A dancer who comes in late needs the downbeat marked

Knowing every metre is slower than knowing none asked how soon a listener who knows the repertoire of additive metres can recognise which one is being played, and found that a small repertoire breaks the floor no cue carried by the notes could break for induction. A listener who knows thirty metres drawn at random recognised 64 per cent of them within the sixteen steps a psychological present holds.

Every one of those streams was heard from its written downbeat. That is the situation of a dancer who starts with the dance, and it is not the situation of anyone who joins one already going, walks into a room where the band is playing, or loses the count and has to find it again. That essay’s last section named the question: how many steps a listener entering at an unknown point of a bar needs to know both the metre and where its bar begins.

Part of the answer requires no search at all, and it is the part that decides the rest.

A metre entered part-way is its own rotations #

Hear the eleven counted 2+2+3+2+2 from the fifth step of its bar and the stream is two, three, two, two, two, two, three, and so on round the cycle. That is also what 2+3+2+2+2 sounds like from its third step, what 3+2+2+2+2 sounds like from its first, what 2+2+2+2+3 sounds like from its ninth and what 2+2+2+3+2 sounds like from its seventh. The onsets are identical for as long as anyone listens. A cycle of groups has only one sequence of gaps, and where the listener came in is the only thing distinguishing its rotations — the same thing the rotation the necklace cannot see found separating a timeline from the even pattern it is a rotation of.

The rotations are not a technicality. A metre is a cycle of groups and a place to start it, and the traditions treat the place as part of the metre: beats of unequal length set rachenitsa’s 2+2+3 against the same seven counted 3+2+2, which is the same cycle counted from a different point, with its long beat in a different place in the steps. The dictionary of 1,820 arrangements the earlier essays enumerated holds every rotation as a metre of its own for exactly that reason.

So the first result is a theorem rather than a measurement. On onsets alone, a listener who did not hear the downbeat can never tell a metre from its rotations. Accenting the long beats does not help, since the long beats rotate with the groups: 2+2+3 and 3+2+2 put their long beat on the same onset of the cycle, one bar line apart. The one cue that separates them is an accent on the bar’s first beat, because that is the one thing a rotation moves.

With the downbeat accented, the long bars stay out of reach #

With the downbeat accented, every metre is eventually found from every step it can be entered at. How long it takes depends on the bar’s length, and the dependence is steep. A nine is found in 10 to 17 steps from inside the bar, and sixteen of the twenty entries sampled fall inside the sixteen-step present. An eleven takes 12 to 21 and half its entries fall inside; a fifteen, 16 to 29, and four entries in thirty-two. From sixteen units up not one entry of any metre is recognised inside the present. A twenty-five takes between 22 and 44 steps.

The sixty-one entries that are recognised inside the present all belong to bars of fifteen units or fewer, and they thin out step by step as the bar lengthens: sixteen of the nines sampled, twelve of the tens, nine of the elevens, eight each of the twelves and thirteens, and four each of the fourteens and fifteens. The line in the figure where the orange bars stop reaching below the present is the same line, near sixteen units, that how long a limping bar can be found bounding a bar a listener can hold at all. A listener coming in part-way meets it sooner, because part of what the present has to hold is the wait for the bar to start.

Heard from its written downbeat with the same accent, a bar of up to twenty-two units is recognised in exactly one step more than its length — as soon as the next downbeat arrives where the metre says it will — and the longest bars in 23 to 25. So coming in part-way roughly doubles the worst case for every length, and the reason is visible in the next figure: a listener who has come in has to wait for a downbeat before the accent can say anything.

The wait for the next downbeat #

For the eleven the shape is exact. Heard from its downbeat it is recognised in twelve steps. Heard from any later step, it takes the steps remaining to the next downbeat plus those same twelve: twenty-two from the second step, thirteen from the last. Coming in part-way costs the wait for a downbeat and then the whole of recognition from it, which means that in a bar that short, what is heard before the first downbeat rules out nothing the downbeat would not have ruled out anyway.

