Acoustics & Music Theory
why it sounds like that
Theme

The second time is different

Almost everything a listener hears has been heard before, a few seconds earlier, in the same piece. Nothing about the sound changes on a repeat; everything about its status does. These essays are about repetition as the material of form — what a distance matrix can find in it, what it costs in bits, and why a literal repeat is the one boundary a change detector cannot see.
thirty-two-bar AABA, every bar against every other bar. A self-similarity matrix of 32 bars of thirty-two-bar AABA. Each cell is the cosine similarity of two bars' pitch-class vectors, with the chord's own notes weighted 1 and the rest of the key 0.5. Similarity is quantised into four bands for drawing and anything under 0.35 is left as paper. Nothing in the computation knows what a section is; the blocks and stripes are what the arithmetic returns. Form and structure

A piece is mostly itself again

Take a piece of music, encode each bar as the notes sounding in it, and compare every bar with every other bar. The picture that comes out has blocks and stripes in it, and those blocks and stripes are the form — arrived at by arithmetic that has never heard of an exposition, a chorus or a refrain.

 Boundaries found by a local operator, at three kernel widths. Foote's checkerboard novelty computed on the self-similarity matrix of thirty-two-bar AABA, at kernel widths of 2, 4, 8 bars. The dashed verticals are where the encoding's sections actually change; nothing about them enters the computation. A peak is a place where the bars before resemble each other, the bars after resemble each other, and the two groups do not resemble each other. Form and structure

The boundary is where the neighbourhood changes

A section boundary can be found by an operator that never sees a section. It walks the diagonal of a similarity matrix asking one local question — do the bars behind me resemble each other, do the bars ahead resemble each other, and do the two groups resemble each other — and where the answer is yes, yes, no, there is an edge. What it cannot find turns out to say more than what it can.

 Redundancy in bits, and why the number needs a length beside it. Left: the LZ78 cost of each scheme divided by the cost of sending the same symbols flat, against how many bars are sent, with each bar coded as its chord and key. Every scheme is above 1 at a single chorus — the coder loses — and every one falls under it as the piece runs. Right: new dictionary phrases per bar at one chorus, which is the statistic that survives at short lengths. Form and structure

How much of this is new

Repetition can be counted rather than looked at. Feed a piece's bars to a compressor and the bits it needs are a measure of how much of the piece is a repeat of an earlier part of itself. The measurement works, the number is real, and it turns out to be a statement about the description rather than about the music — which is the most useful thing it has to say.

 Two ways to fill eight bars, and only one of them accelerates. The period against the sentence, drawn as the lengths of their constituent units against position on a grid of eight bars. The ratio beside each row is the mean unit length in its second half divided by the mean in its first: the period at 1.00, the sentence at 0.67. A ratio below one is an acceleration — the unit shortening as the phrase approaches its arrival — and a ratio of one is a plan whose unit never changes length. Form and structure

One of these eight-bar phrases accelerates

The sentence and the period both occupy eight bars, both end with a cadence, and both are recognised by ear rather than counted. What separates them is arithmetic. One halves its unit halfway through and the other does not, and the difference comes out as a single ratio — 0.67 against 1.00 — computed from nothing but the lengths of the parts.

 The key plan is the shape. The key of a classical sonata-form movement against position in the movement, measured in steps along the chain of fifths from the home key. The positions are the proportions such a movement is described by rather than bar numbers from any one score. The furthest point is 4 steps out, sharing 3 of seven notes with home. Form and structure

The key plan is the form

The large shape of a classical movement is not a shape at all, it is a journey — out to one key and back. Which key is not a matter of taste. Of the two keys that share six of their seven notes with home, only one introduces a note the home key does not use in any of its chords, and that note is the arriving key's own leading note. The departure is audible because of one accidental.

 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.

 Every delay, scored — a round and a tune that is not one. Mean Plomp-Levelt roughness of the simultaneities a tune makes against a copy of itself entering a whole number of bars later, for Frere Jacques and Twinkle, twinkle. The two do not overlap: the worst delay of the first is smoother than the best delay of the second. The model sees roughness and nothing else — no voice leading, no parallels, no distinction between a passing dissonance and a structural one. Harmony and voice leading

A melody that can accompany itself

Whether a tune works as a round is decidable before anyone sings it. Score every simultaneity it makes against a delayed copy of itself, at every delay, with the roughness model already in use, and two nursery tunes separate completely — every delay of one is smoother than every delay of the other. What the scan does not do is pick out the entry the tradition uses, and that refusal is the most informative thing in it.

