Acoustics & Music Theory
why it sounds like that
Theme

Time is a circle

Rhythm drawn as a line looks arbitrary. Drawn as a cycle, the patterns that recur across continents turn out to be the same few shapes.
E(3, 8) as a cycle. A rhythm drawn round a circle, one equal arc per step, with the struck steps filled. On a circle the pattern has no beginning, which is why the same necklace of onsets is several different named rhythms depending on where a listener decides bar one is. Rhythm and metre

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.

 Six rhythms, all from the same construction. Euclidean rhythms generated by Bjorklund's algorithm, which spreads a number of onsets as evenly as a number of steps allows. The patterns were not transcribed from recordings; they are what the algorithm produces, and the names beside them are where each one is already in use. Rhythm and metre

As evenly as possible, which turns out to be a famous rhythm

Ask an algorithm to space five strikes over eight beats as evenly as it can. It produces the cinquillo. Ask for three over eight and it produces the tresillo. Nobody told it about Cuba.

 3 against 2. Two evenly spaced pulses over the same span, one in 3 and one in 2. They agree only where both grids land together, and the gap between agreements is the least common multiple — which is what makes one polyrhythm sound like a shape and another like two unrelated things. Rhythm and metre

Two clocks at once, and where they agree

Three against two repeats every six subdivisions and feels like a figure. Seven against five repeats every thirty-five and feels like weather. The difference is one number.

 How much swing there is, against how fast the music goes. The ratio of the long note to the short note of a swung pair, as a function of tempo, derived from the finding that the short note holds a roughly constant hundred milliseconds. The notated readings are horizontal lines, and each is correct at exactly one tempo. Above about three hundred beats a minute the ratio reaches one and the swing has gone. Rhythm and metre

Swing is a ratio, and it is not two to one

Swung quavers are notated as a triplet figure, which is a ratio of two to one. Measurements of what drummers actually play give anything from three and a half to one down to one to one, and which of those it is depends almost entirely on the tempo.

 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. Rhythm and metre

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.

 3 against 2, from a rhythm to an interval. The same 3:2 pattern at 6 rates, an event rate from 1.5 to 300 per second, on a logarithmic axis. Below about 20 events a second the two streams are counted; above about 40 they are heard as two pitches a 3:2 apart, which for 3:2 is 702 cents. Nothing in the pattern changes across that boundary. The slowest rate drawn is a tempo of 90 events a minute; the fastest is a pitch of 300 Hz against 450 Hz. Rhythm and metre

A rhythm fast enough to be a chord

Three against two is a polyrhythm. Speed the same pattern up until the events arrive faster than about twenty a second and it is a perfect fifth. The ratio never changed, the figure never changed, and the only thing that moved is the rate — which makes rhythm and pitch one continuum with a perceptual boundary across the middle of it.

 One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 -4, 3/4 12, 6/8 12, 12/8 as 3+3+3+3 12, so 3/4 and 6/8 and 12/8 as 3+3+3+3 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener. Rhythm and metre

The beat is inferred, and sometimes wrongly

A bar line is not in the sound. One sequence of onsets supports a reading in four, in three and in six, and which one a listener settles on comes from a small set of preference rules rather than from anything in the signal — which is why the same recording can be heard two ways and why nobody can be talked out of either.

 The metrical levels of 120 bpm, against the window. The range of inter-onset intervals that can be heard as a beat at all, from about 100 to 2000 milliseconds, with the preferred rate near 550. Each mark is one metrical level of a piece at 120 beats a minute. Which of them a listener taps is decided by which falls nearest the preferred rate, not by which one the notation calls the beat. Rhythm and metre

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.

 How often the chord changes, and what a room allows. Chord changes a second implied by each style's stated rate and tempo, on a logarithmic axis, with the rate above which a room leaves more than one earlier chord above 20 dB marked for six rooms. The style rates are conventions rather than corpus measurements and the figure says so; the room rates are arithmetic from the reverberation time. Harmony and voice leading

How often the chord changes

Two pieces can use the same chords in the same order and be nothing alike, because a progression says which chords and not how fast. Harmonic rhythm is the second variable, it runs over a factor of thirty between the styles that use it, and both of its limits are set by things that are not harmony — a listener's memory at the slow end and a building at the fast one.

