Acoustics & Music Theory
why it sounds like that
Concept

Clave — where it appears

A five-onset timeline in sixteen steps, played through a whole piece as a frame the rest of the ensemble is placed against. It is not maximally even, and measurement says what it buys instead: a listener can tell where in the cycle they are after nine steps of it.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Euclidean up to the one thing a timeline is for. Five named timelines, each drawn above the Euclidean pattern with the same number of onsets in the same number of steps, with the rotation between them found by search. 3 of 5 are rotations of the Euclidean pattern and 2 are not Euclidean at any rotation — son clave, 3–2 and rumba clave, 3–2, whose gap sequences are 3·3·4·2·4 and 3·4·3·2·4 against the algorithm's 3·3·3·3·4. Where the match holds it holds only up to rotation, and a rotation is not a small difference: the algorithm has no way to produce a starting position, and a starting position is what a timeline is.

The rotation the necklace cannot see

Ask Bjorklund's algorithm for the world's timelines and the usual answer is that it produces them. Search every rotation of each Euclidean pattern for a match and the answer is more interesting: the tresillo is E(3,8) exactly, the bossa-nova and the standard bell pattern are rotations of theirs, and the son and rumba claves — the two best-known timelines in the world — are not Euclidean at any rotation whatever. Where the match does hold it holds up to a starting position, and a starting position is the one thing a timeline is.

rhythm · Euclidean rhythm 5 onsets in 16, by evenness against locatability. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how much of the cycle has to be heard on average before its position is known. The two objectives are opposed — the perfectly even pattern is the hardest to locate and the most clustered ones are the easiest — so there is no best pattern, only a frontier, and 22 of the 273 are on it. son clave and the bossa-nova pattern are among them. The named timelines are marked.

What the clave buys with its unevenness

The essay before this one found that the two best-known timelines in the world are not Euclidean at any rotation, and asked whether they maximise something else. They do: how quickly a fragment of the cycle says where in the cycle it is. The son clave locates itself in nine of its sixteen steps where the even pattern needs fifteen — and of two hundred and seventy-three patterns, twenty-two are on the frontier between the two objectives and the son clave is one of them.

rhythm · Euclidean rhythm Five rulers on 5 onsets in 16, and what each frontier keeps. The same 273 rotation classes and the same locating cost, with the frontier recomputed against five different definitions of unevenness. The frontier's size varies from 7 to 22 of 273, so how exclusive membership is depends on the ruler. Of the named timelines of this size, son clave is on 4 of the five, rumba clave is on 0 of the five, the bossa-nova pattern is on 4 of the five. The four rulers that measure how large a pattern's departure from even is rank the whole census together, at no worse than 0.89 between any two of them; the one that counts how many distinct gap lengths a pattern uses rather than how large its departures are is anti-correlated with them at -0.26, and its frontier holds none of the named timelines and does not hold the perfectly even pattern.

The frontier and the ruler

A Pareto frontier is a claim about two quantities and only one of them was measured. The locating cost is exact; the unevenness it was traded against was one choice among several. Under four different rulers the son clave stays on the frontier and the frontier shrinks from twenty-two of 273 patterns to seven, so its membership means more rather than less — but the bembe stops being dominated, the two claves stop being equally uneven, and one plausible measure destroys the result entirely.

rhythm · Euclidean rhythm 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 5 censuses: the floor of the longest gap, the ceiling of the syncopation total, and how few patterns reach both. For each census of k onsets in n steps, counted by rotation class: 3 in 8: patterns 7, floor of longest gap 3, at the floor 1, ceiling of total 14, at the ceiling 2, at both 1; 5 in 8: patterns 7, floor of longest gap 2, at the floor 2, ceiling of total 12, at the ceiling 2, at both 2; 5 in 12: patterns 66, floor of longest gap 3, at the floor 6, ceiling of total 28, at the ceiling 6, at both 6; 7 in 12: patterns 66, floor of longest gap 2, at the floor 3, ceiling of total 20, at the ceiling 38, at both 3; 5 in 16: patterns 273, floor of longest gap 4, at the floor 13, ceiling of total 79, at the ceiling 9, at both 9. The rotation-total of syncopation is taken against a metre of 2 × 2 × 2 for 8 steps, 2 × 2 × 3 for 12 steps, 2 × 2 × 2 × 2 for 16 steps. In every census drawn the evenly spaced pattern is at the floor and at the ceiling.

The other censuses keep evenness, not locating

Five onsets in sixteen suggested two rules: a named timeline's longest gap is the shortest its census allows, and its syncopation totalled over every rotation is the largest. Nine of ten named timelines across five censuses meet both — but so does the evenly spaced pattern, in all twenty-one censuses from two onsets in eight to eight in sixteen, and in four of those five censuses the named timeline is the evenly spaced pattern. What the rules describe is evenness. The quantity the frontier was built on points the other way: the evenly spaced pattern is the slowest of its census to locate in nineteen of twenty-one, the standard bell pattern is the slowest of all sixty-six patterns of seven in twelve, and among the orders of its own gaps a named timeline is usually both the most even and the slowest to locate. The son clave, whose most even order is also the fastest, is the exception the locating story was told about.

rhythm · Euclidean rhythm

Named alongside it

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

Euclidean rhythmMaximal evennessRotationTimelineSyncopationCyclic rhythmEntropyInter-onset intervalNecklaceOnset patternEnumerationOptimisation

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