Three metres as trees
Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.
It makes a noise. 52 of its 52 placements carry sound, built from the same numbers as the drawing, offering these buttons: 11/8 kopanitsa, 2+2+2+3+2+2+2+3+2+2+3, downbeat accented, 2+2+2+3+2+2+2+3, equal parts: never told from its cuts, 2+2+3+2+2 from its last step, 2+2+3+2+2 from step 4, downbeat accented, 2+2+3+2+2 from step 4, onsets alone, 2+2+3+2+2 from the step after its downbeat, 2+2+3+2+2+2+2+3+2+2+3, its longest-lasting rival and 73 more.
Called by 11 essays
the blast radius of changing it
Rhythm is a circle, and the bar line is a choice
Draw a rhythm as a line and it looks like a sequence of decisions. Draw it as a cycle and the same pattern turns out to be shared across continents, differing only in where somebody decided to start counting.
Beats of unequal length
A bar of nine in Balkan practice is not nine of anything. It is four beats, three short and one 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.
The beat has a preferred rate, and it is not the notation's
Metre is a hierarchy of levels and only one of them is tapped. Which one is decided by the listener's own clock rather than by the time signature: the window in which a series of events can be a beat at all runs from about a tenth of a second to two seconds, with a marked preference near half a second — so at the extremes of tempo the notated beat and the felt one part company, predictably.
No term for an unequal beat
A Balkan bar of nine is four beats, three short and one long. The induction rules lay their strong positions at a fixed period, so the only readings of nine it can offer are nine, three and one — and for a bar of seven, where seven is prime, the only readings are seven and one. The right answer is not among the candidates.
What the onsets left out
Eight essays induce a metre from a list of ones and zeros, and every failure they recorded was argued about as a failure of the rules. Two of the three are not. Note length is already in that list and the scoring throws it away: put it back and the son clave's two-way tie resolves to the notated downbeat. But the groove with its beat removed cannot be repaired by any cue at any strength, and the reason is arithmetic rather than empirical — the true phase has no onset on any of its strong positions, so there is no credit for a cue to multiply and its line is exactly flat.
How long a limping bar can be
The bound on an unequal beat turned out to be arithmetic, and the tempo window was left with only the tempo to decide. It decides nothing: every metre built from twos and threes gets the same answer, because the window is asked a yes-or-no question. Graded instead, an unequal beat costs 0.135 of the window's own preference at every bar length — and the constraint that does depend on length is the one nobody applied, that the whole bar has to fit inside the psychological present. At the subdivision that suits both beats best, a bar of sixteen units just fits and a bar of seventeen does not, which is where the supply of distinct metres has only started to grow.
A twenty-five is a nine until its last unit
Every account of long additive metres says they are heard as groups of shorter ones, and the metre-induction model had never been pointed at the claim. Pointed at it, the model does not prefer the group — it prefers the long bar outright, and would go on preferring it more the longer anybody listened. What it cannot do is start: the evidence that separates a bar of twenty-five from a bar of nine does not exist until the whole bar has been heard, and at the tempo an unequal metre is best played at the psychological present holds sixteen units.
The accent buys two units, however loud it is
A twenty-five cannot be told from a group of shorter bars on its onsets until more steps have gone by than a listener's present holds, and the obvious objection is that nobody plays an aksak bar as bare onsets: the long beat is louder, and the bar's first beat is marked. So how loud does an accent have to be? The question has a surprising answer. Loudness is not the variable. The existing accent cue changes nothing, and delays the answer where it changes anything. An accent that a reading has to predict works at any strength at all, and at no strength does more than a fixed amount: on the long beat it buys the two steps of a short beat, and on the downbeat it takes every arrangement to one floor — the bar less its last beat — which no cue carried by the notes can break. The longest bar that can be heard as one moves from sixteen units to eighteen.
Knowing every metre is slower than knowing none
A long aksak bar cannot be told from its shorter cuts by induction before one step into its last beat, and no accent carried by the notes moves that floor. The obvious escape is a listener who knows the repertoire and recognises the metre instead. Recognition among all 1,820 arrangements of twos and threes never beats the floor, is never quicker than induction, and is slower for half the metres: a nine induced in 9 steps is recognised in 27. What breaks the floor is a small repertoire that leaves out the metre's own longest cut — with the cut known, no repertoire of any size does.
A dancer who comes in late needs the downbeat marked
Every window for recognising an aksak metre so far started at its written downbeat. A dancer joining a dance already going has not heard the downbeat, and the arithmetic of that is blunt: a metre entered part-way is, onset for onset, each of its own rotations heard from their downbeats, and the rotations are metres too — 2+2+3 and 3+2+2 are counted differently. So on onsets, and with the long beats accented, no metre is ever told from its rotations. Only an accented downbeat tells them apart, and with it a listener who knows thirty metres recognises 54 per cent of them inside the present from a random entry, against 1 per cent without.
A late dancer needs the landmarks, not the rhythm
A listener who joins an additive-metre dance part-way recognises it far more reliably with the downbeat accented. Strip the stream down to its landmarks — the onsets that begin a bar or a long beat, with every other onset removed — and the listener does as well or better: knowing a hundred metres, 23 per cent are recognised within the present against 14 with every onset. Unmarked, the landmarks still beat every onset with the long beats accented. Neither half does it alone; what identifies a metre from a late entry is where its long beats sit relative to its bar.