 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.

 The period, as the piece goes by. The strongest lag of thirty-two-bar AABA computed on only the bars heard so far, against how many bars that is. The final answer is 4 bars; it is revised 6 times on the way, and is not reached for the last time until bar 29 of 32, which is 91 per cent of the way through and 64 seconds at 108 beats a minute. Nothing about the boundary operator is involved: this is the global statistic, and it is the half of the form that a first hearing cannot have. Form and structure

The form a first hearing cannot have

Every figure so far was computed with the whole piece in hand. Run the same methods over only the bars already heard and one of the two methods survives intact — the boundary operator turns out to be causal at a fixed delay of a few bars — while the other collapses. The period of a piece is not knowable until the piece is nearly over, and in two of the six schemes here not until its last bar.

 Sensitivity and specificity on one dial. Aligned similarity — bar i against bar i+L, which is what a return is — for 4 eight-bar comparisons, as the key-invariance dial turns. One comparison is constructed: a literal repeat in the encoding, moved up a fifth, which is a stated manipulation because no scheme encoded here repeats a section in a new key. The shaded band is the margin between the two named comparisons, and it runs from 0.021 to 0.106. Form and structure

The same thing somewhere else

A measure built on which notes are sounding calls a passage that comes back a fifth higher a stranger. There is a dial that fixes this, and turning it is supposed to be a trade — more sensitivity to a transposed return, less specificity against a coincidental one. It is not that trade. Two different statistics answer opposite ways, and the setting that would compromise between them is the worst one available.

 The ranking is settled either side of one narrow band. Remembered repetition — each bar's best match to an earlier bar, discounted by exp(−Δt/τ) with Δt in seconds — for 6 schemes at 108 beats a minute, against the decay constant τ on a logarithmic axis. The order of the schemes changes only between 8 and 13 seconds; outside that band it is fixed, so an estimate of τ wrong by any amount that stays outside it leaves the ranking alone. Perception and the listener

A return has to be remembered

A stripe four bars off the diagonal and a stripe twenty-four bars off it are the same ink and are not the same experience. Convert the lag axis to seconds, discount every comparison by how long ago it was, and the ranking of these six schemes by how repetitive they are changes — and the decay constant and the tempo turn out to enter the arithmetic as one number rather than two.

 thirty-two-bar AABA, as a strip of time. thirty-two-bar AABA laid out one cell per bar, coloured by section, with the roman numeral in each bar. the A section's turnaround is the ii-V every variant keeps; the bridge is a chain of applied dominants. At 108 beats a minute in 4/4 the whole of it lasts 71 seconds. Cut into 4 repeat units of 8 bars, 3 pairs of units agree on more than 50 per cent of their bars. 2 of them are not identical, and 2 of those 2 differ in a run of bars ending at the last bar of the unit; the changed bars are marked in orange. Form and structure

Where a repeat is changed

Cut every scheme into its own repeat unit, compare each unit with every other, and ask where a repeat stops agreeing with what it repeats. The answer is that it stops at the end, in every case the corpus contains — and the number of cases the corpus contains depends entirely on where the threshold for "a repeat" is put. Moving it by nothing at all takes the count from two to twenty and the finding with it.

 How many bars a key change takes to be heard. A twelve-bar progression that moves to G major at bar 6, read by the same correlation against all twenty-four profiles, with a window of 3, 4 and 8 bars. With 3 bars of history the new key is never the answer at all. With 4 bars of history the answer is G major from bar 7, one bar late, and it holds it from there. With 8 bars of history the answer is G major from bar 9, 3 bars late, and it holds it from there. The pivot bar is ambiguous by construction — it belongs to both keys, which is what makes it a pivot — so the lag is not a defect of the algorithm but a statement about how much evidence a key is. Perception and the listener

How much evidence a modulation needs

Run a key-finder bar by bar over a progression that moves to the dominant at bar six. With four bars of history the answer becomes the new key at bar seven and holds. With three bars it never gets there at all, and reports E minor and B minor on the way. The window decides the lag as much as the music does.

 The boundary operator run over only what has been heard. Foote's checkerboard novelty on thirty-two-bar AABA at a kernel width of 4 bars, computed twice: once with the whole piece available, and once using only the bars heard up to and including each bar. The kernel reaches 4 bars forward, so every cell it needs has been heard 3 bars after its centre — the retrospective curve replotted 3 bars to the right lands on the causal one, and the operator turns out to be causal at a fixed delay rather than blind. The dashed verticals are the encoding's real section boundaries and are not an input. Form and structure

An ending that can be heard coming

Two measurements are both called hearing an ending coming and they point in opposite directions. By the halfway mark of an ordinary form almost nothing new arrives — and the cost of coding each bar has not fallen at all. Neither statistic says anything is about to stop, because no statistic over content can: predicting the next event well is not predicting that there will not be one.

