How long a limping bar can be

A bar of nine in Balkan practice is four beats, three short and one long, and the long one is always three subdivisions against two because four and five decompose and two and three do not. That result is arithmetic. It arrived where a perceptual bound was expected, and it left the tempo window holding only one job: to say how fast such a bar can go.

The window does that job by answering a yes-or-no question. A series of events can be a beat only between about a tenth of a second and two seconds, so a metre is playable at the subdivision rates that put every one of its beats inside those edges. For beats of two and three subdivisions that is a subdivision between fifty and 667 milliseconds — a range of more than thirteen to one, and generous.

It is also the same range for every aksak metre there is.

A window that says yes to everything #

That is a genuine limitation rather than a quibble, and it is worth being precise about why.

The window is a membership test. It divides beat durations into those that can carry a pulse and those that cannot, and a metre passes when all of its beats are inside. Membership tests have a characteristic failure: they cannot rank the things that pass. A nine at a subdivision of 218 milliseconds and a nineteen at the same rate both have beats of 437 and 655 milliseconds, both beats are comfortably inside the window, and the test has nothing further to say.

So the account of additive metre this collection had reached was a bound on the ratio, which is exact, and a bound on the tempo, which is loose and length-blind. Nothing in it explains why the metres with names are fives, sevens, nines, elevens and thirteens rather than fives, sevens, nines and thirty-sevens.

There is a graded version of the same published numbers, and it has been sitting in this collection since the beat’s own preferred rate was measured. The window has three edges rather than two: a fast limit near a tenth of a second, a slow limit near two seconds, and a preferred rate near 550 milliseconds where a listener taps most readily and a tempo is most easily held. Scoring a beat by how near it comes to that preference — one at 550, falling linearly in log period to nothing at either edge — turns the yes-or-no into a quantity, and it introduces no parameter the window did not already carry.

One preferred rate, and two beats #

The moment the test becomes a score, the difference between an equal metre and an unequal one stops being a matter of terminology.

A metre with one beat length has one thing to place, and it places it at 550 milliseconds. Three beats of three subdivisions each, at a subdivision of 183 milliseconds, put every beat exactly on the preferred rate and score one.

A metre with two beat lengths cannot do that, and the reason is a counting argument rather than an empirical one. There is one preferred rate and there are two beats. Whatever subdivision rate is chosen, the short beat and the long beat are in a fixed ratio of three to two, so they cannot both sit on 550 milliseconds; one is above it and one below. The best available is the rate at which the worse of the two is as good as it can be, which is where the two curves cross.

The crossing is not found by searching. Writing the window’s fast half as the log distance from a tenth of a second to the preferred rate and its slow half as the log distance from the preferred rate to two seconds, the best subdivision is the one at which the short beat has climbed the same fraction of the fast half as the long beat has descended of the slow half. That is one line of algebra and it gives 218.3 milliseconds, which is what the sweep finds and is the figure’s own check on itself.

So an unequal beat costs 0.135. A metre of twos and threes scores 0.865 against an equal metre’s 1.000, in the units the window itself supplies, and that is the whole of what a rate preference has to say about the difference.

The cost is the same at every length, and that is the problem #

Two things follow immediately, and the second is what this essay is about.

The first is that the cost is a fact about the pair of beat lengths and about nothing else. Two and three are two and three whether the bar holds four beats or nine of them, so every aksak metre scores exactly 0.865, at exactly 218 milliseconds, whatever its length. The graded window is a better instrument than the membership test and it is blind in exactly the same place.

The second is that a metre’s bar is now a computed quantity rather than a free one. If the best subdivision is 218 milliseconds, then a bar of n units lasts 218n milliseconds at the rate that suits its beats best: 1.09 seconds for a five, 1.97 for a nine, 2.40 for an eleven, 2.84 for a thirteen, 3.28 for a fifteen, 3.71 for a seventeen. Those are not arbitrary numbers, and one of them is about to run into something.

The bar is a level too #

The account of the tempo window this ladder has been using names a second constraint and then never uses it: the bar has to fit inside the psychological present.

That is a real and separately measured quantity. A phrase is a number of seconds rather than a number of bars, and the window over which a listener holds a stretch of sound as one present thing runs from about two seconds to eight, with something like three and a half in the middle of it. A bar longer than that is not held as a bar; it is a sequence of things that have to be counted, which is a different act.

Applied to a beat, the present is slack and does nothing. Applied to the bar, it is the one constraint in this whole account that knows how long the bar is.

