Acoustics & Music Theory
why it sounds like that
Concept

Additive metre — where it appears

A bar built by adding beats of unequal length rather than by dividing a whole into equal ones. A bar of seven counted as three plus two plus two is additive; the same seven counted as a subdivided whole is not, and the two are heard differently.

Named by 16 essays across one field — each of them below, with the objects they name alongside it.

Also named here as aksak — the same set of essays touches all of them, so they are one junction rather than several.

Three metres as trees. Three metres — 4/4, 2 + 2 + 2 + 2; 6/8, 3 + 3; 5/8, 3 + 2 — each drawn as the bar dividing into beats and the beats dividing again. Simple and compound metres have the same number of beats and differ in the second division; an additive metre has beats of unequal length, which no single division produces.

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.

rhythm · Additive metre Two metres as trees. Two metres — 9/8 as 3 × 3, 3 + 3 + 3; 9/8 as 2+2+2+3, 2 + 2 + 2 + 3 — each drawn as the bar dividing into beats and the beats dividing again. Simple and compound metres have the same number of beats and differ in the second division; an additive metre has beats of unequal length, which no single division produces.

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.

rhythm · Metre induction How unequal a beat is allowed to be. Five bars — 2+3, 2+2+3, 2+2+2+3, 2+5, 3+4 — with the range of subdivision durations at which every one of the bar's beats stays inside the window where a series of events can be a beat — a tenth of a second to two seconds. The metres made of twos and threes have the widest bands, and they have them for a reason that is not about the window: 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 at all. The only indivisible lengths are two and three, which fixes the ratio between an aksak metre's long and short beats at 3:2 by arithmetic.

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.

rhythm · Additive metre How long a bar of unequal beats can be, at a present of 3.5 seconds. Two quantities against the number of subdivisions in the bar. The bars are how many genuinely distinct unequal metres that length admits — groupings of twos and threes, up to rotation, discarding any that repeats a shorter grouping — and they run from one at 5 to 28 at 23. The line is how good a beat the best of those metres can manage once the whole bar is required to fit inside a psychological present of 3.5 seconds. It is flat at 0.865 up to a bar of 16 units, which is where the bar at the best subdivision first overruns the present, and falls after it: 17 at 0.830, 18 at 0.797, 19 at 0.765, 20 at 0.735. The supply of metres is still growing where the quality has begun to fall, so the lengths a tradition can use are a bounded prefix of an unbounded list.

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.

rhythm · Additive metre Where a bar of 25 and a bar of 9 first disagree. The additive metre 2+2+2+3+2+2+2+3+2+2+3 — 25 units, an onset at the head of every group — with the accents it predicts drawn above the accents predicted by reading it as a repeating bar of 9, which is the cut of it that agrees longest. The two rows are identical for 24 consecutive steps and differ for the first time at step 25, where the shorter reading expects an accent and the metre does not supply one. Nothing before that step distinguishes the two hypotheses, so a listener who has not heard 25 consecutive steps has no evidence either way — whatever they are disposed to hear.

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.

rhythm · Additive metre An accent moves the longest separable bar from 16 units to 18, and no further. For every bar length from 9 to 25 units, over all 1820 arrangements of twos and threes that are not a repeat of a shorter bar, the fewest and the most consecutive steps before the whole bar beats every shorter cut of it. 9: onsets alone 9 to 14, long beat predicted 7 to 12, downbeat predicted 7 to 8; 10: onsets alone 10 to 13, long beat predicted 8 to 11, downbeat predicted 8 to 9; 11: onsets alone 11 to 18, long beat predicted 9 to 16, downbeat predicted 9 to 10; 12: onsets alone 12 to 17, long beat predicted 10 to 15, downbeat predicted 10 to 11; 13: onsets alone 13 to 22, long beat predicted 11 to 20, downbeat predicted 11 to 12; 14: onsets alone 14 to 23, long beat predicted 12 to 21, downbeat predicted 12 to 13; 15: onsets alone 15 to 26, long beat predicted 13 to 24, downbeat predicted 13 to 14; 16: onsets alone 16 to 25, long beat predicted 14 to 23, downbeat predicted 14 to 15; 17: onsets alone 17 to 30, long beat predicted 15 to 28, downbeat predicted 15 to 16; 18: onsets alone 18 to 29, long beat predicted 16 to 27, downbeat predicted 16 to 17; 19: onsets alone 19 to 34, long beat predicted 17 to 32, downbeat predicted 17 to 18; 20: onsets alone 20 to 35, long beat predicted 18 to 33, downbeat predicted 18 to 19; 21: onsets alone 21 to 38, long beat predicted 19 to 36, downbeat predicted 19 to 20; 22: onsets alone 22 to 37, long beat predicted 20 to 35, downbeat predicted 20 to 21; 23: onsets alone 23 to 42, long beat predicted 21 to 40, downbeat predicted 21 to 22; 24: onsets alone 24 to 41, long beat predicted 22 to 39, downbeat predicted 22 to 23; 25: onsets alone 25 to 46, long beat predicted 23 to 44, downbeat predicted 23 to 24. A present of 3.5 seconds holds 16.0 steps at 218 milliseconds a step, so the longest bar some arrangement of which separates inside it is 16 units on onsets alone, 18 with the long beat predicted and 18 with the downbeat predicted.

