Acoustics & Music Theory
why it sounds like that
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Metre induction — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 -4, 3/4 12, 6/8 12, 12/8 as 3+3+3+3 12, so 3/4 and 6/8 and 12/8 as 3+3+3+3 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

    The beat is inferred, and sometimes wrongly

    A bar line is not in the sound. One sequence of onsets supports a reading in four, in three and in six, and which one a listener settles on comes from a small set of preference rules rather than from anything in the signal — which is why the same recording can be heard two ways and why nobody can be talked out of either.

    part 1 · rhythm
  2. The metrical levels of 120 bpm, against the window. The range of inter-onset intervals that can be heard as a beat at all, from about 100 to 2000 milliseconds, with the preferred rate near 550. Each mark is one metrical level of a piece at 120 beats a minute. Which of them a listener taps is decided by which falls nearest the preferred rate, not by which one the notation calls the beat.

    The beat has a preferred rate, and it is not the notation's

    Metre is a hierarchy of levels and only one of them is tapped. Which one is decided by the listener's own clock rather than by the time signature: the window in which a series of events can be a beat at all runs from about a tenth of a second to two seconds, with a marked preference near half a second — so at the extremes of tempo the notated beat and the felt one part company, predictably.

    part 2 · rhythm
  3. 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.

    part 3 · form
  4. One pattern, four metres. The same 8-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are downbeat on step 1 -1, downbeat on step 2 -7, downbeat on step 3 -1, downbeat on step 4 -1, so downbeat on step 1 and downbeat on step 3 and downbeat on step 4 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

    The right period at the wrong phase

    Finding the beat is two problems, not one. How far apart the beats are, and where the first one is. This site's rule set answers the first confidently on a tresillo and returns a three-way tie on the second — the same score for the downbeat on step one, step four and step seven — which is not a near miss but no answer at all.

    part 4 · rhythm
  5. Syncopation against a bar of 8. An 8-step pattern with 3 onsets, against the metrical weights of its bar. A position's weight is zero on the downbeat and one lower at each level down the subdivision tree, drawn here as the depth of the bar hanging beneath it. A note on a weak position followed by a rest on a stronger one costs the difference. the tresillo, in a bar of eight scores 2 — the note at step 4 against the rest at step 5, costing 2.

    Syncopation is a number about the metre

    Longuet-Higgins and Lee price a syncopation at the metrical weight a note skips over. That makes it computable, and it makes it a property of a pair rather than of a rhythm — the son clave scores 4 read from step one, 2 from step three and 8 from step four, and not one onset has moved. Worse, the induction rules pick very nearly the reading that scores lowest.

    part 5 · rhythm
  6. One pattern, four metres. The same 16-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are downbeat on step 1 0, downbeat on step 2 -12, downbeat on step 3 -12, downbeat on step 4 0, so downbeat on step 1 and downbeat on step 4 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

    The beat that is never sounded

    A listener who has heard four bars of a groove and then hears two bars with the downbeats taken out does not move the downbeat. This site's rule set does, every time, on every pattern tried — and the direction it moves in says exactly what kind of model would be needed instead.

    part 6 · perception
  7. Two metres as trees. Two metres — 9/8 as 3 × 3, 3 + 3 + 3; 9/8 as 2+2+2+3, 2 + 2 + 2 + 3 — each drawn as the bar dividing into beats and the beats dividing again. Simple and compound metres have the same number of beats and differ in the second division; an additive metre has beats of unequal length, which no single division produces.

    No term for an unequal beat

    A Balkan bar of nine is four beats, three short and one long. The induction rules lay their strong positions at a fixed period, so the only readings of nine it can offer are nine, three and one — and for a bar of seven, where seven is prime, the only readings are seven and one. The right answer is not among the candidates.

    part 7 · rhythm
  8. A tracker following a tempo that will not stay still. The tracker's error as a fraction of a beat, against beat number, for 3 rates of tempo change with a period-correction gain of 0.2. It settles at 0.025 of a beat behind at 0.5 per cent a beat, 0.092 of a beat behind at 2.0 per cent a beat, 0.206 of a beat behind at 5.0 per cent a beat. It does not lose the beat; it lags, by very nearly the rate divided by the period-correction gain, and the lag reaches a quarter of a beat at 6.3 per cent a beat — at which point the tracker is nearer the wrong onset than the right one.

    A metre has to be able to change its mind

    Replace the scoring function with a phase-corrected oscillator and the two failures reported earlier separate. The beat now survives two silent bars, drifting 23 milliseconds of a 560-millisecond beat. But it does not lose a moving tempo so much as lag behind it, by the rate over the correction gain — and a cadential ritardando that halves the tempo in eight beats is faster than the model can follow.

    part 8 · form
  9. note length against the onsets. Every candidate metre's fit to a 16-step pattern with 4 onsets, plotted against how strongly note length is weighted. At a gain of zero the scoring is the onset-only one every earlier model used, and the winner is step 1 and step 4. At a gain of 0.05 the answer becomes step 4. 2 candidates are exactly flat — step 2 and step 3 have no onset on any strong position, so there is no credit for the cue to multiply and no weighting of it can move the line.

    What the onsets left out

    Eight essays induce a metre from a list of ones and zeros, and every failure they recorded was argued about as a failure of the rules. Two of the three are not. Note length is already in that list and the scoring throws it away: put it back and the son clave's two-way tie resolves to the notated downbeat. But the groove with its beat removed cannot be repaired by any cue at any strength, and the reason is arithmetic rather than empirical — the true phase has no onset on any of its strong positions, so there is no credit for a cue to multiply and its line is exactly flat.

    part 9 · form

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