That makes the entry point a matter of luck with a precise shape. A dancer who comes into the eleven exactly on its downbeat, or anywhere in its last four steps, finds the metre and its bar line inside the present; one who comes in during the six steps after a downbeat does not, and the step immediately after the downbeat is the worst place in the bar to arrive, at twenty-two steps. The late arrival who has just missed a downbeat pays for the whole bar.

The twenty-five, counted 7+7+11, is not so simple. Coming in part-way costs it between two steps less and eight steps more than the twenty-five it takes from its downbeat. From its third step it is found in twenty-three steps, sooner than from the downbeat itself; from its sixteenth step it takes thirty-three. In a long bar, part of the bar heard before the downbeat already does some of the work.

The dashed lines answer a smaller question, the one a listener without the accent can answer: which cycle of groups this is, and where in it they are, without knowing which of its rotations is the bar. From onsets alone that is always answered in the end, and it takes longer than the whole question takes with the accent: 27 to 34 steps for the eleven and 25 to 40 for the twenty-five, never inside the present.

A repertoire with rotations in it #

A dancer does not know every metre, and the essay on recognition found the size of the repertoire deciding almost everything. From a random entry it decides less.

On onsets alone the collapse is immediate. With five metres known, a listener coming in at a random step recognises the metre inside the present about half the time; with thirty, one time in a hundred; and a growing share is never recognised at all — 17 per cent with thirty known and 34 per cent with a hundred — because the repertoire has come to contain a rotation of the metre being played.

Removing the rotations removes the “never” and leaves the slowness. A repertoire in which no two metres are rotations of one cycle is always recognised in the end, however it is entered, and that is checked as well as argued. But it is recognised no faster: 44 per cent inside the present with five known and 1 per cent with thirty, much as when rotations were allowed. What the listener without a downbeat lacks is not only a way to break a tie between rotations; it is the bar line every rival’s prediction is counted from.

The cost of coming in late is easiest to see against the earlier numbers. From the written downbeat, a listener who knew thirty metres recognised 64 per cent of them inside the present on onsets alone. Coming in at a random step, the same listener with the same repertoire recognises 1 per cent — sixty-three points lost to not having heard where the bar began.

The downbeat accent restores most of what the entry cost. With it, five known metres are recognised inside the present 88 per cent of the time from a random step, thirty known 54 per cent, a hundred 14. Nothing is ever left unrecognised.

What each cue buys, coming in late #

Set side by side at thirty metres known, the three cues trade places. Heard from the written downbeat, onsets alone recognise 64 per cent inside the present, the long-beat accent 79 and the downbeat accent 67, so from the downbeat an accent on the long beats is the more useful of the two. Coming in part-way reverses that. Onsets alone fall to 1 per cent, the long-beat accent to 6, with 17 per cent of metres never found under either, and the downbeat accent holds 54.

Which accent matters depends on whether the listener heard the start. The accent buys two units, however loud it is found that for induction from the downbeat an accent on the long beats buys two steps and an accent on the downbeat takes every metre to its floor. For recognition part-way through, the long-beat accent is nearly worthless and the downbeat accent is the whole of what makes a long bar findable at all.

A repertoire of named metres #

The random repertoires stand in for a tradition, and a tradition is not random. The Balkan metres the essays on this subject have named — among them the twenty-five of Sedi Donka that a twenty-five is a nine until its last unit was about — include exactly the kind of pair the arithmetic says will cause trouble.

Heard from the written downbeat, a listener who knows exactly these five recognises each on its onsets in between 9 and 16 steps. Coming in part-way, the two nines — daichovo’s 2+2+2+3 and the nine counted 2+2+3+2, which is the same cycle started a group later — are never told apart from any step on onsets alone, because each is a rotation of the other. Kopanitsa, the fifteen and Sedi Donka, which have no rotation in the repertoire, are found from every step, in at most 23 to 26 steps, which is never quite inside the present.

With the downbeat accented, the small named repertoire is almost entirely recognisable from anywhere. Every metre is found from every step within 17 steps, and 67 of the 69 entries fall inside the present; the two that do not are one step of kopanitsa and one of the fifteen, at 17. A handful of metres with the downbeat marked is recognised inside the present almost however it is entered, which a listener who knows thirty random metres manages only half the time.