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.

rhythm · Additive metre A listener who knows every metre recognises none of them inside the present. For every bar length from nine units to twenty-five, the fewest and the most steps from the downbeat before every other one of the 1820 arrangements of twos and threes has been contradicted by the stream, on onsets alone, with the long beats accented and with the downbeat accented, against the 16 steps a present of 3.5 seconds holds. onsets alone: recognised within the present for 0 of 1820; long-beat accent: recognised within the present for 0 of 1820; downbeat accent: recognised within the present for 85 of 1820. The dashed line is the present.

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.

rhythm · Additive metre Come in part-way with the downbeat accented, and no bar of sixteen units or more is recognised inside the present. For every bar length from nine units to twenty-five, the fewest and the most steps a listener who knows every arrangement of twos and threes needs to recognise the metre and where its bar begins, with the downbeat accented: coming in at a sample of steps inside the bar, against hearing it from its written downbeat. On onsets alone, or with the long beats accented, a metre entered part-way is never told from its rotations. 9: from inside the bar 10 to 17, 16 of 20 inside the present; from the downbeat 10 to 10; 10: from inside the bar 11 to 19, 12 of 20 inside the present; from the downbeat 11 to 11; 11: from inside the bar 12 to 21, 9 of 18 inside the present; from the downbeat 12 to 12; 12: from inside the bar 13 to 23, 8 of 24 inside the present; from the downbeat 13 to 13; 13: from inside the bar 14 to 25, 8 of 28 inside the present; from the downbeat 14 to 14; 14: from inside the bar 15 to 27, 4 of 28 inside the present; from the downbeat 15 to 15; 15: from inside the bar 16 to 29, 4 of 32 inside the present; from the downbeat 16 to 16; 16: from inside the bar 17 to 31, 0 of 32 inside the present; from the downbeat 17 to 17; 17: from inside the bar 18 to 32, 0 of 36 inside the present; from the downbeat 18 to 18; 18: from inside the bar 19 to 32, 0 of 36 inside the present; from the downbeat 19 to 19; 19: from inside the bar 20 to 37, 0 of 40 inside the present; from the downbeat 20 to 20; 20: from inside the bar 21 to 35, 0 of 40 inside the present; from the downbeat 21 to 21; 21: from inside the bar 22 to 40, 0 of 44 inside the present; from the downbeat 22 to 22; 22: from inside the bar 23 to 39, 0 of 44 inside the present; from the downbeat 23 to 23; 23: from inside the bar 24 to 44, 0 of 48 inside the present; from the downbeat 23 to 24; 24: from inside the bar 23 to 39, 0 of 48 inside the present; from the downbeat 23 to 25; 25: from inside the bar 22 to 44, 0 of 52 inside the present; from the downbeat 23 to 25. In all, 61 of 590 entries are recognised within the 16 steps of a 3.5-second present.

