How unequal a beat is allowed to be

A bar of nine in Balkan practice is not nine of anything. It is four beats — short, short, short, long — and the inequality is at the beat level rather than inside it, which is a thing no single division of a bar can produce.

That rung established the object. It did not ask what constrains it, and there is an obvious question sitting there: why is the long beat always three subdivisions against two? Balkan sevens are 2+2+3. Fives are 2+3. Nines are 2+2+2+3. Elevens are 2+2+2+2+3. Nowhere is there a 2+4, or a 3+5, or a bar whose beats stand in any ratio but three to two.

The answer that needs no listener #

A group of four subdivisions is not one beat. It is two beats of two.

That is not a perceptual claim, it is a decomposition: four splits as 2+2, five as 2+3, six as 3+3 or 2+2+2. Every integer above three can be written as a sum of twos and threes, and none below four can be written as anything but itself. So the only groups that are irreducibly one beat are two and three, and any metre whose “beats” include a four is a metre with more beats than it claimed.

Which fixes the ratio. If the beats of a bar are twos and threes and at least one of each, the long beat is three subdivisions and the short is two, and the ratio between them is 3:2. It cannot be anything else, and nothing about hearing was needed to say so.

How many nines there are, and the answer is one #

If a bar of n units is a sum of twos and threes, the whole space of aksak metres of that length can be enumerated, and it is very small.

For every bar length up to nine there is exactly one metre with unequal beats. That is why these metres have names rather than descriptions: a musician saying “a nine” in a Balkan context has said everything, because there is nothing else a nine could be. Ambiguity begins at eleven, where 2+2+2+2+3 and 2+3+3+3 are both available, and both are used.

Which needs the decomposition argument a second time #

That table skips the even lengths, and running the enumeration over all of them turns up a case it would otherwise have to explain. A bar of ten has two unequal groupings, 2+2+3+3 and 2+3+2+3, so on the face of it ambiguity begins at ten rather than eleven.

It does not, and the reason is the argument this essay already made, applied one level up. 2+3+2+3 is 2+3 twice — it is two bars of five, not a bar of ten. A grouping that repeats a shorter grouping is not a new metre for the same reason a group of four is not a new beat: it decomposes, and the decomposition is the thing that is actually being felt.

Filtering the enumeration on that condition:

Ambiguity begins at eleven after all, and it begins there because eleven is the first length at which two genuinely distinct unequal groupings exist. Ten looked like a counterexample and is the confirming case: it has two groupings and one metre.

The filter earns its keep at every even length above ten as well — fourteen loses a grouping that is two sevens, fifteen one that is three fives, sixteen one that is two eights — and it never removes anything from an odd prime length, because a bar of a prime number of units cannot be a repeat of anything. That is why the odd lengths are the ones with names: five, seven, nine, eleven and thirteen are exactly the lengths where every grouping the arithmetic offers is a genuine metre.

So the same one-line observation does two jobs. Downward it says a beat is two or three units, because four and five split. Upward it says a bar is not a repeat of a shorter bar, because a repeat is heard as the shorter bar twice. Both are the statement that a listener takes the shortest available reading, and between them they cut the space of aksak metres down to the list the traditions use.

That is a second derivation of the same grouping and it arrives from an unrelated direction. Bjorklund’s algorithm asked for four strikes distributed as evenly as possible in nine steps and returns 2+2+2+3; the decomposition argument asked which groups are indivisible and returns the same thing. Neither knows about the other, and both give the metre that is played.

The agreement is not a coincidence and it is worth naming why. Maximal evenness puts the odd group at one end because that is the only place a remainder can go; the decomposition argument says a group can only be two or three because everything larger splits. Nine divided into four parts as evenly as possible has exactly one part of three, which satisfies both conditions at once — and for a bar length where the two disagree, the tradition would have to choose. Eleven in four is 3+3+3+2 by evenness and 2+2+2+2+3 by nothing at all, and both are played.

What the tempo window does bound #

The ratio is arithmetic. What is left for perception is the tempo, and there the bound is real and two-sided.

A series of events can be a beat only inside a window running from about a tenth of a second to two seconds. An aksak bar has beats of two lengths, so both have to be inside it: the short beat must be slow enough to be a beat rather than a subdivision, and the long beat fast enough to be a beat rather than a bar.