The quality is flat and then it falls off a cliff. Up to sixteen units the present is slack: the best metre of that length is played at 218 milliseconds a subdivision and scores 0.865, exactly as a five does. At seventeen units the bar would last 3.71 seconds, which does not fit, so the subdivision has to come down to 206 milliseconds and the score with it, to 0.830. At nineteen it is 0.765. At twenty-three it is 0.653, which is to say the metre is being played half again as fast as its beats want in order to be a bar at all.

There is nothing gradual about the mechanism. Below the knee the bar constraint is not binding and costs nothing; above it, it binds and every further unit costs more. A metre longer than about sixteen units is a metre whose beats have been sacrificed to keep the bar inside a listener’s grasp.

And the supply is still growing where the quality has begun to fall #

The other half of the figure is the half that makes the first half matter.

The number of genuinely distinct unequal metres a bar length admits is small at the lengths with names and it does not stay small. There is exactly one at five, seven, eight, nine and ten; two at eleven and twelve; three at thirteen and fourteen; four at fifteen; five at sixteen. Then it goes: seven at seventeen, eight at eighteen, eleven at nineteen, thirteen at twenty, and twenty-eight at twenty-three.

The supply is still accelerating at the length where the quality has already begun to fall. That is the shape of the result and it is worth stating plainly: the arithmetic of twos and threes offers a tradition an unbounded and rapidly widening choice of metres, and the arithmetic of a listener’s present takes almost all of it away. What is left is a bounded prefix of an unbounded list, and the prefix runs to about sixteen units.

The named lengths of the tradition are five, seven, nine, eleven, thirteen and fifteen. Every one of them is inside the flat region, and the region ends immediately after the last of them.

That is one coincidence and it should be read as one. The prefix’s length is not a strong prediction, because it comes from dividing an asserted present by a computed subdivision, and the present is a soft number with a wide published range.

Reading the number backwards #

Which suggests running the argument the other way, and that is a better use of it.

The knee sits at the present divided by 218 milliseconds. At the fast end of the published range for the present — two seconds — the knee is at nine units, and every bar longer than a nine is already paying.

At the slow end — eight seconds — the knee is at thirty-six units and the constraint never bites anywhere in the range of metres anybody plays.

So the observed repertoire is evidence about the present rather than the other way round. A tradition whose longest single-bar aksak metre is a thirteen puts its listeners’ present at 2.84 seconds or more; one that uses fifteens puts it at 3.28 or more. Both of those sit inside the published band and near the middle of it, which is a mild agreement between a psychophysical measurement and a repertoire, arrived at along a path neither was measured on.

It is mild because it is a one-sided bound. A repertoire that uses fifteens says the present is at least 3.3 seconds; it says nothing about how much more. What would sharpen it is the upper end — the longest bar a tradition treats as one bar — and that is exactly where the account becomes interesting.

The metres that are longer, and what happens to them #

Bulgarian dance music has metres a good deal longer than fifteen. Sedi Donka is twenty-five, Yove Male Mome is twenty-two, and there are others. If the argument above were a claim that no bar exceeds sixteen units, those would refute it on the first page of any collection.

They do not, and the reason is the thing the argument actually predicts. A twenty-five is not analysed as a bar of twenty-five. It is analysed as a group of shorter bars — a seven and a seven and an eleven, say — with the grouping named, taught and danced as such. The long metres are not counterexamples to the ceiling; they are what a tradition does when it reaches one.

That is a prediction with a shape rather than a number, and it is the most testable thing here. If the present is what stops a bar, then metres beyond the knee should stop being single bars and start being compounds, and the transition should happen in the region the figure marks rather than at some length fixed by notation. It is the same move the decomposition argument makes downward at the beat — a group of four subdivisions is two beats of two, because a listener takes the shortest available reading — arriving one level up: a bar of twenty-five is two bars and a bit, for the same reason and by the same principle.

What the criterion cannot do, and why not #

Two nulls come out of the same computation and both are worth having, because each is a place where an obvious refinement buys nothing.

The graded window cannot choose between two metres of the same length. At all, and exactly. The two elevens — 2+2+2+2+3 and its evener neighbour 3+3+3+2 — have the same beat lengths present, differ only in how many of each, and therefore have identical curves, identical optima, identical bands and identical scores to every decimal. The membership test could not separate them and neither can the score, and the reason is structural rather than a matter of resolution: the criterion is a function of the set of beat lengths and both metres have the set {2, 3}.