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.

rhythm · Additive metre A listener who hears only the landmarks recognises the metre sooner. The share of metres recognised within the 16 steps of a 3.5-second present after coming in at a random step, against how many metres the listener knows, for five listeners: every onset with nothing marked, every onset with the long beats accented, only the downbeats and long beats with nothing marked, every onset with the downbeat accented, and only the downbeats and long beats with the downbeat marked. Every onset, nothing marked: 5 known, 49% within the present, 3% never; 10 known, 19% within the present, 8% never; 30 known, 1% within the present, 17% never; 100 known, 0% within the present, 34% never. Every onset, long beats accented: 5 known, 68% within the present, 3% never; 10 known, 39% within the present, 8% never; 30 known, 6% within the present, 17% never; 100 known, 0% within the present, 34% never. Landmarks only, nothing marked: 5 known, 69% within the present, 0% never; 10 known, 61% within the present, 3% never; 30 known, 26% within the present, 3% never; 100 known, 9% within the present, 9% never. Every onset, downbeat accented: 5 known, 88% within the present, 0% never; 10 known, 74% within the present, 0% never; 30 known, 54% within the present, 0% never; 100 known, 14% within the present, 0% never. Landmarks only, downbeat marked: 5 known, 91% within the present, 0% never; 10 known, 82% within the present, 0% never; 30 known, 56% within the present, 0% never; 100 known, 23% within the present, 0% never.

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.

rhythm · Additive metre The landmark stream loses its lead when the listener allows for a lost landmark, before any is lost. The share of metres recognised within the 16-step present from a random entry, 10 metres known, against the rate at which played onsets go unheard, for the landmark stream with its downbeat marked and every onset with the downbeat accented. The listener asks for odds of 19 to one against every rival; a silence where a rival predicts an onset is worth 1/p, so one silence settles a rival below p = 0.053 and two are needed up to 0.229. landmarks, downbeat marked, onsets lost: 82% at 0, 77% at 0.05, 47% at 0.1, 15% at 0.4; landmarks, downbeat marked, caution only: 82% at 0, 82% at 0.05, 57% at 0.1, 46% at 0.4; every onset, downbeat accented, onsets lost: 74% at 0, 70% at 0.05, 58% at 0.1, 22% at 0.4; every onset, downbeat accented, caution only: 74% at 0, 74% at 0.05, 69% at 0.1, 69% at 0.4.

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.

rhythm · Additive metre At the same share of time lost, a crowd that hides stretches costs less than one that hides onsets. The share of metres recognised within the 16-step present from a random entry, 10 metres known, with a stated share of the time lost to noise, against the mean length of each stretch lost, in steps — "one by one" is every step lost independently, as in the essay on lost onsets. The listener asks for odds of 19 to one and allows for losses at the true rate. The dashed lines are the same listeners with nothing lost. landmarks, downbeat marked, 10 per cent lost: one by one 46%, 2 46%, 4 46%, 8 47%, 16 48%, 32 49%, 64 50%; nothing lost 56%. landmarks, downbeat marked, 25 per cent lost: one by one 27%, 2 27%, 4 28%, 8 30%, 16 32%, 32 34%, 64 35%; nothing lost 48%. every onset, downbeat accented, 10 per cent lost: one by one 61%, 2 60%, 4 61%, 8 62%, 16 64%, 32 65%, 64 66%; nothing lost 74%. every onset, downbeat accented, 25 per cent lost: one by one 40%, 2 39%, 4 41%, 8 44%, 16 48%, 32 51%, 64 53%; nothing lost 74%.

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.

rhythm · Additive metre A dancer who hears the crowd as noise buys back most of what caution cost. The landmark stream, downbeats marked, 10 metres known, 25 per cent of the time lost to noise in stretches of the stated mean length: the share of metres recognised within the 16-step present by a listener who cannot tell noise from silence and so allows for losses everywhere, and by one who hears the noise and counts a silence only where the room was quiet. Not hearing it: one by one 27%, 2 27%, 4 28%, 8 30%, 16 32%, 32 34%, 64 35%. Hearing the noise: one by one 51%, 2 50%, 4 50%, 8 52%, 16 54%, 32 57%, 64 59%. Nothing lost at all: 81%. Nothing lost but the same caution: 48%. A listener who loses a quarter of the music and knows where does better than one who loses nothing and must allow for losing it.

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.

rhythm · Additive metre In a loud room the accent is worth more, if it is loud enough to get through. Ten metres known, entered at a random step, recognised within the 16-step present by a listener who hears the room's noise: how much marking the downbeat adds to the share recognised from the landmarks, when the noise covers a tenth, a quarter or half of the time in stretches of sixteen steps, every marked downbeat is heard through it, and a stated share of the unmarked strokes are. 10% noise: 0% 23.3, 25% 24.0, 50% 24.4, 75% 24.7, 100% 23.4 points; noise masking everything 21.4. 25% noise: 0% 22.9, 25% 24.6, 50% 25.9, 75% 26.7, 100% 23.4 points; noise masking everything 18.1. 50% noise: 0% 19.8, 25% 23.8, 50% 27.0, 75% 28.9, 100% 23.4 points; noise masking everything 11.3. In a quiet room the accent is worth 23.4 points.

