Acoustics & Music Theory
why it sounds like that
Concept

Information — where it appears

The improbability of an event under a stated model, in bits: minus the log of its probability. It turns a description of a surprising chord into a quantity, and it needs a distribution rather than a list of preferences to exist at all.

Named by 20 essays across 3 fields — each of them below, with the objects they name alongside it.

The surprise of each chord, against the uncertainty it arrived into. The information content of each step — minus the log of its probability under a distribution that multiplies the root-motion weight by how well the destination triad's notes fit the key — with the entropy of the moment before it drawn behind. I – IV – V – I: I→IV 1.71 bits, IV→V 2.80 bits, V→I 1.60 bits, against a mean uncertainty of 2.63; I – IV – V – vi: I→IV 1.71 bits, IV→V 2.80 bits, V→vi 2.61 bits, against a mean uncertainty of 2.63. A surprise larger than the entropy it arrived into is an outcome the model was not expecting even given how uncertain it was; one below it is an outcome the model had already mostly bet on. An earlier essay produced the first of those numbers and had no way to produce the second, because a set of preferences is not a distribution and only a distribution has an entropy.

A chord, given a key and a predecessor

A chord's improbability has been priced from its root motion alone, which left one multiplication unmade: a chord is also improbable because its notes do not fit the key, and that number has been available since the probe-tone profile. Multiplied and renormalised, the two give a conditional distribution — and a distribution has an entropy, which is the quantity a surprise has to be read against and which a list of preferences cannot supply.

harmony · Tonal-expectation How sure the reading is, bar by bar. Every earlier essay reports one best reading. A dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed and has never been printed. Here it is, in bits, for the thirty-two-bar AABA. The mean margin is 1.14 bits and 13 of 32 bars are inside one bit of a rival reading, which is where a listener would be genuinely undecided. The reading itself names C, E, B, D, G; the margin says what that naming is worth, and at the weakest bar — bar 31, C over F — it is worth 0.07.

The margin the dynamic program already had

Nine earlier essays produce a single best reading, and the passages worth arguing about are the ones where two readings are nearly equally good. What is needed for that has been inside the model from early on: a dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed, and printing it turns every analysis here into a measurement of ambiguity.

scales · Key-relations A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players.

A cycle that says where it is

Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke.

rhythm · Cyclic rhythm Expectation as a curve, and what a change costs where it lands. Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous — it sharpens through a bar and collapses when the change arrives — and the two ingredients for it are already here, the harmonic rhythm and the metrical beat weights. The curve is the hazard: given that the chord has not changed yet, the chance that it changes on this beat. It runs from 0.043 on the weakest beat to 0.290 on the downbeat, a ratio of 6.7, against 0.125 if every beat were alike. The marked beats are where the changes actually arrive, and their timing bill is 3.6 bits against 6.0 for a listener with no metre — so these changes are 1.7 times cheaper to expect than a metreless listener would find them. That term is new: an earlier essay prices which chord arrived and this prices when, and a listener meets the sum.

Expectation is a curve, not a list

Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous, sharpening through a bar and collapsing when the change arrives — and the two ingredients for it were already here, in two other accounts. What comes out is a second surprise, for when a chord arrives rather than for which one it is.

harmony · Tonal-expectation How much of the reading comes from what has not happened yet. Every margin reported earlier is two-sided: the best path through a key at a bar is the best score into it plus the best score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line, because the dynamic program already had both halves separately. The mean margin falls from 3.90 bits with hindsight to 1.79 without it, so 54 per cent of this passage's certainty is retrospective. The two passes never disagree about which key is best here, so the hindsight buys confidence rather than a different answer. This is the quantity every earlier essay has assumed and none has measured.

How much of the reading arrives late

Every margin reported earlier is two-sided: the best path through a key at a bar is the score into it plus the score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line. On a thirty-two-bar song it removes more than half the certainty, and on a passage built to be ambiguous it changes the key named at nine bars out of eleven.

scales · Key-relations 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 same eight notes are four times as much to read. How many bits each note of a line carries, taken as minus the log of the probability of the interval that reached it, under the distribution of melodic steps measured over the tunes used throughout. A scale costs 1.76 bits a note and a wide leaps costs 7.02 — a factor of 4.0 at the same number of notes on the page. Every quantity computed until now counts notes, and the page cannot tell these apart: eight quavers are eight quavers of horizontal space whichever line they spell.

A reader does not read notes

Eleven earlier essays count notes, and the page cannot tell one line of eight quavers from another. A reader can: a scale of eight is one object where eight leaps are eight. Measured against the melodic interval distribution, the same eight notes are four times as much to read — and the eye–hand span, the best-measured quantity in the reading literature, is four notes of a tune and one of a leaping line.

scales · Notation Where a cycle of 16 at 16, 8, 4 outruns the listener's memory. The residual uncertainty a listener is left with once the evidence has stopped accumulating, against how long one turn of a 16-step cycle takes. The listener's memory of a step halves after 3.5 seconds throughout; what changes is how many steps that is. At a cycle of 1.6 seconds it is 35 steps and every design reaches certainty, which is the regime a clave is played in. At 60 seconds it is 0.93 steps and none of them does: layers at 16, 8, 4 settles at 1.89 bits, son clave settles at 2.27 bits, the bossa-nova pattern settles at 2.38 bits, the best single line of 7 settles at 1.85 bits. That is the range a gong cycle occupies, and it is the design that wins there.

