Acoustics & Music Theory
why it sounds like that
Concept

Repetition — where it appears

The recurrence of material already heard in the same piece, which is the most common event in music and the material most forms are built from. How much of a piece is a repeat, and how long a listener will wait for one, are both measurable.

Named by 19 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

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.

form · Repetition 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.

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.

form · Repetition 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.

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.

form · Repetition 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.

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.

form · Phrase 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.

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.

form · Cyclic-rhythm 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.

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.

harmony · Voice-leading 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.

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.

form · Repetition 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.

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.

form · Repetition 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.

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.

form · Repetition 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.

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.

perception · Repetition 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.

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.

form · Repetition 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.

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.

form · Repetition 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.

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.

form · Proportion 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 lapse of attention that ends costs a form nothing unless a section is heard twice in a row. How many proportions between 1 : 1 and 3 : 1 a listener who knows the form tells apart across a whole 480-second piece, counted at every level of 2.0 s, 16 s, 64 s, 128 s, with 10 per cent of lapses taking every count at once and lasting 10 seconds on average, after which the listener re-anchors at the section the music says they are in. through-composed, ABCDEFGH: 77.1, 0.0 per cent of returns misplaced; rondo, ABACABAD: 76.4, 0.0 per cent of returns misplaced; sonata, exposition repeated, ABABCDAB: 76.6, 0.0 per cent of returns misplaced; minuet and trio, halves repeated, AABBCCDDAB: 50.2, 5.7 per cent of returns misplaced; thirty-two-bar song, twice, AABAAABA: 47.7, 4.3 per cent of returns misplaced; strophic, AAAAAAAA: 35.4, 10.1 per cent of returns misplaced. With lapses that never end every form reads 4.5, and with every returning listener placed exactly 77.2. The forms that lose most are the ones that play a section twice in a row, not the ones with the most repetition.

A repeat sign loses a listener that a refrain does not

A lapse of attention loses every count a listener is keeping, and when it ends the listener has only the music to say where they are. A listener who knows the form comes back to a rondo's refrain and is almost never misplaced, because the episodes between refrains make only one of them reachable. They come back near the join of a section repeated immediately and are misplaced nearly half the time, and a form built of such repeats loses between a third and a half of the precision a count at every level had restored. One of the most familiar signs in notation does more damage than the most repetitive form.

form · Proportion A first hearing costs the through-composed piece most and the rondo almost nothing. How many proportions between 1 : 1 and 3 : 1 a listener tells apart across a whole 480-second piece of eight one-minute sections, counted at every level, with a tenth of lapses taking every count at once and lasting 20 seconds on average, for a listener who knows the piece, one who knows only its plan of sections, one who knows the style's five forms, and one who knows nothing of the style. through-composed, ABCDEFGH: knows the piece 75.7, knows only the plan 62.7, knows the style 61.4, knows nothing 59.3; rondo, ABACABAD: knows the piece 56.2, knows only the plan 56.0, knows the style 54.6, knows nothing 51.6; sonata, ABABCDAB: knows the piece 61.0, knows only the plan 59.7, knows the style 57.8, knows nothing 53.9; song form, AABAAABA: knows the piece 35.3, knows only the plan 35.3, knows the style 34.5, knows nothing 32.8; strophic, AAAAAAAA: knows the piece 25.9, knows only the plan 25.9, knows the style 25.5, knows nothing 25.0. What a first hearing lacks is not the plan but the ability to recognise a section never heard: new music says only that it is new.

New music says only that it is new

A listener who knows a piece comes back from a lapse of attention and is placed at once by whatever they hear, and a through-composed piece, which never repeats anything, is the easiest of all forms to come back to. A listener hearing it for the first time is not so lucky. What they hear on returning is either something they have heard before, which names itself, or something they have not, which says only that it is new. For a rondo that costs almost nothing. For the through-composed piece it costs its whole lead: at lapses of half a minute or more, a first-time listener is placed no better in it than in a rondo — and being told the piece's plan in advance gives back almost none of the difference.

form · Proportion A function label gives back a first hearing's place only if it does not recur. A through-composed piece of eight one-minute sections, heard for the first time by a listener told its plan, with a tenth of lapses taking every count and lasting 40 seconds on average: proportions told apart across the whole piece, against the chance that on returning into music not yet heard the listener recognises the section's function — opening (O), continuing (C), intensifying (I) or closing (Z) — for four arrangements of the functions. OIOIOIOI, two functions alternating: 0% 29.8, 25% 31.5, 50% 33.6, 75% 36.2, 100% 39.5. OCCCIIIZ, an arc in long runs: 0% 29.8, 25% 31.4, 50% 33.3, 75% 35.7, 100% 38.6. OOCCIIZZ, four in blocks of two: 0% 29.8, 25% 32.1, 50% 35.1, 75% 39.1, 100% 45.0. OCIZOCIZ, four, cycled: 0% 29.8, 25% 33.1, 50% 37.8, 75% 45.4, 100% 60.8. With no label the listener reads 29.8; knowing the piece, 66.1.

