Acoustics & Music Theory
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Series

Proportion — the series

13 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Six named proportions, as blurred as the durations that make them. Six proportions between two parts of a piece — 1 : 1, 4 : 3, 3 : 2, golden section, 2 : 1, 3 : 1 — placed on one axis by the logarithm of the ratio of the longer part to the shorter, and drawn as bars one criterion wide (d′ = 1) for a listener timing both parts with a Weber fraction of 7%, 15%, 35%. Bars that overlap are proportions that listener cannot tell apart. At 7%, 4 of 5 neighbouring pairs stay apart; at 15%, 3 of 5 neighbouring pairs stay apart; at 35%, 0 of 5 neighbouring pairs stay apart.

    A proportion is only as fine as its two durations

    Analyses of form measure proportions in bars and report them to three figures — a climax at 0.618, a section in the ratio 3 : 2. A listener has each part only as an estimate of how long it lasted, and a ratio of two estimates is blurred by both. Timed as well as anyone times a single second, eleven proportions fit between 1 : 1 and 3 : 1; timed from memory over minutes, two do. The golden section is told from 3 : 2 only below a Weber fraction of 5.4 per cent.

    part 1 · form
  2. 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.

    part 2 · form
  3. A count is least reliable at both ends and best in the middle. How precisely a listener knows the length of a section they are counting, against how many units long it is, at three kinds of timing judgement. A slip — one unit miscounted, at 2% a unit — accumulates as a random walk, so its relative cost FALLS as the section lengthens. A lapse — the count lost altogether, at 1% a unit — compounds, so the chance of still having the count falls geometrically and a long enough section is certain to lose it. A listener who has lost the count is back to timing, so the two failures mix into a floor. Against a Weber fraction of 7.5% the count is worth most at 21 units, where it is 1.7 times finer than timing, and falls back under a quarter better by 98 units; Against a Weber fraction of 15% the count is worth most at 10 units, where it is 2.4 times finer than timing, and falls back under a quarter better by 101 units; Against a Weber fraction of 38% the count is worth most at 4 units, where it is 3.7 times finer than timing, and falls back under a quarter better by 102 units. The length at which it stops being worth much is nearly the same in all three, because it is set by the lapse rate alone.

    A count is not an estimate

    Both established routes to a proportion are estimates — a duration timed, blurred by a Weber fraction, and a duration stored, biased by what was new. A listener who has induced a hypermetre has a third, and it is exact until it fails. It fails two ways that pull opposite: a slip miscounts one unit and its relative cost falls as the section lengthens, while a lapse loses the count entirely and its chance compounds. The mixture has a floor at about four units, where counting is 3.7 times finer than timing, and it is worth almost nothing past a hundred.

    part 3 · form
  4. Timing blurs a whole form evenly; counting sharpens it downward. A piece of 480 seconds divided 7 times, each level half the length of the one above, with how many proportions between 1 : 1 and 3 : 1 a listener can tell apart at each. Timed, the answer is 2.1 at the top and 5.2 at the bottom, a spread of 2.4 — because a timing judgement's Weber fraction is a step function of duration and almost every level of a piece falls in one step of it. Counted, in units of 2 seconds, the answer runs 2.2 to 10.3, a spread of 5.5. At no level does timing separate 3 : 2 from the golden section.

    A form is sharp at the bottom and vague at the top

    A movement is divided into sections, each into phrases, each into bars, and every level is a ratio of two estimates. Timed, the hierarchy is almost uniformly blunt — 2.1 distinguishable proportions at the top and 5.2 at the bottom, because a Weber fraction is a step function of duration and six of a piece's seven levels fall in one step of it. Counted, the same hierarchy runs from 2.2 to 12.2 and sharpens monotonically downward. At no level of either does timing separate 3 : 2 from the golden section.

    part 4 · form
  5. The golden section and an equal division are one judgement. Where a boundary falls in a piece, as a share of its length, with the band a listener cannot tell from the golden section shaded. A stretch of minutes is judged with a Weber fraction of about 38%, so one criterion's worth of ratio spread around 0.618 covers everything between 0.492 and 0.730 — a quarter of the piece wide, and containing the halfway point. 1 : 1 and 4 : 3 and 3 : 2 and golden section and 2 : 1 are inside it. A claim that a climax falls at the golden section rather than at the middle is, at this resolution, not a claim about anything a listener could hear.