The three named metres with no rotation in the repertoire show what the accent adds even when nothing is ambiguous in principle. On onsets alone kopanitsa, the fifteen and Sedi Donka are found from every step, but their slowest entries take 26, 23 and 26 steps, all beyond the present. With the downbeat accented the slowest take 17, 17 and 15. A small repertoire without rotations can be recognised part-way from the onsets alone; it cannot be recognised quickly enough to dance to without something marking where the bar starts.

Where the downbeat comes from #

A dance marks its bar line in more than the sound. Beats of unequal length described the metre as a description of a body, with the long beat carrying a weight transfer in rachenitsa and a hop in daichovo, and a dance’s phrase of steps starts somewhere. A dancer joining a line can watch the feet of the dancer beside them, and the feet carry precisely the information this arithmetic says the onsets lack.

It also puts a different light on how the metres are played. A drum pattern that strikes the bar’s first beat with a different stroke, a melody that places its phrase starts on the downbeat, a bass that marks the bar — each of those is the downbeat accent here. The right period at the wrong phase found the same split in finding a beat, where how far apart the beats are and where the first one falls are two problems, and the second is the harder. For an aksak metre heard part-way it is worse than harder: without a cue that marks it, it cannot be solved.

The arithmetic #

The dictionary is the one every essay on this subject has enumerated: every cyclic order of twos and threes from nine units to twenty-five that is not a repetition of a shorter metre, 1,820 in all. Each is tiled from its downbeat, marking onsets, long groups and bar starts. A listener comes in at step e of the metre being played; a rival is any arrangement entered at any step, and it is contradicted at the first step after coming in at which the two streams differ in an onset or, under a cue, in which onsets are marked. The metre is recognised one step after the last rival is contradicted, and a rival never contradicted within 110 steps is one that never will be, since every tiling repeats well inside that. The cycle-only question removes from the rivals every rotation of the metre that matches it onset for onset.

Coming in on the downbeat is checked against the recognition window the earlier essay measured for the whole dictionary, and it agrees. Repertoires are seeded random draws from the dictionary that always hold the metre played, with the entry step drawn at random; the rotation-free draws refuse a second metre from any cycle already drawn. The present is sixteen steps, three and a half seconds at the tempo at which an unequal metre’s beats are jointly best.

What the arithmetic assumes #

That recognition waits for certainty. A listener who commits when the rivals are improbable would find the metre sooner, and a listener who has danced a dance a thousand times probably commits on much less. The rotations do not go away under that change: a listener with no downbeat who commits to a rotation commits to the wrong dance.

That the entry is uniform. A dancer joining a line is more likely to wait for a phrase to end than to step in on an arbitrary beat, and a metre has to be able to change its mind is a reminder that a listener who guessed wrong can revise. Both shorten the times here; neither restores what the onsets cannot say.

And that the stream is exact. Every onset is on its grid, every accent is present on every bar, and nothing is ornamented. A real performance leaves out accents, which is the case where a listener coming in part-way has least to work with.

What rotations cannot establish #

That dancers confuse rotations. The theorem says the onsets cannot distinguish them; it does not say anyone is fooled, because nobody hears only onsets. It says where the information that prevents the confusion has to be — in something that marks the bar’s first beat — and that a repertoire containing rotations makes that marking necessary rather than decorative.

That traditions avoid rotations. Rachenitsa’s 2+2+3 and the seven counted 3+2+2 are both counted, and so are the two nines. A tradition that keeps rotation pairs is one that must mark its downbeats, which is a prediction about practice this arithmetic cannot check.

Still open: a listener who has only a pulse #

Every listener here hears the metre’s own onsets. A dancer in a crowded room, or a listener whose attention is on the melody, may have only the regular subdivision and the strongest accents, and a cycle that says where it is found that a regular layer can carry position when the pattern alone cannot. The computation that follows gives the listener the stream thinned to its downbeats and its long beats, with every other onset removed, and asks how many steps that sparse stream needs to identify metre and bar line from a random entry against a repertoire of stated size. It would say whether the accents alone, without the onsets between them, are already enough to find a long bar, and so whether what a late dancer needs is the rhythm or only its landmarks.