 A cycle has no ending to compute, so it uses density instead. 5 layers over 32 cycles of a 12-step pattern, with each layer's entry and exit marked, and the onsets per step summed underneath. No chord changes and no cadence occurs; the closure vector is zero throughout, and every change a listener hears is a change in how many things are playing. Form and structure

A cycle cannot cadence

Every component of closure is defined by a first time and a last time. Music built on a repeating cycle has neither, so the whole apparatus returns zero on it — not a small value, zero, at every setting. What such music uses instead is how many things are playing, and that is a curve which can be computed from the onsets and nothing else.

 Twelve stages, and the composer chose the rule. Every rotation of a 12-step pattern with 8 onsets against the unrotated original. Each row is one stage of a phase piece: the filled cells are what is heard when the two parts sound together, and the count beside it is how many of the two parts' onsets coincide. The number of stages is the length of the pattern, so the length of the piece is arithmetic. Rhythm and metre

A process that enumerates its own form

Take a twelve-step pattern, play it against itself, and move one copy along by one step at a time. The piece is over when the copy returns to where it started, so its length is twelve — arithmetic, not a decision. What is heard at each stage is the union of the two parts, and nobody composed any of it.

 One construction, a scale on one side and a rhythm on the other. Both rings are the same computation: k things placed as evenly as possible in n positions. On the left it is 7 notes in 12 semitones, which is the major scale; on the right 5 strikes in 8 steps, which is the cinquillo. The filled positions come from the J-function and the open marks from Bjorklund's algorithm, and they are the same set of positions turned by 9 and 6. Scales and modes

The same algorithm made a Cuban rhythm

Ask Bjorklund's algorithm for seven onsets in twelve steps and it returns 101101011010, which read as pitch classes is the natural minor scale. That function has been producing the tresillo and the cinquillo here from the beginning, and nobody has said that the scale field's central object comes out of it unchanged.

 How long until it comes back. Mean similarity along each diagonal of the self-similarity matrix, minus the matrix's own mean off-diagonal similarity, against lag in bars, for 1 case. Lags run to half the length of each scheme, because a longer diagonal holds too few pairs to average. All rows share one vertical scale and the spread of each is printed beside it; the largest is 0.427 and the smallest 0.427. 1 of 1 cases with any spread at all put their strongest lag at the scheme's own repeat unit or a multiple of it. Form and structure

How long until it comes back

A self-similarity matrix has a second reading that nobody looks for. Add up each diagonal instead of walking along one, and out falls repetition as a function of how long ago — a period, in bars, with no segmentation, no kernel width and no bar numbers anywhere in the answer. Five of the six schemes here report the length a listener would have named. The sixth reports something better.

 3 against 2, and the line it adds up to. Two pulse trains over one bar, 3 against 2, and beneath them their union on the common grid of 6 steps. The composite has 4 onsets — 3 + 2 − 1, because the two layers share the downbeat — and its gaps run 2, 1, 1, 2 steps, which uses 2 distinct lengths and reads the same in both directions. No single division of the bar produces that sequence. Rhythm and metre

The third pattern nobody played

Two players play two even pulses and a third rhythm arrives that neither of them played. It has a + b − 1 onsets, its gaps read the same forwards and backwards, and it uses exactly min(a, b) different lengths — which is a better account of why 3:2 is a figure and 7:5 is weather than the number used for the last three essays.

 One pattern, four metres. The same 6-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are the 3 is the beat 8, the 2 is the beat 4, every step is a beat 8, so the 3 is the beat and every step is a beat tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener. Rhythm and metre

Which of the two is the beat

A polyrhythm is notated as a bar of two with three laid across it. Run the metre rules used here over the composite and they choose the three — at every ratio tried, without exception. Add the one other rule the rules have, and the answer changes hands at a bar of 1,013 milliseconds, which is 118 to the minute.

 The bell against every division of its cycle. A 12-step pattern of 7 onsets drawn round a circle, with 4 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 1 of the 2, 2 of the 3, 2 of the 4, 3 of the 6. No division has all of its beats struck and none has none of them, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other. Rhythm and metre

The bell is not a polyrhythm

Five earlier essays have set one pattern against another. In the practice the word cross-rhythm was borrowed from, there is one pattern and it is played against the beat — against several beats at once. Counted against the four divisions of a twelve-cycle, the Ewe bell marks half of the two, two-thirds of the three, half of the four and half of the six, and all of none of them.

