A late dancer needs the landmarks, not the rhythm

A dancer who comes in late needs the downbeat marked asked how many steps a listener who joins an additive-metre dance at an unknown point needs before every other metre they know has been ruled out. On onsets alone, part of the answer is never: a metre entered part-way is, onset for onset, one of its own rotations heard from that rotation’s downbeat, and a listener who knows both — 2+2+3 and 3+2+2 are different dances — cannot tell them apart. With the downbeat accented the ambiguity disappears, and a listener knowing thirty metres recognises 54 per cent of them within the sixteen steps a psychological present holds.

Every listener in that essay heard every onset. The essay’s closing section pointed at a listener who does not: a dancer in a crowded room, or a listener attending to the melody, who has only the strongest parts of the rhythm — the starts of bars and of long beats — and asked whether those landmarks alone, without the onsets between them, are enough to find a long bar.

They are more than enough, and the reason is that the onsets between them were never helping.

A thinner stream #

The metres are the additive metres from nine to twenty-five units built from groups of two and three — every arrangement, every rotation of every arrangement treated as a metre of its own. A listener joins at a random step and hears the stream from there; a metre is recognised at the step after the last rival, entered at any of its own steps, has been contradicted.

The landmark stream keeps only two kinds of onset: the first onset of each bar, and the first onset of each long beat. Everything else is removed. In a bar of 2+2+3+2+2, which is eleven units and the rhythm of a kopanitsa, the full stream has five onsets and the landmark stream two — the downbeat and the head of the three-unit beat. A listener given the landmarks with nothing marked hears two clicks a bar, unevenly spaced; with the downbeat marked, one of the two is accented.

Fewer onsets, more recognised #

On every onset with nothing marked, a listener knowing five metres recognises 49 per cent within the present, and knowing a hundred, none — and leaves 34 per cent never recognised, because the repertoire contains rotations the listener has no way to separate. Accenting the long beats — the cue the accent buys two units however loud it is priced from the downbeat — helps with a small repertoire, 68 per cent at five, and does nothing for a large one: none within the present at a hundred, and still 34 per cent never.

The landmark stream, with nothing marked at all, recognises 69 per cent at five, 61 at ten — against 39 for every onset with long beats accented — 26 at thirty and 9 at a hundred, and it leaves at most 9 per cent never recognised where every onset left 34. A listener told less does better.

With the downbeat marked the same comparison holds at the top of the figure. Every onset with the downbeat accented recognises 88, 74, 54 and 14 per cent as the repertoire grows from five to a hundred. The landmarks with the downbeat marked recognise 91, 82, 56 and 23. Removing every onset that is not a landmark does not cost a listener anything, at any repertoire size tried, and against a hundred metres it improves recognition by nine points.

Why removing onsets helps #

The onsets between landmarks are the ones that make metres look alike. A metre’s short beats all have the same length — how unequal a beat is allowed to be found them never more than two against three — so the onsets that begin them fall into the same regular pattern in every metre that has a run of short beats — and nearly every additive metre does. Those onsets are shared by many rivals and contradict few of them, so a listener hearing them spends steps confirming things that do not discriminate.

The landmarks are different in kind. A long beat’s head is where the unevenness of an additive metre is, and a downbeat is where its bar is, and the gap between the two is specific to the metre. In a stream that contains only those, every onset carries the one piece of information that separates a metre from its rotations — where the long beats sit relative to the bar — and the absence of an onset carries information too, since it says a short beat is passing.

Neither half alone #

That explanation predicts that each kind of landmark on its own should be much weaker than the two together, and it is.

Downbeats alone say how long a bar is and nothing else: a listener knowing five metres recognises 1 per cent within the present and never recognises 48 per cent, and knowing a hundred, 96 per cent are never recognised, because a click every eleven units belongs to every eleven-unit metre there is. Long beats alone leave exactly the metres unrecognised that every onset with the long beats accented leaves — 3, 8, 17 and 34 per cent — and recognise fewer within the present, 57, 29, 3 and 0 per cent against 68, 39, 6 and 0. Both together score 69, 61, 26 and 9 within, and 0, 3, 3 and 9 never.

