Acoustics & Music Theory
why it sounds like that
Concept

Entropy — where it appears

The average information content of an event drawn from a distribution, in bits — how surprising the outcomes are on average. Applied to a chord progression it says how much a listener's expectations narrow what can come next, and it is bounded above by the logarithm of how many things could.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

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 How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire.

Surprise is a number

The chord that did not come was described rather than measured. Its measure is the information content of what did arrive, and a model of the probability has been to hand since the key-finding essays — eight root-motion weights, ordinal and stipulated. Reading them as a distribution prices a deceptive cadence at 2.71 bits against a perfect one's 1.85, and turns up the fact that the largest of the eight had never been read by anything.

harmony · Tonal-expectation 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 A third of the timing surprise is paid for by nothing happening. Every beat of a bar of 4/4 at 1 chord change a bar, with what it costs a listener in expectation. The lower block is the arrival — the chance the chord changes here times what that change costs to be surprised by. The upper block is the hold, which is paid when the chord does not change and is charged at every beat rather than at every event. Over the bar the arrivals come to 2.61 bits and the non-arrivals to 1.29, so the term never spent before is 33 per cent of the total rather than most of it. The two together are exactly the binary entropy of each beat's own hazard, which is why the bar's whole timing bill is 3.90 bits and cannot be raised by rearranging where the changes fall.

The surprise of nothing happening

A hazard charges a listener twice — once when the chord changes and once, quietly, at every beat it does not. The second term was expected to dominate, because there are more beats than changes. It is a third of the bill at one chord a bar and never reaches a half at any rate a metre survives, because the cost of a beat where nothing happens is second-order small.

harmony · Tonal-expectation The term that was owed, and the corpus cannot hold it. For each of the three tunes everything here is measured on, how many of its notes have a duration that differs from the gap to the next onset. The answer is none, in 101 notes: these tunes are stored as a list of pitches and lengths with no rests in them, so a note's duration IS its inter-onset interval and conditioning one on the other leaves exactly zero bits. That is a fact about the representation rather than about music. The prediction was that the term would be small, and it could not have been known that the corpus would make it identically zero — which means the prediction cannot be tested here and the exceptions have to be priced directly.

A note lasts until the next one starts

Pricing where a note is against which note it is left duration as the term it had not, with a prediction that it would be small. Measured on the three tunes these readings are built on, it is exactly zero — and it is zero by construction, because those tunes are stored as pitches and lengths with no rests in them, so every duration is its own inter-onset interval. The prediction cannot be tested on the corpus that produced it. Priced directly, a rest costs 0.67 bits a note where a tenth of the notes have one, which is not well under half a bit.

scales · Notation A tie is charged twice, and the second charge is the larger one. What a tie costs a reader, against the share of noteheads that are the second of a tied pair. The lower curve is the decision itself — is this notehead an event or a continuation? — at 0.52 bits a note where a tenth of them are tied. The upper curve adds what the extra noteheads cost on every other axis: a tied continuation has a pitch and a position and is read like any other notehead before the reader discovers it carries no event, at 4.79 bits each. The total is 1.05 bits a note, which is 2.0 times the decision alone and is a fifth of what a whole note of music costs. A tie is the most expensive mark on the staff per occurrence, and every published account of notational difficulty treats it as a minor one.

The notehead that is not a note

Every quantity so far is charged per notehead, and a tie is the one mark on the staff that puts a notehead on the page carrying no event. Its cost is not the decision that identifies it — that is half a bit where a tenth of the noteheads are continuations. It is the decision plus the whole reading of a notehead that turns out to have been unnecessary, which is 1.05 bits, twice the decision and a fifth of what a note of music costs. Set beside a dot and a longer note value, the tie is five times the price of either and is the only one of the three that can cross a barline.

scales · Notation Leaps do not fall where offbeats do, and a reader gets the difference free. Where each size of melodic move actually lands in the bar, over the 101 moves of the three tunes measured here. The two axes are priced separately everywhere and they are not independent: the mutual information between them is 0.31 bits a note, which is 20 per cent of the smaller of the two. That is the amount the sum over-charges. A reader who has seen where a note falls already knows something about how far it moved, so the joint cost is 3.47 bits rather than the 3.79 the two axes add to — and every reading load computed so far is high by the difference.

