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.

All themes