 What survives a change of encoding: verse and chorus. The same 32 bars of verse and chorus under five encodings, scored on the three things measured here measures. Mean off-diagonal similarity says how alike the piece looks to the arithmetic. Recall and precision are the novelty operator's boundaries against the 3 the section plan has, at a kernel of four bars. The period is the strongest peak of the lag profile, in bars. Under the bag of pitch classes every other figure uses, the piece is 90 per cent self-similar and the operator finds 0 per cent of the boundaries; under how far the root moved it finds 100 per cent. The period is the quantity that does not move. Form and structure

The repeat that is not in the notes

Eight earlier essays compare bars by writing each one as a bag of pitch classes and taking a cosine. Nothing chose that encoding — the first used it and the other seven inherited it. Encode the same six schemes four other ways and one of the three findings survives untouched, one survives with different numbers, and one turns out to have been a statement about the encoding all along: the boundary operator finds none of the section edges in three schemes as a bag of pitch classes and every one of them as tonic, subdominant and dominant.

 How fast two keys can alternate before the finder stops following. The share of bars a moving key-finder names correctly, once its reading is shifted back by its own lag, against how many bars each key holds for. One line per window. Below a block of three bars the second key is never named at all — 2 of the sweep's readings report a single key for the whole passage — and above about twice the window the tracking is over ninety per cent. The lag itself is about half the window: 0 bars at a window of 3, 0 bars at a window of 4, 3 bars at a window of 8. Harmony and voice leading

The alternation a key-finder cannot follow

Two keys sounding together are not in the key-finder's vocabulary, and an earlier essay ended by pointing at the other case and saying what was missing: a passage whose alternation rate can be varied while everything else is held still. Built, it gives a rule with three numbers in it — the second key is never named below a block of three bars, the tracking clears ninety per cent above twice the window, and the reading is late by half the window throughout.

 Crescendo, and what the impression does. A crescendo of 20 dB over 8 seconds, drawn as three loudnesses in sones. The pale line is what is physically sounding, the middle line is the short-term loudness of the moment and the heavy line is the long-term loudness, which is the passage's loudness as a listener would report it. The gap between the last two is the whole of the effect: at its widest the moment is 1.02 times the running impression, and the impression takes 2 seconds to come down against 99 milliseconds to go up. Form and structure

Loud is relative, and it comes down slowly

The account of loudness had a model of a moment and the account of closure asked it for a model of a form. The published one exists and its content is a pair of numbers that are not the same: a listener's running impression of how loud the music is rises to meet a step in a fifth of a second and takes seven seconds to come back down. A twenty-decibel crescendo spread over eight seconds therefore buys almost no contrast at all, and the same twenty decibels taken as a step buys a factor of two.

 Four endings, and the loudness each produces from the page alone. Short-term loudness through the closing 6 bars of a thirty-two bar scheme, computed from the part count of each bar with no performance data of any kind — the parts are realised every way their ranges allow, every partial is placed in its critical band, and the sum is run through the two loudness smoothers. thins to one arrives at 0.764 of the running impression; full final chord arrives at 0.952 of the running impression; unchanged arrives at 1.000 of the running impression; thins then full arrives at 0.929 of the running impression. The result worth the figure is that full final chord is not the loudest: adding parts to a final chord adds power and almost no loudness, because the extra parts land in critical bands the chord already occupies. An ending is made loud by contrast with what preceded it, not by thickness. Form and structure

A final chord is not made loud by adding to it

An earlier essay on closure said the loudest cue an ending has needs a corpus rather than an arithmetic. The arithmetic was built one essay ago, so it does not. Run four ending textures through it and two things come out backwards: a final chord three parts thicker than the rest arrives *quieter* against the running impression than the passage it ends, and a texture that drops a part a bar does not get quieter at all until the bar where there is one part left.

 How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire. Harmony and voice leading

Surprise is a number

The chord that did not come was described rather than measured. Its measure is the information content of what did arrive, and a model of the probability has been to hand since the key-finding essays — eight root-motion weights, ordinal and stipulated. Reading them as a distribution prices a deceptive cadence at 2.71 bits against a perfect one's 1.85, and turns up the fact that the largest of the eight had never been read by anything.

