Acoustics & Music Theory
why it sounds like that
Concept

Progression — where it appears

A sequence of chords, considered as a path from one to the next rather than as a set. Which chords it is made of is decided by a segmentation, and that segmentation depends on the metre rather than on the harmony.

Named by 18 essays across 4 fields — each of them below, with the objects they name alongside it.

The seven chords of a key, by distance from home. Each triad of the major scale placed at a radius equal to how far its voices must move from the tonic chord. The chords that feel closest to home are the ones that are closest, in the plain arithmetic sense.

A progression is a path, and the map can be drawn

Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.

harmony · Progression Where a progression in pure ratios ends up. A short chord sequence taken in exact whole-number ratios, with the running difference from equal temperament measured in cents. The sequence returns to its starting chord and the pitch does not, which is the comma arriving in ordinary music.

The progression that never comes home

Just intonation is usually said to fail because a keyboard has only twelve keys. That is not the reason. Take one of the commonest chord sequences in tonal music, play every chord in exact whole-number ratios, and it arrives back on its starting chord a syntonic comma flat — on any instrument, including a voice.

harmony · Progression thirty-two-bar AABA, every bar against every other bar. A self-similarity matrix of 32 bars of thirty-two-bar AABA. Each cell is the cosine similarity of two bars' pitch-class vectors, with the chord's own notes weighted 1 and the rest of the key 0.5. Similarity is quantised into four bands for drawing and anything under 0.35 is left as paper. Nothing in the computation knows what a section is; the blocks and stripes are what the arithmetic returns.

A piece is mostly itself again

Take a piece of music, encode each bar as the notes sounding in it, and compare every bar with every other bar. The picture that comes out has blocks and stripes in it, and those blocks and stripes are the form — arrived at by arithmetic that has never heard of an exposition, a chorus or a refrain.

form · Repetition How often the chord changes, and what a room allows. Chord changes a second implied by each style's stated rate and tempo, on a logarithmic axis, with the rate above which a room leaves more than one earlier chord above 20 dB marked for six rooms. The style rates are conventions rather than corpus measurements and the figure says so; the room rates are arithmetic from the reverberation time.

How often the chord changes

Two pieces can use the same chords in the same order and be nothing alike, because a progression says which chords and not how fast. Harmonic rhythm is the second variable, it runs over a factor of thirty between the styles that use it, and both of its limits are set by things that are not harmony — a listener's memory at the slow end and a building at the fast one.

harmony · Progression 6 chords in a gothic cathedral. Each chord's reverberant decay in a room with a 8 second reverberation time, at 1 chord a second. Decay is linear in decibels, so each line is straight with a slope of -7.5 dB a second. When a chord arrives, 2 earlier ones are still above 20 dB down.

The room chooses the harmonic rhythm

A chord in a cathedral is still sounding, seven decibels down, when the next one arrives — and the one after that, and the one after that. Reverberation is linear in decibels, so the number of chords audible at once is one number divided by another, and it puts a hard ceiling on how fast a composer writing for that building can change harmony. The ceiling is computable, and the music written for those rooms sits under it.

timbre · Room acoustics 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 What one progression leaves open. Realisations of I–IV–V–I in four parts with no parallel fifths or parallel octaves, counted exactly by a dynamic programme over the voicings rather than sampled. The chord symbols admit 16,100,352,296; the Roman numerals 59,418,496; the figured bass 2,042,672. The three notations differ by four orders of magnitude, and every one of them was in daily professional use.

Three notations, one progression

A figured bass, a Roman numeral and a chord symbol are three professional notations for the same four chords, and the number of four-part realisations each of them admits can be counted exactly rather than argued about. With no parallel fifths or octaves the counts are sixteen billion, fifty-nine million and two million: a factor of eight thousand between the loosest and the tightest. What each one collapses is what its tradition thought a chord was, and the three do not agree.

harmony · Notation Two passages, one note apart, and opposite cadences. Two passages that share eight bars drawn from the six pitch classes C major and G major have in common, and differ only in their last three bars: one cadences in C and one in G, and the single note that separates them is an F against an F sharp. The cadence evidence names C for the first and G for the second, by 9 to 3 and 6 to 4. The profile model names E minor for both, with correlations differing in the third decimal.