 One pattern, four metres. The same 8-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are downbeat on step 1 -1, downbeat on step 2 -7, downbeat on step 3 -1, downbeat on step 4 -1, so downbeat on step 1 and downbeat on step 3 and downbeat on step 4 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener. Rhythm and metre

The right period at the wrong phase

Finding the beat is two problems, not one. How far apart the beats are, and where the first one is. This site's rule set answers the first confidently on a tresillo and returns a three-way tie on the second — the same score for the downbeat on step one, step four and step seven — which is not a near miss but no answer at all.

 The gap that decides whether two clocks are two. The closest approach of the two streams in each ratio, at 100 to the minute in 4-time — a bar of 2.4 seconds. It is the bar divided by the product of the two numbers, which is an identity and is checked here against the measured minimum. 9:5 is the last ratio whose onsets are securely separate at this tempo; past it the two streams' events fall inside the window in which the ear cannot put two onsets in order, and what is heard is one irregular pattern rather than two clocks. The bound has a product in it, which is why 3:2 and 4:3 are everywhere and 11:7 is a notation. Rhythm and metre

The ratio that stops being two

Eight earlier essays have taken two clocks at a rational ratio to be a thing a listener can hold. There is a ratio past which it is not, and the bound is not where anyone would look for it: the cycle is exactly one bar long at every ratio, so the length of the pattern separates nothing. What separates them is the closest the two streams ever come, which is one part in their least common multiple — 400 milliseconds for three against two, and seventeen for thirteen against eleven, which is not two events at all.

 Euclidean up to the one thing a timeline is for. Five named timelines, each drawn above the Euclidean pattern with the same number of onsets in the same number of steps, with the rotation between them found by search. 3 of 5 are rotations of the Euclidean pattern and 2 are not Euclidean at any rotation — son clave, 3–2 and rumba clave, 3–2, whose gap sequences are 3·3·4·2·4 and 3·4·3·2·4 against the algorithm's 3·3·3·3·4. Where the match holds it holds only up to rotation, and a rotation is not a small difference: the algorithm has no way to produce a starting position, and a starting position is what a timeline is. Rhythm and metre

The rotation the necklace cannot see

Ask Bjorklund's algorithm for the world's timelines and the usual answer is that it produces them. Search every rotation of each Euclidean pattern for a match and the answer is more interesting: the tresillo is E(3,8) exactly, the bossa-nova and the standard bell pattern are rotations of theirs, and the son and rumba claves — the two best-known timelines in the world — are not Euclidean at any rotation whatever. Where the match does hold it holds up to a starting position, and a starting position is the one thing a timeline is.

 The same eight notes, read three ways. The scale, eight quavers, scored against every triad and seventh at every root. Barred as written the best reading is C major7 at 0.850; with the barline one quaver later it is D minor7 at 0.850. With no metre — every note weighted the same — 4 readings tie at 0.625 and the passage has no best analysis at all. The notes are identical in all three. What changed is where the bar starts, which is not a fact about harmony. Harmony and voice leading

Which notes are the chord

A progression is a list of chords, and before there is a list something has to decide which of the notes sounding are chord tones and which are passing. Take the eight notes of a scale as eight quavers and score every triad and seventh at every root: barred as written the best reading is C major seventh, with the barline moved by one quaver it is D minor seventh, and with no metre at all three readings tie exactly and the passage has no best analysis. Same eight notes in all three. Harmonic analysis is a function of a variable that is not harmony.

 One pattern, four metres. The same 12-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 3/4 -4, 6/8 12, 2/4 -4, so 6/8 wins. The page prints 3/4, which the model does not prefer — it ranks 6/8 above it. Nothing about the sound differs between these readings; the bar line is supplied by the listener. Rhythm and metre

The time signature is a claim

A bar line is not a measurement. It is a claim about where the accents are, made before the sound exists, and there is a metre-induction model here that can be handed the same onsets and asked whether it agrees. On a hemiola it does not: the page says three and the model says six, by a margin of twelve against minus four. And on a bar of seven the signature is not even a candidate — what decides the reading is the beaming, which the signature does not contain.