 The tempo turns, and almost nothing moves. The earlier arrival reading — what is sounding at the final chord over what the listener has been hearing — swept over bar lengths from 0.5 to 5 seconds, which is 480 down to 48 beats a minute, at 3 closing lengths. Every curve is nearly flat. Across a tenfold change of tempo one gesture's reading moves by a factor of 1.201 and the other's by 1.098, while the gap between the two gestures — which is what that essay was measuring — is 1.228. The expectation was that the tempo would decide the answer, on the grounds that a two-second bar against a two-second release is a comparable pair. The premise is wrong in a way the sweep makes obvious: the thing being compared with the release is not a bar, it is the WHOLE ENDING, which is 2 to 8 bars long and is therefore far longer than the release at every tempo anybody plays. The running impression has caught up with the closing texture before the final chord arrives, at 0.5 seconds a bar and at 5, and what is left is the last bar's own jump. Form and structure

The parameter that did not decide the answer

An earlier essay on closure ended by naming the tempo as the thing every number in it was resting on, and said it was the kind of parameter that had caused trouble before by turning out to decide the answer. Turned across a tenfold range at a closing gesture of fixed length it moves the reading by four per cent, against a twenty-three per cent gap between the gestures it is distinguishing. The parameter beside it in the same figure — how many bars the gesture occupies — moves it by twenty, and nobody had named that one at all.

 Expectation as a curve, and what a change costs where it lands. Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous — it sharpens through a bar and collapses when the change arrives — and the two ingredients for it are already here, the harmonic rhythm and the metrical beat weights. The curve is the hazard: given that the chord has not changed yet, the chance that it changes on this beat. It runs from 0.043 on the weakest beat to 0.290 on the downbeat, a ratio of 6.7, against 0.125 if every beat were alike. The marked beats are where the changes actually arrive, and their timing bill is 3.6 bits against 6.0 for a listener with no metre — so these changes are 1.7 times cheaper to expect than a metreless listener would find them. That term is new: an earlier essay prices which chord arrived and this prices when, and a listener meets the sum. Harmony and voice leading

Expectation is a curve, not a list

Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous, sharpening through a bar and collapsing when the change arrives — and the two ingredients for it were already here, in two other accounts. What comes out is a second surprise, for when a chord arrives rather than for which one it is.

 A ritardando does not spend the diminuendo. The arrival reading under a deceleration into the ending, from no ritardando at all to a final tempo 30 per cent of the starting one — which stretches the closing bars from 12.0 seconds to 20.2. The expectation was that it would matter: a ritardando lengthens exactly the bars the gesture is happening in, so a diminuendo that would have been absorbed at a steady tempo gets more of the smoother's own time to be absorbed in. It moves the reading by 0.00 per cent. Every line here is flat to within the thickness of the line, which is the second time a tempo parameter has been swept here and found to do nothing. Form and structure

The reading was a step response

Sweeping the tempo found it did not decide the answer. This one sweeps the deceleration across a factor of three and finds a null to five figures, and then sweeps the length of the closing gesture across a factor of forty-eight and finds it moves the reading by eight per cent — but not as a function of seconds. Sorted by seconds the twelve runs scatter; sorted by how many bars the instruction covers they fall into three tight groups. One sentence explains the null and the not-null together.

 How much of the reading comes from what has not happened yet. Every margin reported earlier is two-sided: the best path through a key at a bar is the best score into it plus the best score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line, because the dynamic program already had both halves separately. The mean margin falls from 3.90 bits with hindsight to 1.79 without it, so 54 per cent of this passage's certainty is retrospective. The two passes never disagree about which key is best here, so the hindsight buys confidence rather than a different answer. This is the quantity every earlier essay has assumed and none has measured. Scales and modes

How much of the reading arrives late

Every margin reported earlier is two-sided: the best path through a key at a bar is the score into it plus the score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line. On a thirty-two-bar song it removes more than half the certainty, and on a passage built to be ambiguous it changes the key named at nine bars out of eleven.

 A listener has a quarter of an analyst's confidence and the same answer. The mean margin between a passage's best two key readings, against how many bars a listener's memory of the evidence takes to halve. The two-sided reading — the one that uses bars that have not happened yet — sits at 3.90; the forward pass with perfect recall at 1.79; a forward pass whose evidence halves every 6.6 bars at 1.00. The dots' size is how often that reading names the same key as the two-sided one: 100 per cent at perfect recall and 81 at a one-bar half-life. So forgetting costs a great deal of confidence and very little accuracy — the key is robust and the certainty is not. Harmony and voice leading

The listener who forgets

Setting an analyst's reading of a key against a listener's measures what arrives late. Both passes assume perfect recall of their own half — which is as wrong going forward as knowing the future is going back. Put a decay on the forward pass and a listener with a memory of a few bars keeps a quarter of the confidence and nine tenths of the answers.