The cycle that outruns the memory

A timeline and a colotomy were compared at equal strokes and the comparison had no clock in it. A memory span is a number of seconds and a cycle is a number of steps, so the two only meet through a tempo — and at a clave's two seconds a listener's memory covers twenty-eight steps and forgets nothing, while at a gong cycle's forty it covers 1.4 and forgets almost everything. The single line is the better locator up to twenty-three seconds a cycle and the layered code is better after it, which is very close to where each is actually used.

rhythm · Cyclic 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 What a chord change costs where it actually lands. Every beat of a bar of 4/4 at 1 change a bar, priced as a listener meets it: the pale block is what has already been paid waiting through the beats the chord did not come on, and the dark block is the arrival itself. Their sum is what it costs to be surprised by a change here. The last column is the remaining case, never priced before: no change in the bar at all, at 1.61 bits and a probability of 0.33. The 9 costs are a proper distribution — 1.000 — which is the check that this is one model rather than two. And the spread is the finding: the arrival term alone puts a factor of 6.7 between the best and worst beat, and counting the waiting makes it 19.

Where the chord actually lands

Every timing surprise so far is evaluated on a downbeat: the curve spans a factor of 6.7 and every number is read at its peak. Charge the waiting as well as the arrival and the costs over a bar become a proper distribution, the spread between the best and worst beat rises to a factor of 19.5, and a third of the probability sits on a bar in which nothing changes at all.

harmony · Tonal-expectation A notehead in four parts costs 1.30 bits and one in two parts costs 1.89. What one notehead asks of a reader, against how many parts are on the page, for a progression realised by the voice-leading solver used here at 2 semitones of motion a voice a chord. The horizontal rule is a note of a single melody under the measure established earlier, 1.89 bits, which is what a texture costs when its parts have to be read one at a time. The bars are the harmonic reading: the chord, charged at the worst case of 2.81 bits for one of seven diatonic degrees, plus the logarithm of how many voicings of it the previous chord could legally have moved to. At two parts there is no bar, because a duet has no complete voicing of any triad — it cannot state the harmony and has to be read as 3.79 bits of two independent lines. Every thicker texture is cheaper a notehead than the thin one, and the four-part figure is an upper bound.

Four parts are easier to read than two

Twelve earlier essays read one line, and a score is several at once. Measured through the voice-leading model, a notehead of a four-part chorale asks a reader for 1.30 bits and a note of an independent line asks 1.89 — so twice the ink is less than three quarters of the load. The reason is a boundary those essays already established: a duet has no complete voicing of any triad at all, so two parts cannot be read from their harmony and have to be read as two melodies.

harmony · Notation Eighty-one chords the expectation model cannot tell apart. Every voicing of a dominant seventh on G inside the three octaves above its own root, placed by how rough it is and how far its outer voices are apart, and coloured by which member of the chord is at the bottom. The roughness runs from 0.269 to 1.449, a factor of 5.4, computed from each voicing's own spectrum under Plomp and Levelt's roughness model. The identity surprise the expectation model assigns is 3.51 bits for every one of the 81, because it is a function of a scale degree and its predecessor and there is no register anywhere in it. What separates them is spacing rather than inversion: roughness falls as the outer voices spread apart, correlating -0.42 with the span, and is indifferent to which member of the chord is at the bottom at 0.02. The seventh in the bass is not what makes a chord rough; a fourth and a third packed together at the bottom of the range is.

Eighty-one chords, one number

A dominant seventh has eighty-one arrangements inside three octaves and their roughness spans a factor of five and a half. The tonal-expectation model gives every one of them the same 3.51 bits, because its states are scale degrees and there is no register anywhere in them. Conditioning the surprise on the voicing costs no corpus — and the arithmetic says the conditioning belongs beside the probability rather than inside it, for three reasons that can each be computed.

harmony · Tonal-expectation A note on the downbeat costs 1.89 bits and one on the offbeat 5.70. What each position in a bar of 4/4 asks of a reader, by two routes. The solid bar counts where the notes of this collection's own three tunes actually fall — 28, 2, 26, 4, 28, 0, 16, 0 notes at the 8 positions — and takes minus the log of the frequency. The rule across each bar is the same quantity from the stated metrical weights, 1, 0.15, 0.5, 0.15, 0.85, 0.15, 0.5, 0.15, normalised and logged the same way. Nothing makes the two agree. They put the eight positions in the same order, and they price the tunes' own rhythm a fifth of a bit apart — while differing by more than a whole bit about the quaver after the downbeat, which two notes in a hundred and four ever use. The two positions these tunes never touch at all are drawn at the floor, which is the same floor the melodic measure gives an interval nobody plays. A weight was always a probability waiting to be read as one.