For a metre of twos and threes that gives a subdivision duration between 50 and 667 milliseconds — a range of more than thirteen to one, which is generous. For a 2+5, the five-group has to stay under two seconds, so the range shrinks at the top; and the shrinkage is the second reason such a metre is uncomfortable, after the first reason that it is not one beat.

Where the real constraint bites, and it is not either of these #

Both bounds above are loose. A ratio of 3:2 with a thirteen-to-one tempo range does not sound like a tight design space, and yet aksak metres are rare — most of the world’s music has equal beats.

The cost is somewhere else, and this ladder’s neighbour has already found it. The induction machinery lays its strong positions at a fixed period, so the only readings of nine a periodic model can offer are nine, three and one — and the right answer is not among them. An unequal beat is not hard to play; it is hard to infer. A listener meeting one for the first time has no periodic hypothesis that fits, and the metre has to be learned as a pattern rather than found as a period.

Maximal evenness is necessary and not sufficient, and the cases where it fails are the ones that show what is missing. Five beats distributed as evenly as possible in twelve gives 3+2+3+2+2 — beats of two and three, in a legal order, at a bar length traditions use constantly — and no tradition plays it. So the arithmetic produces more metres than the repertoire contains, and what selects among them is not a further arithmetical condition but a history of dances.

So the three bounds sort neatly. The ratio is fixed by arithmetic; the tempo is bounded by the beat window and generously; and what makes these metres a minority tradition rather than a universal one is that they are outside the hypothesis space of the ordinary way of finding a beat.

The same argument outside rhythm #

The decomposition result has a twin in this collection and the two were written for entirely different subjects.

The diatonic set is two sizes of step arranged as evenly as the twelve allow — five whole tones and two semitones, and the semitones as far apart as they can be. That is 2+2+1+2+2+2+1, which is a maximally even distribution of seven in twelve with two group sizes, one large and one small, in a ratio of two to one.

A Balkan nine is 2+2+2+3: a maximally even distribution of four in nine with two group sizes, one large and one small, in a ratio of three to two.

The two are the same construction on different numbers, which is a thing this site has said before about scales and rhythms and has not said about unequal beats specifically. What the rhythmic case adds is the decomposition bound, and the pitch case does not have one: there is nothing about a scale step of three semitones that makes it decompose into smaller steps, which is why harmonic minor exists and a beat of four does not.

So the analogy is exact where it holds and its failure is informative. Time has a floor that pitch does not — a subdivision fast enough to stop being countable — and that floor is what makes the beat lengths of an aksak metre a finite list of two rather than an open choice.

Which computation produced the numbers #

The partitions are an exhaustive enumeration of the ways of writing n as an ordered sum of twos and threes, deduplicated by rotation — because a bar and the same bar started elsewhere are the same necklace, which is the argument the Euclidean rung makes about timelines and it applies unchanged here.

The second filter is a test for periodicity on the same list: a grouping is discarded when its sequence of parts repeats at some divisor of its own length, which is what makes 2+3+2+3 two fives rather than a ten. It is a stricter filter than the rotation one and it is applied afterwards, so the two columns of the table are the same enumeration counted twice rather than two enumerations. Nothing in either step is a judgement about music; both are properties of a list of integers, which is what lets the essay’s one perceptual assumption — that a listener takes the shortest available reading — do all of the work in both directions.

The tempo bands come from the published window this ladder already uses — 100 ms at the fast end and 2,000 at the slow, with a preference near 550 — applied to each of the bar’s beat lengths and intersected. The bar is also required to fit inside the psychological present, which binds only for very long bars.

The claim that four decomposes is a claim about hearing dressed as arithmetic, and it is worth saying so. That 2+2 is preferred to 4 is the assertion that a listener offered two readings takes the shorter beat, and this ladder’s own rung establishes the preference rather than deriving it. What the arithmetic contributes is that once the preference is granted, the 3:2 ratio follows with no further assumptions.

What an unequal beat costs a performer, and it is nothing #

One more expectation is worth disposing of, because it is the usual one and the arithmetic refuses it.

An unequal beat is often described as difficult, and it is difficult to read and difficult to count. It is not difficult to produce. A player keeping a 2+2+2+3 nine is keeping a steady subdivision and grouping it, which is exactly what a player keeping 3/4 at the same subdivision rate is doing; the two differ in where the accents fall and not in what the hands are doing. Nothing in the motor problem changes.