The cadence as evidence

Every model so far infers a key from a bag of notes, and the thing that most obviously declares a key is a cadence — an ordered pair of chords in which the order is the whole content. The essays on closure built a five-component cadence vector long before this and nobody has used it for this. Two passages differing in one note, one cadencing in C and one in G, get opposite answers from the ordered pairs and the same answer from the profile.

harmony · Progression Two kinds of evidence, each measured against its own chance. Every key as a point: across, how many standard deviations its cadence count is above what the same bars resampled would give; up, the same for its pitch-class profile correlation. The winner on cadences is key C at 6.2 standard deviations and on profile C at 1.1, and they agree. Standard deviations above chance are the same unit whatever produced them, which is the commensuration this figure is for — and it costs something: it assumes the two chances are equally interesting, which is a weighting in disguise. The obvious null does not work at all for one of the two: shuffling the bars leaves a pitch-class histogram exactly as it was, so its spread is zero and every key scores nothing against it.

A count and a correlation

Cadence evidence is a count of ordered pairs and profile evidence is a correlation with a template, and the cadence essay refused to total them because they are not in the same units. Score each against its own chance and they are — standard deviations above chance are the same unit whatever produced them. Then the trouble moves: the obvious null does nothing at all to one of the two, because shuffling the bars leaves a pitch-class histogram exactly as it was.

harmony · Progression 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 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 The passage built so the two statistics disagree. A body of chords diatonic to C major, of growing length, ended by a ii–V–I in G. The bag of notes says one key and the ordered pair says the other, which is the case the earlier figures never contained. The line is the cadence evidence for G, and under the axis is what each reading actually names at each length. The cadence reading holds G up to a body of 6 bars and is overturned at 8, so one explicit cadence is worth about that many bars of profile evidence — and it is overturned by the body's OWN incidental root motions rather than by the histogram at all. That is the finding the earlier essay could not have: on real material the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.

The passage built to make them disagree

A cadence count and a key-profile correlation have been put on one scale and run on a passage where the two agree, which tells nobody anything. Building one where they disagree — the notes of one key and the cadences of another — measures the exchange rate at about six bars per cadence, and finds something the agreement case concealed: the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.

harmony · Progression 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 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 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 The change reading follows the chords, not the bar. How far above the other candidates the true barline stands, in standard units, for the reading that scores how much the pitch-class content changes at each candidate — at three harmonic rhythms. At 2 chords a bar the margin is 0.12 and the reading finds the barline 12 per cent of the time; At 1 chord a bar the margin is 1.66 and the reading finds the barline 42 per cent of the time; At a chord every two bars the margin is 0.47 and the reading finds the barline 27 per cent of the time, against a chance rate of 13 per cent. The passages read earlier all changed chord once a bar, which is the middle column and the only one where the reading has anything. Two chords a bar puts a change at the half-bar as well and the reading cannot tell the two apart; a chord every two bars leaves half the barlines with no change at all and the margin halves exactly.

The change reading follows the chords, not the bar

Every passage read until now changes chord exactly at the barline, which is the one harmonic rhythm at which 'the chords change here' and 'the bar starts here' are the same sentence. Pull them apart and the reading goes with the chords: at one chord a bar it stands 1.52 standard units above the other candidates and finds the barline half the time, at two chords a bar it stands 0.01 above them and is at chance, and at a chord every two bars its margin is exactly half — because half the barlines then carry no change at all.

harmony · Progression Asked for the rate, it answers a multiple of it. The change reading asked its own question — what period do the chords change at — over passages built at three harmonic rhythms, with its standardised score for each candidate period. Given 2 chords a bar it recovers the rate 33 per cent of the time and answers too slow 65; Given 1 chord a bar it recovers the rate 58 per cent of the time and answers too slow 38; Given a chord every two bars it recovers the rate 93 per cent of the time and answers too slow 0. It never errs fast in the way it errs slow, and the reason is structural: a chord change every four slots also produces a change at every eighth slot, so a slower grid inherits a faster rate's evidence and a faster grid cannot inherit a slower one's. That ambiguity is why the reading looked like a barline detector in the first place — the bar is a multiple of every harmonic rhythm that fits inside it.

Asked for the rate, it answers a multiple

A reading that follows the chord rate rather than the bar can be asked what the rate is, and the shape of its errors is the whole of why it looked like a barline detector. Given two chords a bar it returns the right period a third of the time and something slower two thirds; given a chord every two bars it is right nine times in ten. It errs slow and essentially never fast, because a change every four slots also falls on every eighth slot and a slower grid inherits a faster rate's evidence — which is the same asymmetry that makes a pitch detector report an octave too low.

harmony · Progression

Named alongside it

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

SegmentationCadenceHarmonic rhythmKey-findingTonal functionMetreExpectationHarmonic analysisInformationMetrical weightTonal hierarchyChord segmentation

All concepts