 The same steps, counted by the clock. The step distribution of Ode to Joy and Twinkle, twinkle counted two ways: once per interval, which is what every earlier figure did, and once weighted by how long the note it leaves is held. The two disagree because a tune's long notes are not distributed evenly over its interval sizes — in Ode to Joy the 2-semitone step is 55.2 per cent of the moves and 60.0 per cent of the time. Which of the two a claim about melodic motion means has never been stated here, and the answer matters most exactly where a tune slows down, which is at the ends of its phrases. Rhythm and metre

The note that has a length

Every melodic figure so far reads a table where each note is a pair — a pitch and a duration — and throws the second number away. A step between two minims and a step between two quavers have been one event in every histogram it has drawn. Weighting the same statistics by time moves the step distribution by up to seven points, changes forty-four of a hundred and one contour signs, and turns up an off-by-one in the one figure that did use the durations: it took the length of the note arrived at where the time between two onsets is the length of the note left.

 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. Rhythm and metre

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.

 5 onsets in 16, by evenness against locatability. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how much of the cycle has to be heard on average before its position is known. The two objectives are opposed — the perfectly even pattern is the hardest to locate and the most clustered ones are the easiest — so there is no best pattern, only a frontier, and 22 of the 273 are on it. son clave and the bossa-nova pattern are among them. The named timelines are marked. Rhythm and metre

What the clave buys with its unevenness

The essay before this one found that the two best-known timelines in the world are not Euclidean at any rotation, and asked whether they maximise something else. They do: how quickly a fragment of the cycle says where in the cycle it is. The son clave locates itself in nine of its sixteen steps where the even pattern needs fifteen — and of two hundred and seventy-three patterns, twenty-two are on the frontier between the two objectives and the son clave is one of them.

 A duration category has a tempo range of its own. Each simple ratio's short note is the beat divided by one more than the ratio, so at a high enough tempo it falls under the fastest interval that can be a beat at all — 100 milliseconds. Each bar here runs from the slowest tempo at which the ratio's long note still belongs to a beat to the fastest at which its short note is still a note: 1:1 ends at 300 bpm, 2:1 ends at 200 bpm, 3:1 ends at 150 bpm, 4:1 ends at 120 bpm. The line is this site's swing curve, and where it crosses a category boundary the category it is leaving has already ceased to exist. Rhythm and metre

The short note is sitting on the floor

Swing is modelled here as a short note of constant absolute length, which was measured from drummers and left as a fitted parameter. The tempo window's fast edge — the shortest interval a series of events can be a beat at — was measured from listeners tapping. Both are a hundred milliseconds, and if that is not a coincidence then the swing ratio has no free parameter in it at all: the short note is not held constant, it is resting on the floor.

 Twinkle, twinkle, phrased at the level each tempo selects. The number of boundaries the model finds when its smoothing scale is set by the psychological present rather than chosen, against the tempo the tune is taken at. The scale in notes is the present's 3.5 seconds divided by the mean note length, so a fast tempo puts more notes inside the present and smooths harder. The page's own phrasing has 5 boundaries, drawn as the flat line; the model matches it best at 160 beats per minute, where the present holds 8.2 notes. The same tune at two tempos is read at two levels, which is the prediction and is not a free parameter. Form and structure

The level the tempo chooses

The boundary detector has a scale parameter and an earlier essay left it free, ending with the sentence that names this one: the scale is in notes and the psychological present is in seconds. The psychological present is two to eight seconds, a tempo converts one to the other, and the level a listener reads then stops being a parameter at all — which is a prediction with teeth, because the same tune at two tempos should be phrased differently at levels the arithmetic names in advance.

 A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players. Rhythm and metre

A cycle that says where it is

Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke.

 Where a cycle of 16 at 16, 8, 4 outruns the listener's memory. The residual uncertainty a listener is left with once the evidence has stopped accumulating, against how long one turn of a 16-step cycle takes. The listener's memory of a step halves after 3.5 seconds throughout; what changes is how many steps that is. At a cycle of 1.6 seconds it is 35 steps and every design reaches certainty, which is the regime a clave is played in. At 60 seconds it is 0.93 steps and none of them does: layers at 16, 8, 4 settles at 1.89 bits, son clave settles at 2.27 bits, the bossa-nova pattern settles at 2.38 bits, the best single line of 7 settles at 1.85 bits. That is the range a gong cycle occupies, and it is the design that wins there. Rhythm and metre

The cycle that outruns the memory

A timeline and a colotomy were compared at equal strokes and the comparison had no clock in it. A memory span is a number of seconds and a cycle is a number of steps, so the two only meet through a tempo — and at a clave's two seconds a listener's memory covers twenty-eight steps and forgets nothing, while at a gong cycle's forty it covers 1.4 and forgets almost everything. The single line is the better locator up to twenty-three seconds a cycle and the layered code is better after it, which is very close to where each is actually used.