The measurements built for periodic metre report otherwise, and they report it in a way worth naming. Score the nine’s onsets against the weights a fixed period produces for nine positions and the onsets and the weights disagree almost everywhere, which a syncopation measure reads as heavy syncopation — of a metre that is not syncopated at all. A measure built on a periodic grid describes a non-periodic grouping as a deviation from a metre nobody is playing, so the instrument reports difficulty where the difficulty is its own.

That is consistent with what the traditions look like from outside. Aksak metres are dance metres, played fast, by musicians who learned them before they could read — the conditions under which a pattern is acquired whole and the inference problem never arises. The difficulty belongs entirely to the listener meeting the metre cold and to the notation trying to write it down, and both of those are the same failure: an assumption of periodicity where the material is a pattern.

What the enumeration does not explain #

Two things it leaves open are worth naming, because they are the parts a listener would ask about first.

It does not say why the long beat goes last. A Balkan nine is almost always 2+2+2+3 and hardly ever 3+2+2+2 or 2+3+2+2, and the enumeration counts those as the same necklace — which is correct as combinatorics and wrong as music, because a timeline is a rotation and the rotation is the thing it is for. Where the long beat sits is exactly the information the necklace throws away, and it is a decision each tradition makes.

And it does not say why some lengths are used and others are not. Eights exist as 2+3+3 and are far rarer than nines and sevens. Thirteens are rare despite offering three groupings. Nothing in the arithmetic prefers a nine, and the frequency with which lengths occur is a fact about repertoire.

There is one pattern in the enumeration that points at an answer without supplying it. The lengths with names — five, seven, nine, eleven, thirteen — are the odd ones, and the odd ones are precisely the lengths at which no grouping is a repeat of a shorter bar, because an odd number of units cannot be several copies of anything with an even total and the parts are twos and threes. So a tradition that uses odd bar lengths never has to distinguish a metre from a pair of shorter metres, and one that uses even lengths does. Whether that is why the odd ones are the named ones, or a coincidence of two facts about small numbers, is exactly the sort of question the corpus this essay does not have would settle.

Both of those are the same kind of gap as the one at the end of this essay: the arithmetic bounds the space and the tradition picks inside it, and this collection can compute the first and not the second.

Whose music, and when #

Aksak metres are Balkan, Turkish, Greek and Bulgarian above all, with relatives across the Middle East, in Indian tala and in some Latin American practice. The word is Turkish and means “limping”. The nines, sevens and fives are traditional dance metres with names and repertoires attached, and the enumeration above says nothing about which of the available ones a tradition chose — only that at each bar length there was almost no choice to make.

Western art music has the same metres as occasional devices, notated as 5/8 or 7/8 with a beaming that specifies the grouping. The beaming is doing real work: the grouping is the metre, and without it the same nine quavers admit both readings the enumeration found.

What the picture cannot show #

The subdivisions are treated as equal. They are not: measurements of Bulgarian dance music find the long beat is often nearer 2.3 or 2.4 to one than 3 to 2 — the same kind of systematic deviation the swing ratio turned out to be. The 3:2 result here is about the notated structure, and the performed ratio is a different measurement this site does not have.

Only twos and threes are admitted. A tradition that used a group of four as a genuine beat would falsify the decomposition argument, and the argument’s defence — that the four would then be heard as two — is not testable from inside this model.

The tempo window is quoted. Its edges are from the literature, they are soft, and both the fast and the slow end vary between listeners and with training.

And nothing here is measured on a performance. The metres are the notated ones, the tempi are plausible rather than recorded, and no dance was timed.

Where this ladder goes next #

This ladder had one rung and now has two: the object, and what bounds it. The bound turns out to be arithmetic where it was expected to be perceptual, and perceptual where nobody was looking — in the inference rather than in the execution.

The rung after it is the one the enumeration makes obvious and this site cannot yet write. If a bar of eleven admits two aksak groupings and a bar of thirteen admits three, then a tradition that uses elevens has made a choice, and which one it made is evidence about what the tradition wants. That is a question about repertoire, it is answerable by counting, and it needs a corpus of Balkan dance metres that this collection does not have.