Acoustics & Music Theory
why it sounds like that
Generator

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.
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.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 92 of its 92 placements carry sound, built from the same numbers as the drawing, offering these buttons: 160 seconds a cycle with two alike points at steps 7 and 13 — the first fourteen steps, 160 seconds a cycle with two points alike at steps 7 and 13 — the first fourteen steps, 160 seconds a cycle with two points told apart at steps 6 and 13 — the first fourteen steps, 160 seconds a cycle with two points told apart, at steps 6 and 13 — the first fourteen steps, 160 seconds a cycle, points at steps 6 and 13 heard alike — the first fourteen steps, 3 onsets spread over 16, three times round, 3 onsets spread over 8, three times round, 80 seconds a cycle, two alike points at steps 4 and 12 — the first thirteen steps and 58 more.

Called by 22 essays

the blast radius of changing it

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 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.

rhythm · Cyclic-rhythm 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.

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.

rhythm · Euclidean rhythm 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.

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.

harmony · Progression 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.

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.

rhythm · Polyrhythm 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.

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.

rhythm · Polyrhythm 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.

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.

rhythm · Metre induction 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.

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.

rhythm · Euclidean rhythm 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 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.

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.

rhythm · Euclidean rhythm 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.

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.

rhythm · Cyclic-rhythm 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 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.

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.

rhythm · Cyclic-rhythm 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.

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.

rhythm · Euclidean rhythm 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.

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.

rhythm · Euclidean rhythm 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.

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.

rhythm · Cyclic-rhythm 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.

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.

rhythm · Cyclic-rhythm Timed from every layer, a slow colotomic cycle is placed almost at once. Bits of position a listener with a 3.5-second memory is still missing over the first cycle, against how long the cycle takes, for a 16-step colotomy with layers every 16, 8, 4 steps: a newcomer, a listener timing from the last gong, a listener timing from the last stroke of every layer, and a listener timing from the gong to ten per cent; with son clave, timed from its own cycle, for comparison. a colotomy, newcomer: 1.6 s 0.71, 2 s 0.71, 3 s 0.72, 4 s 0.73, 6 s 0.75, 8 s 0.78, 12 s 0.85, 16 s 0.93, 24 s 1.14, 32 s 1.37, 40 s 1.60, 48 s 1.83, 60 s 2.14; a colotomy, timing a whole cycle: 1.6 s 0.27, 2 s 0.45, 3 s 0.45, 4 s 0.45, 6 s 0.46, 8 s 0.46, 12 s 0.47, 16 s 0.48, 24 s 0.54, 32 s 1.21, 40 s 1.45, 48 s 1.70, 60 s 2.02; a colotomy, timing from every layer: 1.6 s 0.00, 2 s 0.00, 3 s 0.00, 4 s 0.00, 6 s 0.00, 8 s 0.00, 12 s 0.01, 16 s 0.01, 24 s 0.01, 32 s 0.03, 40 s 0.11, 48 s 0.17, 60 s 0.20; a colotomy, from the gong to ten per cent: 1.6 s 0.34, 2 s 0.34, 3 s 0.34, 4 s 0.34, 6 s 0.34, 8 s 0.35, 12 s 0.35, 16 s 0.35, 24 s 0.37, 32 s 0.42, 40 s 0.49, 48 s 0.60, 60 s 0.80; son clave, timing the cycle: 1.6 s 0.32, 2 s 0.68, 3 s 0.68, 4 s 0.68, 6 s 0.68, 8 s 0.69, 12 s 0.70, 16 s 0.73, 24 s 0.85, 32 s 1.63, 40 s 1.90, 48 s 2.12, 60 s 2.38. At 40 seconds the clock leaves 0.11 bits, against 1.45 timing from the gong and 0.49 for a ten-per-cent timer.