 Five rulers on 5 onsets in 16, and what each frontier keeps. The same 273 rotation classes and the same locating cost, with the frontier recomputed against five different definitions of unevenness. The frontier's size varies from 7 to 22 of 273, so how exclusive membership is depends on the ruler. Of the named timelines of this size, son clave is on 4 of the five, rumba clave is on 0 of the five, the bossa-nova pattern is on 4 of the five. The four rulers that measure how large a pattern's departure from even is rank the whole census together, at no worse than 0.89 between any two of them; the one that counts how many distinct gap lengths a pattern uses rather than how large its departures are is anti-correlated with them at -0.26, and its frontier holds none of the named timelines and does not hold the perfectly even pattern. Rhythm and metre

The frontier and the ruler

A Pareto frontier is a claim about two quantities and only one of them was measured. The locating cost is exact; the unevenness it was traded against was one choice among several. Under four different rulers the son clave stays on the frontier and the frontier shrinks from twenty-two of 273 patterns to seven, so its membership means more rather than less — but the bembe stops being dominated, the two claves stop being equally uneven, and one plausible measure destroys the result entirely.

 A third of the timing surprise is paid for by nothing happening. Every beat of a bar of 4/4 at 1 chord change a bar, with what it costs a listener in expectation. The lower block is the arrival — the chance the chord changes here times what that change costs to be surprised by. The upper block is the hold, which is paid when the chord does not change and is charged at every beat rather than at every event. Over the bar the arrivals come to 2.61 bits and the non-arrivals to 1.29, so the term never spent before is 33 per cent of the total rather than most of it. The two together are exactly the binary entropy of each beat's own hazard, which is why the bar's whole timing bill is 3.90 bits and cannot be raised by rearranging where the changes fall. Harmony and voice leading

The surprise of nothing happening

A hazard charges a listener twice — once when the chord changes and once, quietly, at every beat it does not. The second term was expected to dominate, because there are more beats than changes. It is a third of the bill at one chord a bar and never reaches a half at any rate a metre survives, because the cost of a beat where nothing happens is second-order small.

 What a chord change costs where it actually lands. Every beat of a bar of 4/4 at 1 change a bar, priced as a listener meets it: the pale block is what has already been paid waiting through the beats the chord did not come on, and the dark block is the arrival itself. Their sum is what it costs to be surprised by a change here. The last column is the remaining case, never priced before: no change in the bar at all, at 1.61 bits and a probability of 0.33. The 9 costs are a proper distribution — 1.000 — which is the check that this is one model rather than two. And the spread is the finding: the arrival term alone puts a factor of 6.7 between the best and worst beat, and counting the waiting makes it 19. Harmony and voice leading

Where the chord actually lands

Every timing surprise so far is evaluated on a downbeat: the curve spans a factor of 6.7 and every number is read at its peak. Charge the waiting as well as the arrival and the costs over a bar become a proper distribution, the spread between the best and worst beat rises to a factor of 19.5, and a third of the probability sits on a bar in which nothing changes at all.

 The longest silence against the spread of the gaps. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how long its longest gap is. The two rank the census together at a Spearman correlation of 0.94, so the longest silence is not a second thing to know about a timeline; it is the same thing measured more crudely. Against the other axis on the frontier — how much of the cycle has to be heard before a listener knows where in it they are — it correlates at -0.02, which is nothing. The named timelines are marked, and all of them sit at 4, the shortest longest-gap any pattern of this size can have. Rhythm and metre

The longest silence is not a third axis

The frontier between evenness and locatability was drawn twice and both times against two quantities, with the third job a timeline does left uncomputed. Computed, neither candidate for it is a third quantity: the longest silence ranks the whole census with the spread of the gaps at 0.94, and syncopation is not a property of a cycle at all — every one of the 273 patterns changes its count when the bar line moves. What the request was actually asking for is a cap rather than an axis, and under the tightest one the census admits thirteen patterns and every named timeline is among them.

