Acoustics & Music Theory
why it sounds like that
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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 metre scores 2 — the note at step 4 against the rest at step 5, costing 2.
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 metre scores 2 — the note at step 4 against the rest at step 5, costing 2.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 15 of its 15 placements carry sound, built from the same numbers as the drawing, offering these buttons: the pattern, the same onsets, accented as 2 × 2 × 3, the same onsets, accented as 2 × 3 × 2, the same onsets, accented as 3 × 2 × 2, the same onsets, accented every step struck, the same onsets, accented from step 1, the same onsets, accented from step 12, the same onsets, accented from step 2 and 10 more.

Called by 7 essays

the blast radius of changing it

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.

rhythm · Metre induction 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.

rhythm · Metre induction 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.

perception · Metre induction One pattern, four metres. The same 12-step onset pattern read under 3 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 3/4 -4, 6/8 12, 2/4 -4, so 6/8 wins. The page prints 3/4, which the model does not prefer — it ranks 6/8 above it. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

The time signature is a claim

A bar line is not a measurement. It is a claim about where the accents are, made before the sound exists, and there is a metre-induction model here that can be handed the same onsets and asked whether it agrees. On a hemiola it does not: the page says three and the model says six, by a margin of twelve against minus four. And on a bar of seven the signature is not even a candidate — what decides the reading is the beaming, which the signature does not contain.

rhythm · Notation What the joint search changes, and what it never changes. Over 552 constructed passages of eight slots with rests, how often the joint reading differs from the pipeline's. The chord differs in 29 per cent and the barline in 31, with both differing in 20. The key differs in 0 per cent — never — because the key is read from a pitch-class histogram, which does not know where the bar starts or which notes are chord tones. Two of the three decisions are entangled and the third is not.

Three decisions that constrain each other

Every model here decides one thing at a time — the key from the pitch classes, the metre from the onsets, the chords from the metre — and an earlier essay ended by saying a listener does all three at once. Resolving them jointly costs a hundred and fifty-seven times the search and changes the reading of two passages in five. It never once changes the key, and the reason it cannot is the reason the whole account is built the way it is.

harmony · Progression The number nobody has moves the size and not the order. The mean total surprise per chord change, against how much the two surprises share. At zero they are independent and the total is their sum; at one they are the same event and the total is the larger of the two. The mean falls by a factor of 1.53 across that whole range, which is the size of the thing a corpus would settle. The ordering of the events by total surprise does not move at all until the very end: 5 of the 6 correlations swept give exactly the ordering independence gives, and only perfect dependence changes it, by 3 places out of 7. The most surprising event in the passage is the same one at every correlation. So the corpus three separate accounts have recorded wanting would change what this figure reports and not what it concludes.

Two surprises and one event

A chord change is surprising twice over — in which chord it is, and in when it comes — and a listener meets one event. Adding two surprises needs to know how much they share, which is a fact about a repertoire nobody has. Sweeping it instead: the total moves by half, the ordering does not move at all, and the most surprising moment in a passage is the same one whatever the answer turns out to be.

harmony · Tonal-expectation The longest silence against the spread of the gaps. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how long its longest gap is. The two rank the census together at a Spearman correlation of 0.94, so the longest silence is not a second thing to know about a timeline; it is the same thing measured more crudely. Against the other axis on the frontier — how much of the cycle has to be heard before a listener knows where in it they are — it correlates at -0.02, which is nothing. The named timelines are marked, and all of them sit at 4, the shortest longest-gap any pattern of this size can have.

The longest silence is not a third axis

The frontier between evenness and locatability was drawn twice and both times against two quantities, with the third job a timeline does left uncomputed. Computed, neither candidate for it is a third quantity: the longest silence ranks the whole census with the spread of the gaps at 0.94, and syncopation is not a property of a cycle at all — every one of the 273 patterns changes its count when the bar line moves. What the request was actually asking for is a cap rather than an axis, and under the tightest one the census admits thirteen patterns and every named timeline is among them.

rhythm · Euclidean rhythm

All figures · What can be heard