There is a simple way to see why the halves fail on their own terms. A stream of downbeats is a metronome ticking once a bar: every metre of the same length produces it, from every entry, forever. A stream of long-beat heads is irregular, but a rotation of the same cycle produces the same irregular pattern shifted, so a listener who has not heard the bar’s start cannot place it. Put the two together and the pattern of gaps between landmarks is anchored — one of the landmarks is where the bar begins — and a shift of the pattern is no longer the same stream.

The combination is not additive: two streams that each recognise almost nothing at a hundred metres recognise 9 per cent together and leave 91 per cent recognisable. What identifies a metre from a late entry is a relation, not a feature — the position of the long beats within the bar — and a relation is only present in a stream that contains both of its terms.

Against every metre there is #

A repertoire of a hundred is still a small part of the dictionary. The last comparison sets each metre against every arrangement from nine to twenty-five units, entered at every step.

Here the landmarks with the downbeat marked do not beat every onset; they tie. The median windows are identical from nine units to sixteen — 13, 15, 17, 19, 18, 20, 22 and 24 steps — and from seventeen units up the landmarks are within one step either way, and a step or two faster on the longest bars: 26 steps against 28 at twenty-three units, 26 against 27 at twenty-five. Against the whole dictionary, the onsets between landmarks carry nothing a marked downbeat does not already supply.

The unmarked landmarks do not survive the whole dictionary. Their medians run a few steps longer, and at most bar lengths some entries are never resolved — half of them at ten, thirteen, fourteen and seventeen units, three quarters at twenty-four. With every metre in play, two metres can share a landmark pattern exactly, and nothing unmarked separates them. That is the same wall the earlier essay found for onsets alone, arriving later, and it is sharpest on the long bars that a twenty-five is a nine until its last unit and how long a limping bar can be found hardest to tell from their cuts.

What a late dancer is waiting for #

The medians hide a simple structure, which the windows for single metres show entry by entry.

For the kopanitsa the window has a sawtooth shape. A listener who comes in on the downbeat needs 12 steps. One who comes in a step later needs 22, and each step later than that needs one step fewer — 21, 20, 19 and so on down to 13 from the last step of the bar. The shape says what the listener is doing: waiting. From any entry but the downbeat, nothing is settled until the next marked downbeat arrives, and after it arrives the recognition takes the same twelve steps it takes from the downbeat itself. Coming in one step late is the worst entry there is, because it is the furthest from the next downbeat.

The two marked listeners have exactly the same sawtooth, step for step. For this metre the landmarks lose nothing and gain nothing: the downbeat is the whole of what a late listener needs, and the onsets between, whether present or removed, do not change when it comes. The unmarked landmarks follow the same sawtooth four steps higher — 16 from the downbeat, 26 from the step after — because without a mark the listener has to hear one long beat’s position relative to the downbeat landmark before either is known to be which.

The twenty-three-unit bar is where the two marked listeners part company, and they part in both directions. Entering at the second step, every onset with the downbeat accented needs 25 steps and the landmarks with the downbeat marked need 34. Entering at the twelfth step, the landmarks need 24 and every onset 26. In a long bar the onsets between landmarks sometimes settle a rival before the next downbeat and sometimes do not, and which happens depends on which long beat the listener heard first. On average, as the comparison over the whole dictionary showed, the two come out level.

The sawtooth also puts a number on how long joining takes. Sixteen steps is the three and a half seconds of the present, so a step at this tempo lasts about 0.22 seconds. A kopanitsa entered on its downbeat is recognised in 12 steps, about 2.6 seconds, comfortably inside the present. Entered one step late it takes 22 steps, about 4.8 seconds — longer than the present holds, with the downbeat accented and every onset heard. Averaged over the eleven possible entries, the window is 17 steps, a little over the present. The twenty-three-unit bar, entered at its worst step, takes 35 steps, about 7.7 seconds, which is two presents and a bar and a half of its own.