 A long note and a strong note disagree, and the winner is neither. The same 8 notes scored against every triad and seventh at every root, with the weighting run from the metrical one always used to a durational one never drawn. On the left each note counts for its metrical weight; on the right, for how long it is held. The long notes here are on beats 2, 4, 6, 8, which are the weak ones. The two cues point at different chords — C major7 on the left and D minor7 on the right — turning over at a mixture of 40 per cent. And at the crossing the winner is A minor7, which is neither cue's answer — a chord that shares three notes with each and is not the reading either rule asks for. Nothing about the notes changed. What changed is which of two cues a theorist would call obvious is being believed. Harmony and voice leading

The long note and the strong note

The segmentation that produces every object connected here has carried a free parameter since the day it was written: whether a note counts for its metrical weight or for how long it is held. Only the first has ever been drawn. The two name different chords on sixteen per cent of passages where the cues agree about the notes and forty-three per cent where they do not — and where they disagree most sharply a mixture of them picks a third chord neither one asks for.

 The same forms by the clock and by what is stored. Six forms, each drawn twice: its sections sized by their share of the bars, and sized by their share of what a listener has to store when a bar counts only if it is recognised from 1 bar of context. Returns are drawn pale with a dashed edge. twelve-bar blues: returns take 67 per cent of the clock and 13 per cent of the storage; thirty-two-bar AABA: returns take 25 per cent of the clock and 5 per cent of the storage; rondo, ABACA: returns take 40 per cent of the clock and 10 per cent of the storage; verse and chorus: returns take 50 per cent of the clock and 13 per cent of the storage; two eight-bar phrases: returns take 0 per cent of the clock and 0 per cent of the storage; a four-bar ostinato: returns take 88 per cent of the clock and 0 per cent of the storage. Form and structure

A return is shorter than its first hearing

A rondo's refrain takes three fifths of the clock and a verse-and-chorus song is balanced to the bar. Count instead the bars a listener could not have predicted when they arrived, and the returns shrink to between a tenth and a quarter of what is kept — so a song equal by the clock is between three and seven times heavier in its first half. A coder that learns repeats one bar at a time says the halves are equal, and the two memories disagree by more than any proportion a listener could confuse.

 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 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.

 An honest drum that comes late is too late, however long it plays. Two bars of ground and twenty-four of a tune that marks the bar, over a walking bass whose root arrives late in every bar, with an honest drum on every downbeat from a stated bar to the end: the share of passages a listener choosing among three sets of habits ends on the true barline, for four starting credits for the drum. Credited 0.75: from bar 1 100%, from bar 2 0%, from bar 3 0%, from bar 4 0%, from bar 5 0%, from bar 6 0%, from bar 7 0%, from bar 9 0%. Credited 0.9: from bar 1 100%, from bar 2 99%, from bar 3 29%, from bar 4 0%, from bar 5 0%, from bar 6 0%, from bar 7 0%, from bar 9 0%. Credited 0.95: from bar 1 100%, from bar 2 100%, from bar 3 88%, from bar 4 0%, from bar 5 0%, from bar 6 0%, from bar 7 0%, from bar 9 0%. Credited 0.99: from bar 1 100%, from bar 2 100%, from bar 3 100%, from bar 4 88%, from bar 5 0%, from bar 6 0%, from bar 7 0%, from bar 9 0%. At the credit the drum has always had, three in four, a drum that misses the first bar never recovers the reading, though it then plays twenty-five bars running. Each bar the late root is believed before the drum arrives raises the credit it would need; at 0.99, ninety-nine to one, the drum must still start by the fourth bar. Harmony and voice leading

An honest drum has to come first

Two strokes of an honest drum at the very start of a passage protect a listener from a bass whose root arrives late in every bar. The same two strokes a bar later do nothing at all, and neither does a drum that then plays every downbeat to the end. A listener that keeps a record of its witnesses scores the late drum against the reading the root has already made, and convicts it within two bars. Every bar the lie is believed first multiplies the trust the drum would need, and by the seventh bar no trust short of certainty is enough.

