns the one that fits best; it is the same computation that finds the bar above the bar and the same one that prices syncopation once a reading has been chosen. Pointed at a constructed long metre, it answers the question directly, because “one bar of twenty-five” and “a bar of nine, repeating” are two periods over one onset stream, which is the only form of the question a model can take.

The answer is not the one the tradition gives, and it is not the opposite either.

The window says the same thing about a nine and a twenty-five #

Before the induction model, the obvious instrument is the one the ladder already carries, and it is worth showing again that it has nothing to say here.

The tempo window asks whether every beat of a metre falls between about a tenth of a second and two seconds. A metre built from twos and threes has beats of two lengths whatever else is true of it, so the shortest beat sets the fast edge and the longest sets the slow one, and the number of beats — and therefore the length of the bar — never enters.

The graded version of the window is no better for this purpose. Scoring each beat by how near it comes to the preferred rate near 550 milliseconds gives a twenty-five and a nine the same 0.865 at the same subdivision of 218 milliseconds, because the score depends only on which beat lengths are present, and a seven, a nine and a twenty-five have the same two. The window prices the beat. It has no term for the bar.

The inner bands are where a term for the bar can come from, and they are worth reading carefully, because they are the whole argument in miniature. They are the same question asked with the bar also required to fit inside a psychological present of eight seconds — the generous end of the published band. That requirement costs the seven and the nine nothing at all; it cuts the twenty-five’s range to 44 per cent of what the window alone allows. Bar length enters this ladder only through memory, never through rate.

So the question has to go to the model that reads the accents rather than the durations.

The rules do choose, and they choose the long bar #

The preference-rule set is three numbers, and they are the ones Lerdahl and Jackendoff’s account is usually implemented with: three points for a strong position that carries an onset, minus two for a strong position with nothing on it, minus one for an onset that lands off every strong position. A candidate is a period together with the positions inside it the metre calls strong, so the twenty-five’s own reading has a period of twenty-five and eleven strong positions, and the nine’s has a period of nine and four.

Handed one complete cycle of the constructed twenty-five, the model is not ambiguous at all.

This is worth stating plainly, because it refutes the received account rather than refining it. The model never prefers the group. The whole cycle is the true period of the onset stream, so every strong position it names carries an onset and every onset falls on one; it scores the maximum the rule set allows, and no cut of it can do better than tie. A reading that is right cannot be beaten by a reading that is wrong.

The nine loses by exactly one empty strong position — the accent it expects at step twenty-five, which never arrives.

The evidence arrives when the bar ends, and not before #

That single empty position is the entire case, and its location is the finding.

Take the same computation and vary not the candidate but the evidence: score both readings on a window of consecutive steps, and ask how many steps have to be in the window before the whole cycle beats every cut of it.

Zero is a strong word and it is the right one. Below the separation point the two hypotheses are not merely close; they are identical predictions, and no scoring function whatever — not this rule set, not a different weighting, not a Bayesian one — can prefer either, because there is nothing in the signal for it to prefer on. The eleven-unit reading, the eighteen-unit reading and the twenty-five-unit reading all say: accent, gap of two, accent, gap of two, accent, gap of three, and so on, and they go on saying it in unison until one of them runs out of bar.

Measured over every arrangement of every unequal metre from nine units to twenty-five — 1,820 of them — no arrangement separates in fewer steps than the bar has units. The bar length is a floor on the evidence, and it is a floor that is never broken. It is reached exactly by 267 of the 1,820; the rest need more, and the Bulgarian twenty-five needs thirty-nine, more than half as long again as the bar it is about.

The shorter reading is not a rival, it is one that decays #

There is a second thing the arithmetic refuses, and it concerns what the alternative actually is.

The received account offers “three shorter bars” as if it were a competing analysis a listener might reasonably settle into. It is not a competing period. A twenty-five counted 7+7+11 has no repeating sub-bar at all: the seven is right twice and then wrong, and a listener holding it does not hold a metre that fits the music slightly less well — they hold one that walks steadily out of phase with it.