That is the practical content of the recognition window for a dancer. Joining a short aksak dance with its downbeat marked is a matter of waiting for one downbeat and then one bar; joining a long one is a matter of waiting through several bars, and a dancer who wants to join sooner than that needs something the stream does not give, such as watching the feet of the line.

So a late dancer’s real problem is the downbeat. The landmarks are enough because the downbeat is one of them, and they are better than every onset against a limited repertoire because the other onsets make rivals look alike while the listener waits.

Which computation produced the numbers #

Metres are every arrangement of twos and threes summing to nine through twenty-five units that is a genuine additive metre, with every rotation counted as a metre of its own. A stream is generated for a hundred and ten steps from an entry; two metres entered at two steps are contradicted at the first step where their streams disagree — an onset against a silence, or, when a mark is given, a marked onset against an unmarked one. For the landmark streams an onset exists only where a bar or a long beat begins. A metre’s window is one step past the latest contradiction of any rival entered at any step, and never if some rival is never contradicted within a hundred and ten steps.

Repertoires of 5, 10, 30 and 100 metres are drawn at random from the dictionary, always including the metre played, with a random entry, over every twenty-third arrangement twice; the present is sixteen steps, three and a half seconds at the essays’ standard aksak tempo. The comparison against the whole dictionary samples up to four arrangements of each bar length and every third entry step.

What the arithmetic assumes about a listener #

Perfect perception of what is given. A landmark is heard exactly when it sounds, and a missing onset is heard as missing. A listener in a noisy room who hears only the landmarks also misses some of them, and a missed landmark is far more costly than a missed short-beat onset, since every landmark carries information.

That the landmarks are what survives. The premise is that the loudest parts of a rhythm are its downbeats and long beats. In much Balkan dance music the long beat is marked by a step and often by the drum, and the downbeat by harmony or melody; but which onsets survive a noisy room is a question about that music’s instrumentation, and beats of unequal length found the long beat defined by duration, not by accent.

One tempo. Every step is converted to seconds at the aksak tempo the metre essays have used throughout, at which a sixteen-step present is three and a half seconds. At a faster dance the same windows are shorter in seconds and a larger share of them fits inside the present; at a slower one the reverse. The comparison between listeners is in steps and does not depend on the tempo; the statement that joining a kopanitsa one step late takes longer than the present does.

No learning within the dance. Each window is computed as if the listener starts cold at the entry step, which is the right model for joining a dance and the wrong one for a listener who has been dancing and loses count.

What recognition counts cannot say #

Whether a dancer recognises a metre or only its feel. A dancer who falls into step without naming the metre has not done what the arithmetic counts, and may need far less: only where the long beat is, and not which of the thousands of possible metres is sounding.

Whether listeners attend to landmarks. The finding says a stream of landmarks is sufficient and a stream with onsets between them is no better. It does not say listeners ignore the onsets between, and a cycle that says where it is found that in cyclic music the regular layers carry position that the pattern alone does not, and knowing every metre is slower than knowing none that a listener’s repertoire changes what counts as evidence.

Whose dances #

The dances these metres come from — the kopanitsa’s 2+2+3+2+2 and the long aksak bars of the southern Balkans — are danced in lines where joining part-way is normal. Their accompaniment is often described as marking the long beat with the dancers’ own step and a bass drum, while the short beats are filled in by higher instruments, which are the first thing a crowded room drowns. On this arithmetic that arrangement is almost exactly the landmark stream with its downbeat marked, and it is the stream from which a late dancer recognises a metre soonest.

Still open: the landmark that goes missing #

The computation assumed every landmark is heard. A listener in a crowd does not hear every landmark; a step is sometimes lost under talk and a drum stroke under a trumpet. Because every landmark carries information and every onset between them carries almost none, the cost of missing a landmark should be far higher than the cost of missing an ordinary onset — and it is the landmark stream, not the full stream, that a listener relying on landmarks would be exposed to.

The recognition computation extends directly: each landmark is dropped with a stated probability, and a rival is contradicted only by a landmark that was heard. The question is at what drop rate the landmark stream stops being as good as the full stream with its downbeat marked, and whether that rate is anywhere near the share of steps a crowded dance floor actually hides.