Acoustics & Music Theory
why it sounds like that
Concept

Hypermetre — where it appears

Metre one level up, in which the bar is the beat and a group of bars is the unit that carries the strong and weak positions. It is what makes a four-bar phrase feel like a single measure, and it is inferred by the same rules one level down.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Five signals, computed separately, and no total. The five components of closure for 4 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted.

An ending that exists so a bigger one can

Half of the cadences in tonal music are built to fail. A phrase that stopped convincingly at bar four would be a piece four bars long, so the ending at bar four is engineered to arrive and not to settle — and the components it withholds are exactly the ones its partner at bar eight supplies. Closure is nested, and the nesting is what turns two phrases into one thing.

form · Closure The same induction, one level up. Bar-level onsets from thirty-two-bar AABA — a bar is marked where a section or a key begins — scored against hypermetres of 2, 3, 4, 6, 8 bars with the identical function the beat-level figures use. The best-fitting period is 8 bars, which at 108 beats a minute lasts 17.8 seconds.

The bar above the bar

A four-bar group is a bar whose beats are bars. That is not an analogy — it is the same computation, and the same metre-induction model produces one when it is handed bars instead of beats, unchanged. What decides where the hierarchy of levels stops is not in the arithmetic at all, and it is a number the phrase essay already measured.

form · Metre induction 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.

form · Proportion 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.

form · Proportion 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.

form · Proportion 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.

form · Proportion

Named alongside it

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

PhraseMusical formProportionWeber fractionDurationPerceptual presentHierarchyMemory decayAntecedent consequentCadenceClosureDownbeat

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