So the two readings stand in a peculiar relation. On one cycle of evidence they are tied. On two the long bar is ahead; on four the short reading has lost two-thirds of its score; by eight it is below zero and the accents it predicts are landing on silence more often than on onsets. The short reading is not where a listener settles. It is where a listener starts, and it is refuted by anybody who stays.

Which makes the tradition’s claim strange in a specific way. It cannot be a claim that the group is the better analysis, because the group is a worse one at every amount of evidence past a single cycle. It has to be a claim about what a listener can get to.

Evidence needed against evidence available #

The two quantities are now in the same units, which is the only reason any of this decides anything.

How much evidence the question needs is a number of consecutive steps, and it is at least the length of the bar. How much evidence a listener has is also a number of steps: the psychological present — Fraisse’s perceived group, Pöppel’s integration interval — is a duration, and a duration divided by the length of one subdivision is a count.

Sixteen units. The same sixteen that came out of asking whether the bar fits inside the present at all, and for a reason that is not a coincidence but is not quite a tautology either: a bar that fits inside the present has, by definition, room for as many steps as it has units, and the evidence needs exactly that many. The two constraints coincide because both count the same thing.

What can be bought, and at what price, is worth being precise about. The twenty-five can be squeezed inside three and a half seconds by playing it faster — a subdivision of 140 milliseconds rather than 218, beats of 280 and 420 milliseconds, about 214 short beats a minute. At that speed the whole bar is inside the present and the evidence does arrive. It costs 0.261 of the beat’s own quality: 0.604 against the 0.865 the same metre scores when it is allowed to choose its tempo freely. That is close to a third of the window’s preference given up, and it is given up on every beat of every bar, to buy a discrimination that happens once per cycle.

No tradition makes that trade, and the arithmetic says why it would be a bad one.

The threshold is a measurement of the present #

Run the argument backwards and it stops being an explanation and becomes an instrument.

A tradition that treats bars up to some length as single bars and longer ones as combinations is not expressing a preference about grouping. It is reporting how many subdivisions its listeners can hold at once — and since the subdivision at which an unequal metre is best played is fixed at 218 milliseconds by the beats alone, that count converts straight into seconds.

The band is two to eight seconds — Fraisse’s reviews put the upper bound of a perceived group near five, Pöppel’s integration interval nearer three — and the reading is 3.49. That is the surprise in this essay: a rule of thumb from Balkan dance pedagogy, stated in units of a bar and never intended as psychophysics, turns out to be a sharper estimate of the psychological present than the band it is being compared against. It is sharp because the subdivision it is implicitly measured at is not free — the twos and threes fix it — and because the threshold is a count rather than a judgement of duration, which is the quantity people are bad at.

The direction of that inference deserves care. It is not evidence that the present is 3.5 seconds; the present is measured independently and by other means. It is evidence that the threshold and the present are the same fact, which is what the whole argument has been claiming, and the agreement to two decimal places is a stronger form of that claim than anything the models produce on their own.

The one case where the group really does win #

There is a version of the compound in which the tradition is not merely right but right in a much stronger way, and it is the version the ladder has already ruled out for a different reason.

A bar of unequal beats is not a repeat of a shorter bar, because a repeat is heard as the shorter bar twice — which is why 2+3+2+3 is two fives and not a ten. Applied to the long metres, that says an eighteen counted 9+9 is not an eighteen at all.

The induction model agrees, and it agrees absolutely rather than on balance. For a compound of equal halves the long reading and the short one predict the same accents for ever: there is no last unit at which they part, because the second half repeats the first exactly. The margin between them is zero at every window of every length, and the position measure that asks how much of a cycle has to be heard before a listener knows where in it they are returns no answer at all — an eighteen made of two nines has nine distinct rotations rather than eighteen, so the phase within the long cycle is not merely hard to find, it is not determined by the onsets.

