Additive metre — the series
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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.
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How unequal a beat is allowed to be
A Balkan bar of nine is four beats, three short and one long, and the long one is always three subdivisions against two. Never four against two, never five against three. The usual explanation is that this is what the tradition does; there is a better one and it needs no perceptual measurement at all. A group of four subdivisions decomposes into two twos and a group of five into a two and a three, so neither is a beat — it is two. The only indivisible lengths are two and three, which fixes the ratio at three to two before anybody listens, and leaves the tempo window to decide only how fast the bar can go.
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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.
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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.
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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.
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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.
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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.
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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.
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A crowd costs a late dancer the silences, not the steps
A dancer who hears only the landmarks of an aksak rhythm — the heads of bars and long beats — recognises it from a late entry better than one who hears every onset, as long as everything played is heard. Let a share of what is played be lost in a crowd and the lead goes, but not for the reason expected. Most of the cost arrives before a single landmark is missed: a listener who allows that one might be can no longer treat a silence as proof, and the landmark stream is built almost entirely of silences. At one lost onset in ten, the caution alone takes 25 points from the landmarks and 5 from the full stream.
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A crowd that hides bars is kinder than one that hides onsets
A dancer joining an additive-metre dance in a noisy room loses some of what is played, and the essay before this one lost it one onset at a time. A crowd does not. It hides stretches — a burst of laughter covers a bar. The prediction was that a stretch as long as a bar would hurt the stream of landmarks most, since it could remove every landmark in a cycle, and that long bars would survive best. Holding the share of time lost fixed, longer stretches help every stream, help the full stream more than the landmarks, and leave long bars as hard to place as ever.
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A dancer who hears the crowd can trust the quiet
A late dancer in a noisy room who cannot tell noise from silence has to treat every silence as weak evidence, and that caution was the largest cost in every figure before this one. A dancer who hears the laugh that covers a bar knows which silences to discount and can trust the rest. Losing a quarter of the music, that dancer recognises more metres than one who loses nothing but must allow for losses — and the landmarks, which a noisy room had made the worse stream to follow, lead again once the noise comes in bursts. The quiet has to be trusted to one onset in nineteen.
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In a loud room the accent has to be loud
A late dancer who hears the crowd as noise ignores everything played under it. Real noise is not that thorough. It hides the soft strokes and lets the loud ones through, and an accented downbeat is a loud one. Let the noise pass loud strokes and some soft ones, and marking the downbeat is worth more to the dancer than in a quiet room: with half the time under noise, 29 points of recognition against 23. Noise that swallows the accent halves its worth. In a quiet room no amount of loudness bought an accent anything extra; in a loud one, loudness is the whole of what the accent is for.
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In a crowd the downbeat belongs to the drum
In a loud room a marked downbeat is worth more to a late dancer than in a quiet one, so long as the mark gets through. Whether it does depends on where its sound lies, because talk is not loud everywhere: it is loudest from 250 to 400 hertz, exactly where a melody's notes are, and fifteen decibels quieter at the bottom of a drum. A louder melody note is heard half the time through talk ten decibels over the band; a low drum survives a crowd twelve decibels louder than that. With the talk fifteen decibels over the band, a downbeat marked in the melody is worth two-fifths less to the dancer than one marked by the drum.
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A felt drum reaches only where the heard one already does
A low drum near a stage is felt in the body as well as heard, and talk cannot mask a vibration. Given that second channel, a drum-marked downbeat always arrives — and is then worth 19.8 points to a late dancer, exactly what an always-heard drum was worth, because the crowd has already taken the soft strokes that made a mark worth more. The felt channel pays only under crowds twenty decibels over the band. And it is driven by the drum's own level, which in a reverberant room is high enough to be felt only where the drum is already loud enough to be heard.
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The felt channel carries the inner beats, not the downbeat
A low drum felt in the body gets the downbeat through any crowd, and is then worth less than it was in a quiet room, because the crowd has taken the soft strokes between. Put a kick drum on every landmark of the dance and let all of it be felt, and in a crowd thirty decibels over the band a late dancer recognises the dance in seventy per cent of entries against thirty-seven for the felt downbeat — with no felt accent at all. The accent, felt or not, is worth about ten points on top. The worry that a stream felt alike has no downbeat turns out to be the small half of the arithmetic.