A section that says what it is for places a first hearing

A first-time listener who loses count in a through-composed piece comes back into music they have never heard, and new music says only that it is new. Suppose instead each section announces what it is doing — opening, continuing, intensifying, closing — and the listener recognises it. How much of their place that gives back depends hardly at all on how many functions there are and almost entirely on how soon the same function comes back. Four functions cycled restore 85 per cent of what a first hearing costs at forty-second lapses; the same four in the familiar arc, two sections each, restore 42; two functions alternating restore 85 per cent at twenty seconds and 14 at a minute.

form · Proportion Four functions cycled lose their advantage the moment neighbours are confused. A through-composed piece of eight one-minute sections, a first hearing, a listener told the plan of functions, lapses of forty seconds on average: proportions told apart across the whole piece against the chance that a recognised function is heard as its neighbour in the arc opening, continuing, intensifying, closing, for five arrangements. OCIZOCIZ, four, cycled: 0% 60.8, 5% 47.7, 10% 44.1, 15% 41.3, 20% 39.0, 30% 36.2, 40% 35.1, 50% 34.9. OOCCIIZZ, four in blocks of two: 0% 45.0, 5% 40.0, 10% 35.3, 15% 34.4, 20% 33.0, 30% 32.3, 40% 31.3, 50% 31.5. OIZOIZOI, three, cycled: 0% 51.4, 5% 43.2, 10% 41.9, 15% 40.8, 20% 39.8, 30% 37.9, 40% 37.1, 50% 36.7. OZOZOZOZ, opening and closing alternated: 0% 39.5, 5% 39.5, 10% 39.5, 15% 39.5, 20% 39.5, 30% 39.5, 40% 39.5, 50% 39.5. OCOCOCOC, opening and continuing alternated: 0% 39.5, 5% 37.3, 10% 35.3, 15% 33.7, 20% 31.8, 30% 31.0, 40% 30.4, 50% 30.6. Unlabelled, 29.8; knowing the piece, 66.1.

Two functions far apart outlast four that blur

A first-time listener who loses their place in a through-composed piece gets most of it back if each section announces what it is doing and the same function does not come back too soon: four functions cycled restore 85 per cent of what a first hearing costs. That rests on hearing each function exactly. Let a listener mistake a function for its neighbour in the arc — a continuation for an intensification — one time in ten, and the cycle keeps less than half of its gain. Two functions at the opposite ends of the arc, opening and closing alternated, lose nothing to such confusion, and from a little over one confusion in five no arrangement does better.

form · Proportion A listener who has to learn the plan loses most where the plan said most. Eight one-minute sections, a first hearing, lapses of forty seconds: proportions told apart across the whole piece, against the chance a recognised function is heard as its neighbour in the arc, for a listener told the plan of functions (dashed) and one who must infer it from the functions heard before each lapse, among five plans it thinks equally likely (solid). four, cycled: told 60.8, 44.1, 39.0, 34.9; learning 44.2, 38.5, 35.4, 33.3; four in blocks of two: told 45.0, 35.3, 33.0, 31.5; learning 39.1, 33.2, 32.4, 30.7; opening and closing alternated: told 39.5, 39.5, 39.5, 39.5; learning 37.1, 37.1, 37.1, 37.1 at confusions of 0, 0.1, 0.2 and 0.5. A listener who knows the piece reaches 66.1; one with no functions, 29.8.

A listener who learns the plan keeps only the far pair's immunity

A first-time listener who recognises what each section of a through-composed piece is doing gets most of their place back after a lapse — if they know the plan of functions. A listener who has to infer the plan from the sections heard so far loses most where the plan said most: four functions cycled give up more than half of what their labels were worth, nearly all of it to lapses in the first two minutes, before the second function has settled which plan the piece has. Opening and closing alternated lose a little and stay immune to confusing neighbouring functions. The far pair's immunity was never knowledge of the plan; it was the arc.

form · Proportion

Named alongside it

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

RefrainMemory decaySelf-similaritySegmentationHierarchyMusical formOstinatoExpectationNoveltyProportionDescription lengthPeriodicity

All concepts