The strokes between the gongs are a clock

An experienced listener arriving in a forty-second gong cycle was expected to be no better placed than a newcomer, because a whole cycle is timed in the band where a Weber fraction is near forty per cent. But nobody times a gong cycle as one interval. A listener who resets at the last gong, the last kempul and the last kenong times three short intervals instead of one long one, and arrives knowing their position to a tenth of a bit — better than a listener timing the whole cycle to ten per cent. A single timeline, whose strokes are all one instrument, offers nothing to reset on.

rhythm · Cyclic-rhythm At every number of strokes, the clock that halves at each level places a slow listener best. Bits of position unknown over the first cycle of a sixteen-step cycle at 40 seconds, for a listener timing from the last stroke of every layer, for every nesting of layers whose periods divide sixteen, grouped by how many strokes a cycle the layers make together. 1 stroke: 16 1.61; 2 strokes: 16 : 8 0.44; 4 strokes: 16 : 8 : 4 0.11, 16 : 4 0.39; 8 strokes: 16 : 8 : 4 : 2 0.10, 16 : 8 : 2 0.16, 16 : 4 : 2 0.37, 16 : 2 0.90. In every group the best nesting is the one in which each layer is twice as fast as the one above it.

The best gong cycle halves at every level

A colotomic cycle is a clock whose layers are its hands, and a clock can be built many ways from the same number of strokes. Searched over every nesting of a sixteen-step cycle, the one that places a slow listener best is, at every budget of strokes, the one in which each layer is exactly twice as fast as the layer above it. A gong with one other layer does best at a ratio of four, and three halving layers with the same four strokes beat it by a factor of three and a half. A layer finer than the one whose interval is already timed to half a step adds almost nothing.

rhythm · Cyclic-rhythm An elaboration's most useful boundary is one the skeleton does not have. Bits of position unknown over the first cycle of a sixteen-step cycle with gong, kempul and kenong, timing from every layer, when an elaborating part adds a figure a whole cycle long whose one recognisable boundary falls at the step drawn, at 40, 80, 160 seconds a cycle. Dashed: the skeleton alone. Dotted: the skeleton with a kethuk every two steps. 40 s: skeleton 0.112, with a kethuk 0.104; boundary at 2 0.029, 4 0.004, 6 0.004, 8 0.006, 10 0.005, 12 0.004, 14 0.028. 80 s: skeleton 0.329, with a kethuk 0.301; boundary at 2 0.125, 4 0.094, 6 0.067, 8 0.087, 10 0.057, 12 0.072, 14 0.089. 160 s: skeleton 0.646, with a kethuk 0.457; boundary at 2 0.360, 4 0.340, 6 0.256, 8 0.323, 10 0.251, 12 0.295, 14 0.292. A boundary on the gong is the gong's own timer and adds nothing. Anywhere in the middle of the cycle it re-times the digit the gong reads worst, and does more than a whole layer of kethuk strokes.

An elaboration is a clock hand where the skeleton has none

A colotomic cycle is a clock whose hands are its punctuating layers, and the finest hand a slow cycle can use runs out as the cycle slows. The elaborating parts that fill the cycle were predicted to supply that finer hand. They can, and a two-step figure recognised at every boundary is a kethuk exactly. But the figure that does most is not a fine one. It is a long one with a single recognisable point in the middle of the cycle, where no skeleton stroke falls: at eighty seconds a cycle it leaves a listener a fifth as uncertain as the skeleton alone, where a kethuk takes off a tenth — and it still does more than a kethuk when a listener catches that point one time in ten.

rhythm · Cyclic-rhythm A second recognisable point is worth a great deal if it can be told from the first. Bits of position unknown over the first cycle of a sixteen-step cycle with gong, kempul and kenong, when the elaborating parts add none, one, two or three recognisable points, each set at its best placement, read by a listener who can tell the points apart and by one who cannot, at 80 and 160 seconds a cycle. 80 s: none 0.329; 1 told apart 0.057 at 10, alike 0.057 at 10; 2 told apart 0.001 at 6/13, alike 0.005 at 6/12; 3 told apart 0.000 at 6/10/14, alike 0.007 at 4/9/14. 160 s: none 0.646; 1 told apart 0.242 at 11, alike 0.242 at 11; 2 told apart 0.094 at 6/13, alike 0.180 at 7/13; 3 told apart 0.038 at 5/11/15, alike 0.117 at 7/10/13. Points told apart add like checkpoints, each taking away most of what is left. Points alike add a little, and at eighty seconds a third alike point leaves the listener worse placed than two.

