Acoustics & Music Theory
why it sounds like that
Concept

Metrical weight — where it appears

A number attached to each position in a bar saying how strong the metre makes it. It is supplied by the reading rather than by the sound, and syncopation is measured as the cost of putting an onset where the weight is low.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

The same eight notes, read three ways. The scale, eight quavers, scored against every triad and seventh at every root. Barred as written the best reading is C major7 at 0.850; with the barline one quaver later it is D minor7 at 0.850. With no metre — every note weighted the same — 4 readings tie at 0.625 and the passage has no best analysis at all. The notes are identical in all three. What changed is where the bar starts, which is not a fact about harmony.

Which notes are the chord

A progression is a list of chords, and before there is a list something has to decide which of the notes sounding are chord tones and which are passing. Take the eight notes of a scale as eight quavers and score every triad and seventh at every root: barred as written the best reading is C major seventh, with the barline moved by one quaver it is D minor seventh, and with no metre at all three readings tie exactly and the passage has no best analysis. Same eight notes in all three. Harmonic analysis is a function of a variable that is not harmony.

harmony · Progression 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 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.

form · Metre induction 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 How much earlier an accent is heard, by mechanism. An accented note on an instrument with a 90 millisecond attack, drawn against how many decibels louder it is, with the three ways it can arrive early separated. A criterion tied to the note's own peak on an unchanging envelope gives exactly nothing. The same criterion on the shorter rise a harder-driven instrument has gives 5.3 milliseconds at 12 decibels. A criterion at a fixed level gives 23.0. Both together give 24.0, and the rise at that dynamic is 73 milliseconds rather than 90. The rise-shortening exponent is stipulated at 0.15 rather than measured, and the two upper curves would separate further if it were smaller.

Playing louder is playing earlier

An accent has two effects on when its note is heard and neither is a timing decision. A harder-driven instrument has a shorter attack, and a criterion set by the surrounding music is crossed sooner by a bigger rise — so a twelve-decibel accent on a bowed note is heard twenty-four milliseconds early with no change whatever in when the bow was put down. It is also the measurement that tells the two competing models apart.

perception · Perceptual-centre 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 Which bars the key is decided by, and which bars it is believed on. Every bar of a 32-bar scheme removed in turn, with what its absence costs. The column is how far the passage's mean margin falls without that bar — how much of the model's certainty it supplies. The dot is how many bars are then read as a different key — how much of the answer it supplies. The 18 bars an analysis would point at — a section opening or closing, a dominant, the chord a dominant resolves to — average 0.079 bits of certainty and 0.50 bars moved; the 14 ordinary bars average 0.047 and 0.07. So the structural bars carry 1.7 times as much certainty as the ordinary ones, and 7.0 times of the answer. Those are different quantities, and the second is the one an analysis is about: an ordinary bar can carry a great deal of a passage's certainty and none of its reading.

The bars a key is made of

A discount that treats every bar alike is the wrong shape for a memory, so take each bar away in turn and see what it was worth. The bars an analysis points at carry seven times as much of the answer as the ordinary ones and slightly less of the certainty — and a memory built only of them reads the passage worse than a memory with no structure in it at all.

harmony · Key-relations 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 long note and a strong note disagree, and the winner is neither. The same 8 notes scored against every triad and seventh at every root, with the weighting run from the metrical one always used to a durational one never drawn. On the left each note counts for its metrical weight; on the right, for how long it is held. The long notes here are on beats 2, 4, 6, 8, which are the weak ones. The two cues point at different chords — C major7 on the left and D minor7 on the right — turning over at a mixture of 40 per cent. And at the crossing the winner is A minor7, which is neither cue's answer — a chord that shares three notes with each and is not the reading either rule asks for. Nothing about the notes changed. What changed is which of two cues a theorist would call obvious is being believed.

The long note and the strong note

The segmentation that produces every object connected here has carried a free parameter since the day it was written: whether a note counts for its metrical weight or for how long it is held. Only the first has ever been drawn. The two name different chords on sixteen per cent of passages where the cues agree about the notes and forty-three per cent where they do not — and where they disagree most sharply a mixture of them picks a third chord neither one asks for.

harmony · Progression The reading the joint search was never offered. The best chord at each mixture of the two segmentation cues, and what the same weighting gives the same notes shuffled into a different order. Both fall along the axis, and most of the fall is the ruler rather than the music: a metrical weighting over a bar of eight spans a factor of eight and a three-to-one duration spans three, so the weighted note mass is 2.1 times more concentrated at the left of the figure than at the right, and a concentrated mass is easier for four notes to cover. What is not the ruler is the gap. It is widest at a mixture of 0.75, where the reading is D minor7 at 2.15 standard deviations above its own null, against 1.12 for C major7 at a mixture of nought. The joint search holds this axis at nought, so D minor7 is not among the hypotheses it considers.

A fourth decision, and two that were never made

The joint search resolves key, metre and segmentation together and holds the segmentation's cue mixture at zero. Adding the mixture is one loop, and reading the search in order to add it turns up something worse than a missing axis: on the passages it is drawn on, the key it reads is the same key at all forty-eight of its hypotheses and the metre scores every barline identically. The fourth axis then cannot be ranked at all until each reading is measured against its own null, because a mixture changes the ruler and not only the answer.

harmony · Progression 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 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 How often the metre, the chords and their product find the barline, chords at 1. Constructed passages of four bars of eight quavers, 100 at each setting, with the barline at the first slot. Rhythm regularity is how much likelier a note is on a strong slot than a weak one; chord regularity is how much likelier a note is to be a tone of its bar's chord than a random scale tone. rhythm 0: metre finds it 10%, chords find it 41%, product finds it 16%; rhythm 0.25: metre finds it 34%, chords find it 35%, product finds it 56%; rhythm 0.5: metre finds it 49%, chords find it 21%, product finds it 66%; rhythm 0.75: metre finds it 50%, chords find it 17%, product finds it 56%; rhythm 1: metre finds it 50%, chords find it 11%, product finds it 45%.

The chords never move the barline

Every hypothesis the joint search had drawn was one bar long, and on one bar with a note in every slot the metre cannot choose a barline at all. Four bars with rests in them make the barline a decision the metre and the chords both have an opinion about, and the prediction was that the chords would move the barline more often than the barline moves the chords. It is the other way round, completely: whenever the two prefer different barlines the search takes the metre's, on up to 72 per cent of passages, and in fifteen hundred passages the chords never once move it. What the chords decide is the one thing the metre cannot see — whether the bar starts on the downbeat or half a bar later — and they decide it right a little over two times in three at best.

harmony · Progression

Named alongside it

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

MetreSyncopationInformationSegmentationHarmonic analysisHarmonic rhythmNotationChord toneDownbeatDurationInferenceKey-finding

All concepts