 A break lets a late drum win only if the listener comes out of it holding no reading and almost no record. Two bars of ground and twenty-four of a tune that marks the bar, over a walking bass whose root is late in every bar, the choosing listener, and an honest drum on every downbeat from bar 5, 9 or 13 to the end, entering after a break at which the listener keeps a stated share of its witnesses' records. Solid: the break also wipes its reading. From bar 5: 96% keeping 0, 96% keeping 0.02, 96% keeping 0.05, 91% keeping 0.1, 46% keeping 0.2, 42% keeping 0.3, 30% keeping 0.5, 8% keeping 0.75, 7% keeping 1. From bar 9: 95% keeping 0, 92% keeping 0.02, 47% keeping 0.05, 43% keeping 0.1, 11% keeping 0.2, 2% keeping 0.3, 0% keeping 0.5, 0% keeping 0.75, 0% keeping 1. From bar 13: 93% keeping 0, 57% keeping 0.02, 44% keeping 0.05, 10% keeping 0.1, 1% keeping 0.2, 0% keeping 0.3, 0% keeping 0.5, 0% keeping 0.75, 0% keeping 1. Dashed, on the floor: the same breaks leaving half the reading's margin, where the drum wins no passage at any share of the record kept. The horizontal axis is drawn as the square root of the share. Harmony and voice leading

A break rescues a late drum only by ending the reading

A drum that enters after a late root has made the reading needs odds in the hundreds to undo it. Bring it in after a break in the texture, and the obvious guess is that the break helps by making the listener forget whom it trusted. It barely does. What rescues the drum is the listener forgetting what it heard: with the old reading gone, a drum on bar 9 needs odds of nine to one, and with the record gone too it needs less than the three to one it has always had. A break that leaves even a hundredth of the old reading behind rescues nothing at all.

 A silent break ends the listener's reading only after as many bars as the reading has votes in hand. The choosing listener, every root late, and a break of silence before bar 3, 5, 9 or 13 through which it goes on expecting a downbeat once a bar at the reading it holds, each failure charging that reading 1 vote: the share of passages whose reading the silence ends, against the break's length in bars. Before bar 3, mean margin 2.9 votes: 1 bar 0%, 2 bars 0%, 4 bars 100%, 8 bars 100%, 16 bars 100%; half ended by 3 bars. Before bar 5, mean margin 6.8 votes: 1 bar 0%, 2 bars 0%, 4 bars 0%, 8 bars 88%, 16 bars 100%; half ended by 8 bars. Before bar 9, mean margin 20.7 votes: 1 bar 0%, 2 bars 0%, 4 bars 0%, 8 bars 0%, 16 bars 0%; half ended by 24 bars. Before bar 13, mean margin 38.5 votes: 1 bar 0%, 2 bars 0%, 4 bars 0%, 8 bars 0%, 16 bars 0%; half ended by 48 bars. The horizontal axis is logarithmic in one plus the length. Harmony and voice leading

A silent break has to outlast the reading's lead

A listener waiting through a break goes on expecting downbeats at the reading it holds, and each one that fails to come can be charged against that reading. Charged at a witness's vote apiece, the silence ends the reading only after as many bars as the reading has votes in hand: three bars after two bars of tune, eight after four, twenty-one after eight. A one-bar break would need a missing downbeat to cost twenty votes. And a silence that does end the reading leaves the record standing, which convicts the drum anyway.

 An announcement of the plan pays the cycle only when it is nearly always right. Recognised exactly: proportions told apart across the whole piece, averaged over announcements that name the true plan with the chance on the axis and otherwise one of the four wrong plans, for a listener who puts exactly that chance of its prior on the plan named; dashed, the listener told the plan. four, cycled: 0.2 44.2, 0.3 44.2, 0.4 44.2, 0.5 44.4, 0.6 46.6, 0.7 46.2, 0.8 46.6, 0.9 56.1, 0.95 56.8, 0.99 60.6; told 60.8, learning 44.2; four in blocks of two: 0.2 39.1, 0.3 39.1, 0.4 39.1, 0.5 37.3, 0.6 37.8, 0.7 40.9, 0.8 41.6, 0.9 42.0, 0.95 42.5, 0.99 44.8; told 45.0, learning 39.1; opening and closing alternated: 0.2 37.1, 0.3 37.1, 0.4 37.1, 0.5 38.6, 0.6 38.2, 0.7 38.6, 0.8 38.9, 0.9 38.8, 0.95 39.2, 0.99 39.4; told 39.5, learning 37.1. The cycle recovers half of what being told gives it only at 90 per cent reliability. Form and structure

An announced plan pays the cycle and punishes the blocks

A slow introduction, a title or a programme note can tell a first-time listener the plan of a piece before the first lapse — and be wrong. Trusted as far as it deserves, an announcement gives back half of what being told the plan is worth to four cycled functions only once it is right nine times in ten. But how far to trust it depends on the piece: a cycle rewards believing even a coin-toss announcement completely, while functions in blocks of two are made worse by an announcement right half the time, and a confident wrong one can cost a listener forty points of the first minute.