 5 censuses: the floor of the longest gap, the ceiling of the syncopation total, and how few patterns reach both. For each census of k onsets in n steps, counted by rotation class: 3 in 8: patterns 7, floor of longest gap 3, at the floor 1, ceiling of total 14, at the ceiling 2, at both 1; 5 in 8: patterns 7, floor of longest gap 2, at the floor 2, ceiling of total 12, at the ceiling 2, at both 2; 5 in 12: patterns 66, floor of longest gap 3, at the floor 6, ceiling of total 28, at the ceiling 6, at both 6; 7 in 12: patterns 66, floor of longest gap 2, at the floor 3, ceiling of total 20, at the ceiling 38, at both 3; 5 in 16: patterns 273, floor of longest gap 4, at the floor 13, ceiling of total 79, at the ceiling 9, at both 9. The rotation-total of syncopation is taken against a metre of 2 × 2 × 2 for 8 steps, 2 × 2 × 3 for 12 steps, 2 × 2 × 2 × 2 for 16 steps. In every census drawn the evenly spaced pattern is at the floor and at the ceiling. Rhythm and metre

The other censuses keep evenness, not locating

Five onsets in sixteen suggested two rules: a named timeline's longest gap is the shortest its census allows, and its syncopation totalled over every rotation is the largest. Nine of ten named timelines across five censuses meet both — but so does the evenly spaced pattern, in all twenty-one censuses from two onsets in eight to eight in sixteen, and in four of those five censuses the named timeline is the evenly spaced pattern. What the rules describe is evenness. The quantity the frontier was built on points the other way: the evenly spaced pattern is the slowest of its census to locate in nineteen of twenty-one, the standard bell pattern is the slowest of all sixty-six patterns of seven in twelve, and among the orders of its own gaps a named timeline is usually both the most even and the slowest to locate. The son clave, whose most even order is also the fastest, is the exception the locating story was told about.

 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. Rhythm and metre

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.

 With a fading memory the slowest order of the gaps 1 1 2 2 2 2 2 is not the one a perfect memory finds. For each of the 3 cyclic orders of the gaps 1, 1, 2, 2, 2, 2, 2 in 12 steps, the bits of position still unknown once listening has settled, against how many steps a listener's memory of a step takes to halve, with a mismatch costing 6. 2 2 2 1 2 1 2: 12 → 4e-9, 6 → 4e-5, 4 → 0.007, 3 → 0.070, 2 → 0.540, 1.5 → 1.130, 1 → 1.786; 2 2 2 2 1 1 2: 12 → 2e-5, 6 → 0.035, 4 → 0.335, 3 → 0.795, 2 → 1.405, 1.5 → 1.694, 1 → 2.005; 2 2 1 2 2 1 2 (the standard bell pattern): 12 → 2e-5, 6 → 0.027, 4 → 0.231, 3 → 0.535, 2 → 1.028, 1.5 → 1.368, 1 → 1.819. With perfect memory the slowest to locate is the standard bell pattern; at a half-life of 6 steps the highest floor is the order 2 2 2 2 1 1 2, and at 1 it is the order 2 2 2 2 1 1 2. Rhythm and metre

The bell pattern is slowest only to a perfect memory

Among the orders of its own gaps, a named timeline is usually both the most even and the slowest to locate — for a listener who never forgets. Give the listener a memory that halves and the result comes apart. Of six timelines slowest among their orders with perfect memory, only the fume-fume stays slowest for every forgetting listener, and the standard bell pattern, which is the fume-fume with onsets and rests exchanged and settles at exactly the same floors, is second of its three orders for every memory of half its cycle or less. The census ranking survives better, and in fourteen of twenty-one censuses it was the arithmetic of a pattern that repeats.