    A golden section is a coin toss with six coins

    An analysis that reports a climax at 0.618 of a piece has not tested one prediction; it has looked at a piece with several defensible boundaries and reported whichever landed nearest. The rate at which that happens under no hypothesis is one line of arithmetic, and the tolerance it needs is not a number chosen on the page — it is the blur a listener's own timing puts on the judgement. Over a stretch of minutes that blur covers everything from 0.492 to 0.730 of the piece, which contains the halfway point, and six candidate boundaries produce a hit eighty per cent of the time.

    part 5 · form
  6. Checkpoints sharpen the middle of a form and leave its top vague. A piece of 480 seconds divided 7 times, with how many proportions between 1 : 1 and 3 : 1 a listener tells apart at each level: timed, counted in 2-second units, and counted with a second count of 16-second phrases that can mend a lapse in the first. 480 s: 2.1 timed, 2.2 counted, 3.9 with 0 per cent of lapses shared and 2.5 with 50 per cent of lapses shared; 240 s: 2.1 timed, 2.5 counted, 7.1 with 0 per cent of lapses shared and 3.1 with 50 per cent of lapses shared; 120 s: 2.1 timed, 3.1 counted, 13.2 with 0 per cent of lapses shared and 4.0 with 50 per cent of lapses shared; 60 s: 2.1 timed, 4.1 counted, 19.4 with 0 per cent of lapses shared and 5.4 with 50 per cent of lapses shared; 30 s: 2.1 timed, 5.4 counted, 18.1 with 0 per cent of lapses shared and 7.2 with 50 per cent of lapses shared; 15 s: 5.2 timed, 12.2 counted, 12.2 with 0 per cent of lapses shared and 12.2 with 50 per cent of lapses shared; 7.5 s: 5.2 timed, 10.3 counted, 10.3 with 0 per cent of lapses shared and 10.3 with 50 per cent of lapses shared. With the two counts failing independently, the level of 60 seconds goes from 4.1 to 19.4, and the whole piece only from 2.2 to 3.9.

    Checkpoints sharpen the middle of a form, not its top

    A listener who counts bars loses the count somewhere in a long section and is thrown back on timing the whole of it. A listener who also counts phrases can mend the lapse at the last phrase. If the two counts fail independently, the level a minute long goes from four distinguishable proportions to nineteen; the whole eight-minute piece goes only from two to four, because thirty phrases are long enough to lose a count as well. And if a fifth of lapses take both counts at once, three quarters of the gain is gone.

    part 6 · form
  7. A count at every level turns a form's vague top into its sharpest level. A piece of 480 seconds divided 7 times, with how many proportions between 1 : 1 and 3 : 1 a listener tells apart at each level, counted three ways: in 2.0 s units alone, with one checkpoint count every 16 s, and with a count at every level of 2.0 s, 16 s, 64 s, 128 s, each mending the one below, every count failing independently. 480 s: 2.2, 3.9 and 76.4; 240 s: 2.5, 7.1 and 54.0; 120 s: 3.1, 13.2 and 38.1; 60 s: 4.1, 19.4 and 19.4; 30 s: 5.4, 18.1 and 18.1; 15 s: 12.2, 12.2 and 12.2; 7.5 s: 10.3, 10.3 and 10.3. The whole piece goes from 2.2 with one count and 3.9 with one checkpoint to 76.4 with a count at every level.

    Only a lapse of attention keeps a form's top vague

    A listener who keeps a count at every level of a form — bars, phrases, periods, sections — each mending the one below when it lapses, does not have a vague top at all. The whole eight-minute piece becomes the sharpest level, at seventy-six distinguishable proportions against two. Failures shared between neighbouring counts barely dent it, because a shared failure has to climb every level to reach the top. What puts the vagueness back is a lapse that takes every count at once, and one lapse in a hundred of that kind is enough.

    part 7 · form
  8. 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.

    part 8 · form
  9. 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.

    part 9 · form
  10. 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.

    part 10 · form
  11. 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.

    part 11 · form
  12. 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.

    part 12 · form
  13. 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.

    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.

    part 13 · form

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