 After a silence long enough to end the reading, a bar of eight fill strokes hands the next reading to the drum. The choosing listener, every root late, a silent break before bar 3, 5 or 9 at one vote a failed downbeat, then a bar of 8 fill strokes for the true barline heard as the drum, then the drum on every downbeat: the share of passages ending on the barline against the silence's length (solid), with the silence alone dashed. Before bar 3: 0 bars 86%, 2 bars 100%, 4 bars 100%, 8 bars 100%, 12 bars 100%, 24 bars 100%; silence alone 4 bars 44%, 8 bars 44%, 12 bars 44%, 24 bars 44%. Before bar 5: 0 bars 0%, 2 bars 0%, 4 bars 53%, 8 bars 99%, 12 bars 100%, 24 bars 100%; silence alone 4 bars 0%, 8 bars 5%, 12 bars 7%, 24 bars 7%. Before bar 9: 0 bars 0%, 2 bars 0%, 4 bars 0%, 8 bars 0%, 12 bars 0%, 24 bars 99%; silence alone 4 bars 0%, 8 bars 0%, 12 bars 0%, 24 bars 0%. The horizontal axis is logarithmic in one plus the length. Harmony and voice leading

A fill cannot end a reading, but it can win the next one

A drum fill in the break before a late drum is a witness that speaks while the old reading still stands, so every stroke it plays is scored as disagreeing, and its credit falls to a coin toss within two or three strokes: at three to one it can cast 1.37 votes in all, against a lead of twenty by the middle of a tune. A fill cannot end a reading. But once a silence has ended it, the same fill sets the next one, and the record that convicted the drum through every length of silence starts convicting the bass instead: after twelve bars of silence before bar 5 the drum goes from 7 per cent of passages to all of them.

 A cadence hands the next section to a two-stroke fill only by emptying the reading, not by weakening it. The choosing listener, every root late, an honest drum from the bar after a cadence that leaves the share on the axis of the reading's lead (logarithmic, with "none" an emptied reading), with a bar of 2 fill strokes heard as the drum between them: the share of passages ending on the barline. Before bar 3 (mean lead 2.9 votes): 100% 7%, 50% 92%, 30% 100%, 20% 100%, 15% 100%, 12% 100%, 10% 100%, 7% 100%, 5% 100%, 3% 100%, 2% 100%, 1% 100%, 0% 100%. Before bar 5 (mean lead 6.8 votes): 100% 0%, 50% 0%, 30% 0%, 20% 26%, 15% 64%, 12% 87%, 10% 89%, 7% 90%, 5% 93%, 3% 95%, 2% 95%, 1% 96%, 0% 100%. Before bar 9 (mean lead 20.7 votes): 100% 0%, 50% 0%, 30% 0%, 20% 0%, 15% 0%, 12% 0%, 10% 0%, 7% 0%, 5% 1%, 3% 1%, 2% 1%, 1% 2%, 0% 100%. Before bar 13 (mean lead 38.5 votes): 100% 0%, 50% 0%, 30% 0%, 20% 0%, 15% 0%, 12% 0%, 10% 0%, 7% 0%, 5% 0%, 3% 0%, 2% 0%, 1% 0%, 0% 84%. Harmony and voice leading

A cadence has to empty the reading, not weaken it

A drum fill after a long enough silence wins the next section for an honest drum, because it speaks into a reading that is empty. A cadence might do the silence's work in a bar. It does, but only on one condition: before bar 9, a cadence that removes ninety-eight per cent of the listener's conviction leaves a two-stroke fill winning one passage in a hundred, and one that removes all of it lets the fill win every one. The fill turns the reading either way. What the last sliver decides is whether the fill's first stroke is recorded as a disagreement — and that record is what the late root then uses to win the reading back.