And the obvious repair does not repair it. Scoring a metre by the average preference over its beats rather than by the worst does make the two elevens different, because they have different numbers of long beats. It makes them different by 0.003 — 0.9403 against 0.9372, with the evener eleven marginally ahead. Both are played, extensively, in the same repertoire. A criterion that separates two equally common metres by a third of a per cent has not separated them; it has found the rounding error in its own model.

So the question the second rung left open — which of the available metres of a given length a tradition will use — is not answered by making the tempo criterion finer. It is not a tempo question. Where the long beat sits in the bar is exactly the information a necklace throws away, and it is decided somewhere this collection cannot compute.

Which computation produced the numbers #

The enumeration is the one the ladder already uses, with the three filters it already has: the parts are twos and threes; at least one of each, so that the metre is unequal; and no grouping that repeats a shorter grouping at any rotation, which is what makes 2+3+2+3 two fives rather than a ten.

The rate preference is linear in log period between the window’s published edges and one at its published preferred rate, so it introduces no fitted parameter — a smooth bell would need a width nothing in the literature supplies. A metre’s score is the worst of its beats’ preferences, because a metre needs all of its beats to be beats, and the best subdivision is the one that maximises that worst case.

The bar constraint is an upper bound only. A bar has to fit inside the present; it does not have to fill it. An earlier version of this arithmetic required a bar to be at least as long as the present’s lower edge and produced subdivisions of 400 milliseconds for a bar of five, which is three times slower than a paidushko is ever danced.

What the picture cannot show #

The present is a soft number and the knee is proportional to it. Everything about where the ceiling falls depends on a quantity published as a range from two seconds to eight, and the position of the knee moves from nine units to thirty-six across that range. The existence of the knee does not depend on it, and its position does.

The subdivisions are treated as equal. They are not: measurements of Bulgarian dance music find the long beat nearer 2.3 or 2.4 to one than three to two, which is the same kind of systematic deviation the swing ratio turned out to be. A performed metre whose beats are 437 and 620 rather than 437 and 655 scores slightly differently, and by an amount nothing here computes.

The preference is about tapping, not about dancing. The 550-millisecond preferred rate comes from studies of spontaneous tempo and of synchronised tapping. Whether the rate a dancer’s step prefers is the same rate a finger’s tap prefers is a question this collection does not have an answer to, and a dance metre is chosen by feet.

And a bar is not the only unit. The hierarchy above the bar is a real level with its own rate, and a tradition that groups its bars in twos has a four-second object whether the bar is a nine or not. Nothing here asks what the present does to that, and it should: if the constraint is on the largest unit a listener holds whole, the bar may not be the unit it binds.

Whose music, and when #

The metres are Balkan, Turkish and Greek above all, with relatives across the Middle East and in Indian tala. Paidushko is a five, rachenitsa a seven, daichovo a nine, kopanitsa an eleven, buchimish a fifteen; every one of them is a dance with a name, a repertoire and a region, and none of them was chosen by anybody computing anything.

The claim here is not that a tradition solved an optimisation. It is that a metre nobody can hold as one bar cannot survive as a dance metre, that the length at which that happens is computable from two published numbers, and that the repertoire’s longest single bars sit just under it.

The longer metres — the twenty-twos and twenty-fives — are genuinely played and genuinely long, and their existence is the strongest available test of all of this. They are taught as compounds, counted as compounds and danced as compounds. Whether that is because they have to be, or because that is how somebody once wrote them down, is a question about how they are heard rather than about how they are notated, and it is the sort of question an induction model can be asked and a page cannot.

Where this ladder goes next #

Three rungs. A bar of nine is four beats of unequal length, which no single division of a bar produces; the ratio between the beats is fixed at three to two by arithmetic and not by hearing; and the length of the bar is bounded by the one perceptual quantity the first two arguments named and never used, with the supply of available metres still growing where the bound begins to bite.

What is owed after this is the compound. Every metre above about sixteen units is analysed as a group of shorter ones, and this essay has taken that on the tradition’s word — which is exactly the kind of claim a metre-finding model was built to test and has never been pointed at. The question is whether a listener presented with a twenty-five of twos and threes recovers a single cycle of twenty-five or a group of three shorter bars, and at what tempo the answer changes. That is a question about an inference rather than about a repertoire: it needs the induction machinery this collection already has, run on constructed input rather than on a corpus, and it is the first debt this ladder has recorded that arithmetic alone can pay.