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.

rhythm · Additive metre Talk buries a melody's accent ten decibels before it buries a low drum. A band at 70 decibels marking its downbeats four ways, each mark at the band's level, under a crowd whose talk has the long-term spectrum of speech, at a stated level above the band: the chance the marked downbeat is heard, band by band at its clearest, with the crowd's level fluctuating three decibels either way. a louder melody note: -10 dB 100%, -5 dB 100%, 0 dB 100%, +5 dB 94%, +10 dB 47%, +15 dB 4%, +20 dB 0%; a bass note an octave below: -10 dB 100%, -5 dB 100%, 0 dB 100%, +5 dB 100%, +10 dB 100%, +15 dB 90%, +20 dB 34%; a low drum: -10 dB 100%, -5 dB 100%, 0 dB 100%, +5 dB 100%, +10 dB 100%, +15 dB 100%, +20 dB 82%; a high bell: -10 dB 100%, -5 dB 100%, 0 dB 100%, +5 dB 100%, +10 dB 100%, +15 dB 93%, +20 dB 43%; the melody's unmarked strokes, eight decibels under the accent: 100%, 99%, 72%, 14%, 0%, 0%, 0%. Talk is loudest from 250 to 400 hertz, where a melody's fundamentals are; it is fifteen decibels quieter at 63 hertz and twelve quieter at 3 kilohertz.

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.

rhythm · Additive metre A drum that is always felt is worth no more than one that is always heard, until the crowd drowns the hearing. Ten metres known, entered at a random step, recognised within the 16-step present, the crowd covering half the time at a stated level above the band: the points marking the downbeat adds, for each way of marking it heard through the crowd, and for a low drum that is felt as well and so always arrives. a louder melody note: +0 28.9, +5 20.4, +10 14.7, +15 11.6, +20 11.3, +25 11.3, +30 11.3; a bass note an octave below: +0 28.9, +5 22.1, +10 19.8, +15 18.0, +20 13.7, +25 11.4, +30 11.3; a low drum: +0 28.9, +5 22.1, +10 19.8, +15 19.8, +20 17.3, +25 12.9, +30 11.4; a high bell: +0 28.9, +5 22.1, +10 19.8, +15 18.2, +20 14.3, +25 11.5, +30 11.3; a low drum, felt as well: +0 28.9, +5 22.1, +10 19.8, +15 19.8, +20 19.8, +25 19.8, +30 19.8. A quiet room's mark is worth 23.4.

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.

rhythm · Additive metre Felt on every landmark, the stream recovers most of a quiet room's recognition even when no accent is felt. Ten dances known, entered at a random step, the crowd covering half the time at the level on the axis above the band: the share of entries in which the dance is recognised within the present, for four ways of carrying the landmarks — a drum on the downbeat and the melody's softer strokes between, heard; the same with the downbeat felt as well; a kick drum on every landmark eight decibels softer than the downbeat, heard; and the same felt, with a felt accent recognised 0 per cent of the time. A quiet room gives 80.9. drum downbeat, melody between: 0 dB 58.4, 5 dB 41.0, 10 dB 37.4, 15 dB 37.3, 20 dB 34.9, 25 dB 30.4, 30 dB 28.9; the downbeat felt: 0 dB 58.4, 5 dB 41.0, 10 dB 37.4, 15 dB 37.3, 20 dB 37.3, 25 dB 37.3, 30 dB 37.3; a kick on every landmark, heard: 0 dB 80.9, 5 dB 80.9, 10 dB 68.1, 15 dB 50.2, 20 dB 35.7, 25 dB 30.4, 30 dB 28.9; a kick on every landmark, felt: 0 dB 80.9, 5 dB 80.9, 10 dB 80.9, 15 dB 80.9, 20 dB 75.8, 25 dB 71.6, 30 dB 70.1.

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.

rhythm · Additive metre

Named alongside it

The objects these essays reach for when they reach for this one.

AksakPerceptual presentDownbeatInferenceEvidenceMetreMetrical levelSubdivisionEnumerationIdentificationBeatMasking

All concepts