 A goal degree that returns is better marked once than twice, and best told apart. A cycle of 160 seconds, sixteen skeleton steps, a swell of beating in the step before each of four goal strokes; two of the four goals share a slendro degree and so a beat rate, at every pair of positions. Bits of position left unknown with both of that degree's swells, with the first or the second silenced, with neither, and with the two told apart by some other mark. steps 3 and 7: both 0.034, first silenced 0.049, second 0.062, neither 0.143, told apart 0.003; steps 3 and 11: both 0.146, first silenced 0.049, second 0.082, neither 0.128, told apart 0.003; steps 3 and 15: both 0.046, first silenced 0.049, second 0.067, neither 0.126, told apart 0.003; steps 7 and 11: both 0.066, first silenced 0.062, second 0.082, neither 0.202, told apart 0.003; steps 7 and 15: both 0.101, first silenced 0.062, second 0.067, neither 0.127, told apart 0.003; steps 11 and 15: both 0.077, first silenced 0.082, second 0.067, neither 0.188, told apart 0.003. Scales and modes

A goal tone that returns should swell only once

When two of a gamelan cycle's four goal strokes land on the same degree, their swells of beating sound alike, and a listener who hears one cannot tell which goal is coming. Across all 625 ways five degrees can fill four goals, 365 are served better by an elaboration that anticipates fewer goals than four. The repeated pair costs most at slow cycles, can cost more than every swell is worth at moderate ones, and is spared only when its first swell comes straight after the gong.

 A cadence that votes as a witness of its own moves the line from an emptied reading to a fiftieth of one. The choosing listener, every root late, a cadence before bar 9 leaving the share on the axis of the reading's lead (logarithmic, "none" an emptied reading); at the cadence its arrival casts one vote for the barline, heard as the tune's long note, as the drum, or as a witness of its own, or is silent; then 1 fill stroke heard as the drum, then the honest drum: the share of passages ending on the barline. a silent cadence: 100% 0%, 50% 0%, 30% 0%, 20% 0%, 15% 0%, 12% 0%, 10% 0%, 7% 0%, 5% 0%, 3% 0%, 2% 0%, 1% 0%, 0% 6%. heard as the tune: 100% 0%, 50% 0%, 30% 0%, 20% 0%, 15% 0%, 12% 0%, 10% 0%, 7% 0%, 5% 1%, 3% 1%, 2% 1%, 1% 10%, 0% 11%. heard as the drum: 100% 0%, 50% 0%, 30% 0%, 20% 0%, 15% 0%, 12% 0%, 10% 0%, 7% 0%, 5% 1%, 3% 1%, 2% 1%, 1% 2%, 0% 100%. heard as its own witness: 100% 0%, 50% 0%, 30% 0%, 20% 0%, 15% 0%, 12% 0%, 10% 0%, 7% 0%, 5% 1%, 3% 51%, 2% 90%, 1% 93%, 0% 93%. Harmony and voice leading

A cadence that votes must not vote as the drum

A cadence lands on a downbeat, so it could cast a vote for the barline at the moment it loosens the listener's grip. Whether that vote helps depends on whose it is taken to be. Heard as the tune's long note, it speaks with credit the tune spent long ago and is worth almost nothing. Heard as the drum, it is exactly one more stroke of fill, and is convicted as the fill's first stroke was. Heard as a witness of its own, it carries a third of a vote and no record to lose — and that moves the line from an emptied reading to a fiftieth of one.

 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. Rhythm and metre

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.

 A mark does little until it is nearly always heard, and then it does everything, whichever stroke was moved. The standard bell pattern with a varied cycle every 3, both pulses heard half the time, a listener who learns how often strokes are lost: the surprise a step once placed, averaged over the second to sixth cycles, against how reliably each stroke's mark is heard. The plain bell reads 0.0078 and the other order of the gaps 0.0016. the bell shifted: 50% 0.0252, 60% 0.0238, 70% 0.0204, 80% 0.0154, 90% 0.0092, 95% 0.0062, 100% 0.0027; a new figure: 50% 0.0055, 60% 0.0055, 70% 0.0052, 80% 0.0048, 90% 0.0041, 95% 0.0036, 100% 0.0025; nothing moved: 50% 0.0076, 60% 0.0076, 70% 0.0075, 80% 0.0070, 90% 0.0057, 95% 0.0047, 100% 0.0026. Rhythm and metre

A bell can mark a cycle without moving a stroke

A lead drummer's varied cycle helps a listener or misleads one, according to which stroke it moves, because a moved stroke can make the bell look as if it had started somewhere else. Give the varied cycle a different sound as well and the distinction goes. Heard nine times in ten and a misleading variation is no longer one; heard always, it no longer matters which stroke moved, or whether any did. A mark heard seven times in ten does almost nothing, and the arithmetic of that threshold is the same as a coin's.

 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. Rhythm and metre

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.

All themes