Where the note is costs more than which note it is

Thirteen earlier essays measure a page, and the three that price a reader price only its pitches — every line they measure is a run of equal notes. A metrical weight normalised by its own sum is a probability, and minus its logarithm is bits — the same substitution made earlier for intervals. Measured over the tunes used throughout it comes out at 2.23 bits a note against the pitches' 1.89, so the larger half of a reader's load is where the note is.

rhythm · Notation 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 A cycle already known, against a cycle just arrived at. How many bits of uncertainty about position a listener has, against how long the cycle takes, for a single timeline and for a layered colotomy — each drawn twice, once as a listener arriving and once as a listener who has been hearing it long enough to settle. At 1.6 seconds a cycle the timeline goes 0.94 bits arriving and 0.00 settled, and the colotomy 0.71 and 0.00; At 16 seconds a cycle the timeline goes 1.10 bits arriving and 0.08 settled, and the colotomy 0.93 and 0.23; At 60 seconds a cycle the timeline goes 2.47 bits arriving and 2.27 settled, and the colotomy 2.14 and 1.89. The gap between each pair is what the repetitions are worth, and it narrows as the cycle slows. The two designs are drawn at their own step counts rather than at equal strokes, so the levels here are not the earlier ones and the gaps are.

Repetition buys least where it is needed most

Every locating figure so far is a listener arriving — the uncertainty averaged over the first cycle heard. Cyclic music comes round dozens of times, and the same model already carries the answer for a listener who has settled: a floor of uncertainty that nothing had read. At two seconds a cycle the repetitions close the whole gap. At sixty they close eight per cent for a single timeline and twelve for a layered code. A slow cycle is worse on the first hearing and gains less from the second, and the two disadvantages compound.

rhythm · Cyclic rhythm Against a pulse, the bell pattern is the quickest of its orders to place. The bits of position a listener is still missing, averaged over the first cycle heard, for each cyclic order of the gaps 1 1 2 2 2 2 2 in 12 steps, heard alone, against a pulse every three steps and against a pulse every four, at perfect memory, half-life 3 steps, half-life 1.5 steps. 2 2 2 1 2 1 2: alone 1.08, 1.23, 1.73; against a pulse every 3 steps 0.69, 0.75, 0.96; against a pulse every 4 steps 0.63, 0.67, 0.85. 2 2 2 2 1 1 2: alone 1.22, 1.63, 2.02; against a pulse every 3 steps 0.60, 0.73, 0.92; against a pulse every 4 steps 0.83, 1.07, 1.36. 2 2 1 2 2 1 2 (the standard bell pattern): alone 1.25, 1.45, 1.82; against a pulse every 3 steps 0.54, 0.56, 0.67; against a pulse every 4 steps 0.55, 0.57, 0.68. Alone, the bell pattern is not the quickest order to place at any memory. Against either pulse it is the quickest at every memory.

Against a pulse the bell pattern is the easiest to place

Heard alone, the standard bell pattern is not the quickest order of its own gaps to place in its cycle, for a listener with any memory. Heard against a pulse every three steps or every four — which is how anyone hears it — it is the quickest, at every memory and at every alignment of pulse and bell, and by a wide margin: at a memory of a quarter of the cycle, 0.56 bits unplaced over the first cycle against 0.73 for either rival against a pulse in threes. Six of eight named timelines do the same. A timeline's order of gaps looks chosen for how it sits against the beat, not for how it sounds alone.

rhythm · Euclidean rhythm A timed expectation would erase a slow cycle's cost, and a listener cannot time a slow cycle that well. Bits of position a listener with a 3.5-second memory is still missing over the first cycle of son clave, against how long the cycle takes, for a newcomer with no expectation, a listener timing the cycle with the Weber fraction a duration that long is judged with, and a listener timing it to ten per cent. a newcomer, no expectation: 2 s 0.94, 8 s 0.97, 24 s 1.41, 40 s 2.02, 60 s 2.47; timing as well as listeners do: 2 s 0.68 (w 0.150), 8 s 0.69 (w 0.150), 24 s 0.85 (w 0.150), 40 s 1.90 (w 0.375), 60 s 2.38 (w 0.375); timing the cycle to ten per cent: 2 s 0.47, 8 s 0.48, 24 s 0.55, 40 s 0.73, 60 s 1.02. At ten per cent even a sixty-second cycle is placed about as well as a newcomer places a two-second one. At the precision a listener actually has for durations of half a minute or more, the expectation is worth a tenth of a bit.

An expectation cannot rescue a cycle too slow to time

A listener who knows a piece arrives with an expectation of where in the cycle they are, and the size of that expectation was the number the last essay said nobody had. It can be given one: a listener who has been timing the cycle carries a spread of their Weber fraction times the cycle, which is the same number of steps at any tempo. Timed to ten per cent, a forty-second cycle would be placed better than a newcomer places a two-second one. But forty seconds is judged in the band where the Weber fraction is nearer forty per cent, and there the expectation is worth a tenth of a bit.

rhythm · Cyclic rhythm

Named alongside it

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

ExpectationMetreNotationSight-readingEntropyMemory decayMetrical weightSurpriseHarmonic rhythmDurationEntrainmentInference

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