 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. Rhythm and metre

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 cycle already known, against a cycle just arrived at. How many bits of uncertainty about position a listener has, against how long the cycle takes, for a single timeline and for a layered colotomy — each drawn twice, once as a listener arriving and once as a listener who has been hearing it long enough to settle. At 1.6 seconds a cycle the timeline goes 0.94 bits arriving and 0.00 settled, and the colotomy 0.71 and 0.00; At 16 seconds a cycle the timeline goes 1.10 bits arriving and 0.08 settled, and the colotomy 0.93 and 0.23; At 60 seconds a cycle the timeline goes 2.47 bits arriving and 2.27 settled, and the colotomy 2.14 and 1.89. The gap between each pair is what the repetitions are worth, and it narrows as the cycle slows. The two designs are drawn at their own step counts rather than at equal strokes, so the levels here are not the earlier ones and the gaps are. Rhythm and metre

Repetition buys least where it is needed most

Every locating figure so far is a listener arriving — the uncertainty averaged over the first cycle heard. Cyclic music comes round dozens of times, and the same model already carries the answer for a listener who has settled: a floor of uncertainty that nothing had read. At two seconds a cycle the repetitions close the whole gap. At sixty they close eight per cent for a single timeline and twelve for a layered code. A slow cycle is worse on the first hearing and gains less from the second, and the two disadvantages compound.

 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. Rhythm and metre

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.

 Against a pulse, the bell pattern is the quickest of its orders to place. The bits of position a listener is still missing, averaged over the first cycle heard, for each cyclic order of the gaps 1 1 2 2 2 2 2 in 12 steps, heard alone, against a pulse every three steps and against a pulse every four, at perfect memory, half-life 3 steps, half-life 1.5 steps. 2 2 2 1 2 1 2: alone 1.08, 1.23, 1.73; against a pulse every 3 steps 0.69, 0.75, 0.96; against a pulse every 4 steps 0.63, 0.67, 0.85. 2 2 2 2 1 1 2: alone 1.22, 1.63, 2.02; against a pulse every 3 steps 0.60, 0.73, 0.92; against a pulse every 4 steps 0.83, 1.07, 1.36. 2 2 1 2 2 1 2 (the standard bell pattern): alone 1.25, 1.45, 1.82; against a pulse every 3 steps 0.54, 0.56, 0.67; against a pulse every 4 steps 0.55, 0.57, 0.68. Alone, the bell pattern is not the quickest order to place at any memory. Against either pulse it is the quickest at every memory. Rhythm and metre

Against a pulse the bell pattern is the easiest to place

Heard alone, the standard bell pattern is not the quickest order of its own gaps to place in its cycle, for a listener with any memory. Heard against a pulse every three steps or every four — which is how anyone hears it — it is the quickest, at every memory and at every alignment of pulse and bell, and by a wide margin: at a memory of a quarter of the cycle, 0.56 bits unplaced over the first cycle against 0.73 for either rival against a pulse in threes. Six of eight named timelines do the same. A timeline's order of gaps looks chosen for how it sits against the beat, not for how it sounds alone.

 A timed expectation would erase a slow cycle's cost, and a listener cannot time a slow cycle that well. Bits of position a listener with a 3.5-second memory is still missing over the first cycle of son clave, against how long the cycle takes, for a newcomer with no expectation, a listener timing the cycle with the Weber fraction a duration that long is judged with, and a listener timing it to ten per cent. a newcomer, no expectation: 2 s 0.94, 8 s 0.97, 24 s 1.41, 40 s 2.02, 60 s 2.47; timing as well as listeners do: 2 s 0.68 (w 0.150), 8 s 0.69 (w 0.150), 24 s 0.85 (w 0.150), 40 s 1.90 (w 0.375), 60 s 2.38 (w 0.375); timing the cycle to ten per cent: 2 s 0.47, 8 s 0.48, 24 s 0.55, 40 s 0.73, 60 s 1.02. At ten per cent even a sixty-second cycle is placed about as well as a newcomer places a two-second one. At the precision a listener actually has for durations of half a minute or more, the expectation is worth a tenth of a bit. Rhythm and metre

An expectation cannot rescue a cycle too slow to time

A listener who knows a piece arrives with an expectation of where in the cycle they are, and the size of that expectation was the number the last essay said nobody had. It can be given one: a listener who has been timing the cycle carries a spread of their Weber fraction times the cycle, which is the same number of steps at any tempo. Timed to ten per cent, a forty-second cycle would be placed better than a newcomer places a two-second one. But forty seconds is judged in the band where the Weber fraction is nearer forty per cent, and there the expectation is worth a tenth of a bit.

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