That is the flat line in the evidence figure, and it is the sharpest result here. The compound rule is exactly right for equal parts and exactly wrong for unequal ones, and the tradition states one rule for both cases. Where the parts are equal, the long bar is not a hypothesis a listener could hold; where they are unequal, it is the only hypothesis that survives contact with more than one cycle.

Where the model stops #

It cannot show a listener who is not already counting. Every candidate here is a period counted from an assumed downbeat, so the cuts available are the prefixes of the metre from wherever the bar is taken to begin. That is why the separation window depends on the arrangement: a twenty-five counted 9+9+7 separates at twenty-five and the same twenty-five counted 7+7+11 at thirty-nine. A listener who has not yet found the downbeat has a larger hypothesis space than any of these figures draws, and would need more evidence rather than less.

It cannot show anything but onsets. The cues a listener actually has include note lengths, accents and harmonic change, and a played aksak bar marks its groups with all of them — the long beat is louder, the drum stroke on it is different, the dance step lands there. Any of those would supply the missing evidence earlier, and the honest statement of this result is that it is about the onsets alone, which is the weakest input the question can be asked on.

It cannot show a listener who forgets gradually. The present is treated here as a hard edge: sixteen steps are available and the seventeenth is not. Memory is not like that, and a decaying memory changes what a cycle can say about its own position rather than simply truncating it. A graded version would blur the crossing rather than move it, but it would blur it.

It cannot show learning. A dancer who has heard a twenty-five ten thousand times is not inferring it from a window; the metre is a stored pattern and the question is recognition rather than induction. That is a different model and probably the right one for the repertoire — which makes the result here a statement about a first hearing, and about what a tradition could have discovered by ear.

And the subdivision is held at its optimum. Everything above is computed at 218 milliseconds, the rate at which an unequal metre’s two beats are jointly best. Real performances are faster and slower, and the crossing moves with the tempo: at 140 milliseconds it sits at twenty-five and at 300 it sits at eleven.

Whose metres these are #

The long metres are Balkan and they are specific. Bulgarian and Macedonian dance practice uses an eleven counted 2+2+3+2+2, a fifteen counted 2+2+2+2+3+2+2, a twenty-two and a twenty-five; the twenty-five most often cited is the dance Sedi Donka, counted 7+7+11. Turkish usul practice builds much longer cycles still, and describes them explicitly as chains of named shorter units.

The claim tested here is a claim about that repertoire and about how it is taught. It is not a claim about Western notated metre, which has no bars of twenty-five, and it is not a claim about the West African and Cuban cycles the timeline ladder is about — those are twelve and sixteen steps, comfortably inside the present, and the question does not arise for them.

What the arithmetic supports, in the end, is the practice rather than the explanation. Teachers are right that a twenty-five is counted as a seven and a seven and an eleven. They are wrong about why: not because a listener prefers the shorter unit, but because the unit that would refute it is longer than a listener can hold, and the tradition has been reporting the size of that limit in the only units it has.

Where this ladder goes next #

Four rungs. A bar of nine is four beats of unequal length, which no single division of a bar produces; the ratio between those beats is fixed at three to two by arithmetic and not by hearing; the length of the bar is bounded by the psychological present, with the supply of available metres still growing where the bound begins to bite; and the received account of what happens past that bound is right about the practice and wrong about the mechanism, because the model never prefers the group and simply cannot reach the evidence that would settle it.

What is owed is the accent. Every figure above scores a bare list of onsets, and that is the weakest input the question admits, as the account of where the model stops says at length. An aksak bar in performance marks its long beat with more than an onset: it is louder, it is longer than the notation says, and in the drum patterns it is a different stroke. The machinery for weighting a second cue against the onsets already exists and has a gain on it, and the question it answers here is sharp: how strong does an accent on the long beat have to be before the separation window falls inside the present at a bar of twenty-five? That is a single number, it is a number a recording could be measured against, and it decides whether the result above is a fact about listeners or an artefact of feeding them less than they get. It is arithmetic, and it needs no corpus — though a corpus would then be able to say whether real performances supply the accent the arithmetic asks for.