Leaps do not fall where offbeats do

Every reading load computed so far is a sum of two terms priced as though the axes were independent, and an earlier essay named the interaction it could not reach. Measured on the same hundred and one notes every other essay uses, the mutual information between how far a note moves and where it falls in the bar is 0.31 bits — a fifth of the smaller axis, and a sixth of a note's total load. Every reading load published so far is high by that amount, and the quantity saturates at exactly the grid the tunes are notated on, which is the check that it is measuring the music rather than the grid.

scales · Notation One number a page, and what a hard rhythm buys against a hard tune. Every combination of six kinds of line and seven kinds of rhythm, placed by what each axis costs a reader. The duration term (0.67 bits) and the interaction (0.31) are the same for every cell, so the diagonals are pages of equal difficulty and the exchange rate between the two axes is the slope of one. The pitch axis spans 5.26 bits across the six lines and the position axis 4.46 across the seven rhythms, so a composer choosing between the hardest line and the hardest rhythm is choosing between quantities within 18 per cent of each other. The hardest page is wide leaps in off the beat at 14.0 bits a note and the easiest is a scale on the beat at 4.2.

One number for a page

Four terms and an interaction give a single bit rate per note, and with it the exchange rate a long run of essays has been pointing at. Six kinds of line span 5.26 bits and seven kinds of rhythm span 4.46, so a composer trading a harder tune against a harder rhythm is trading quantities within eighteen per cent of each other — and pages that look nothing alike sit on the same contour. The hardest page on the grid costs 13.96 bits a note and the easiest 4.24, a factor of three and a half, and the subject closes there.

scales · Notation With a fading memory the slowest order of the gaps 1 1 2 2 2 2 2 is not the one a perfect memory finds. For each of the 3 cyclic orders of the gaps 1, 1, 2, 2, 2, 2, 2 in 12 steps, the bits of position still unknown once listening has settled, against how many steps a listener's memory of a step takes to halve, with a mismatch costing 6. 2 2 2 1 2 1 2: 12 → 4e-9, 6 → 4e-5, 4 → 0.007, 3 → 0.070, 2 → 0.540, 1.5 → 1.130, 1 → 1.786; 2 2 2 2 1 1 2: 12 → 2e-5, 6 → 0.035, 4 → 0.335, 3 → 0.795, 2 → 1.405, 1.5 → 1.694, 1 → 2.005; 2 2 1 2 2 1 2 (the standard bell pattern): 12 → 2e-5, 6 → 0.027, 4 → 0.231, 3 → 0.535, 2 → 1.028, 1.5 → 1.368, 1 → 1.819. With perfect memory the slowest to locate is the standard bell pattern; at a half-life of 6 steps the highest floor is the order 2 2 2 2 1 1 2, and at 1 it is the order 2 2 2 2 1 1 2.

The bell pattern is slowest only to a perfect memory

Among the orders of its own gaps, a named timeline is usually both the most even and the slowest to locate — for a listener who never forgets. Give the listener a memory that halves and the result comes apart. Of six timelines slowest among their orders with perfect memory, only the fume-fume stays slowest for every forgetting listener, and the standard bell pattern, which is the fume-fume with onsets and rests exchanged and settles at exactly the same floors, is second of its three orders for every memory of half its cycle or less. The census ranking survives better, and in fourteen of twenty-one censuses it was the arithmetic of a pattern that repeats.

rhythm · Euclidean rhythm

Named alongside it

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

InformationNotationSight-readingEuclidean rhythmMaximal evennessRotationTimelineClaveCyclic rhythmDurationExpectationInter-onset interval

All concepts