Two landmarks in a cycle help only if they can be told apart

One recognisable point inside a slow gong cycle, placed where the skeleton has no stroke, did more for a listener than a whole layer of kethuk strokes. A second point does as much again, a third nearly as much again — provided the listener can tell which point they have just heard. If the points sound alike, a second one is worth a fraction of that, seventy of the hundred and five ways of placing it leave the listener worse off than one point would, and two alike points half a cycle apart undo most of what the first one gave. Whether the listener keeps one timer or several turns out not to matter at all.

rhythm · Cyclic-rhythm A newcomer to a slow cycle needs about four hearings to tell two marks apart, and the best pair is not the quickest to learn. A sixteen-step cycle of 160 seconds with two recognisable points that sound alike, and a listener who learns that they are two events by hearing them at different distances from the gong: the bits of position left unknown in each of the first ten cycles, for four pairs of points, with each pair's clock once fully learned dashed. steps 6 and 13 (d′ 1.30 a hearing, nine listeners in ten naming them after 3.7 cycles): 0.219, 0.147, 0.117, 0.103, 0.098, 0.095, 0.094, 0.094, 0.094, 0.094; learned 0.094; steps 7 and 13 (d′ 1.08 a hearing, nine listeners in ten naming them after 4.9 cycles): 0.180, 0.146, 0.126, 0.116, 0.110, 0.106, 0.105, 0.104, 0.103, 0.103; learned 0.102; steps 4 and 12 (d′ 1.69 a hearing, nine listeners in ten naming them after 2.6 cycles): 0.445, 0.252, 0.206, 0.195, 0.192, 0.191, 0.191, 0.191, 0.191, 0.191; learned 0.191; steps 1 and 15 (d′ 2.49 a hearing, nine listeners in ten naming them after 1.7 cycles): 0.301, 0.255, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253, 0.253; learned 0.253.

A newcomer learns a cycle's landmarks in four hearings

Two recognisable points in a slow gong cycle help a listener only if they can be told apart, and a newcomer at first cannot: two marks that differ only by where they fall in the cycle sound alike until the listener has heard enough cycles to notice. Timed from the gong, the best pair of points is learned by nine listeners in ten after about four cycles — the same four at any slow tempo, so ten minutes at 160 seconds a cycle. The pairs learned fastest sit right after the gong and are poor clocks once learned; for the first four cycles a newcomer is best served not by the best clock but by a pair whose confusion costs less while it lasts.

rhythm · Cyclic-rhythm With nothing but position to go on, four marks are worse than two for as long as a newcomer has patience, and a difference of sound turns that round. What is still unknown about the position in a 160-second cycle of sixteen steps, in bits, cycle by cycle, for the best two, three and four marks with a d′ of 0 between each mark's sound and the next. two marks: 1 0.219, 2 0.147, 3 0.117, 4 0.103, 5 0.098, 6 0.095, heard alike 0.219, told apart 0.094; three marks: 1 0.194, 2 0.180, 3 0.158, 4 0.138, 5 0.121, 6 0.108, heard alike 0.194, told apart 0.038; four marks: 1 0.290, 2 0.284, 3 0.262, 4 0.235, 5 0.208, 6 0.185, heard alike 0.290, told apart 0.000.

More marks are worse until they sound different

Two points in a slow gong cycle take a newcomer four hearings to tell apart, and the obvious way to shorten that is to make them sound unlike each other. For two marks it barely matters: the saving is a sixth, the first hearing is lost whatever they sound like, and the best pair stays the best pair. The surprise is what happens to more than two. Marks told apart by position alone are worse the more of them there are, for dozens of cycles. Past a very small difference of sound, four marks beat two inside the first four hearings.

rhythm · Cyclic-rhythm

All figures · What can be heard