Acoustics & Music Theory
why it sounds like that
Series

Progression — the series

17 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · harmony
  2. 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.

    part 2 · harmony
  3. 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.

    part 3 · harmony
  4. 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.

    part 4 · harmony
  5. 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.

    part 5 · harmony
  6. 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.

    part 6 · harmony
  7. 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.

    part 7 · harmony
  8. 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.

    part 8 · harmony
  9. 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.

    part 9 · harmony
  10. 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.

    part 10 · harmony
  11. 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.

    part 11 · harmony
  12. The product and the sum of standard scores, finding 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: product finds it 16%, sum of z finds it 22%, metre finds it 10%, chords, z 35%; rhythm 0.25: product finds it 56%, sum of z finds it 58%, metre finds it 34%, chords, z 38%; rhythm 0.5: product finds it 66%, sum of z finds it 61%, metre finds it 49%, chords, z 19%; rhythm 0.75: product finds it 56%, sum of z finds it 58%, metre finds it 50%, chords, z 14%; rhythm 1: product finds it 45%, sum of z finds it 48%, metre finds it 50%, chords, z 12%.

    The chords are a weak witness to the barline

    Scaled by its own range, the metre overrules the chords every time the two disagree about where a bar begins. The obvious repair is to score each reading against its own chance — the metre against the same number of notes placed at random, the chords against the passage's notes shuffled across its bars — and add the standard scores. It changes very little: the search finds the barline within six points of where the product found it, and the chords gain the power to move the barline only on passages whose rhythm says nothing, where random notes move it nearly as often. The null's real result is the size of the two witnesses. At the written barline the metre stands up to 5.9 standard deviations above chance, and the chords, with every note a tone of its bar's chord, stand 1.55 above it at best.

    part 12 · harmony
  13. Four bars read by where the chords change, barline by barline. A constructed passage of four bars of eight quavers, its barline at the first slot and its chords C, F, Em, Dm. Notes: slot 1 C, slot 3 E, slot 4 G, slot 5 E, slot 6 C, slot 9 C, slot 10 C, slot 11 F, slot 13 A, slot 16 C, slot 17 B, slot 19 B, slot 20 E, slot 21 E, slot 24 G, slot 25 F, slot 29 F. For each of the eight places the barline could fall: as written metre, z 3.97, chords, z 2.00, change, z 4.30, metre + change, z 8.27; 1 quaver late metre, z -2.45, chords, z 2.14, change, z -0.87, metre + change, z -3.32; 2 quavers late metre, z -1.38, chords, z 2.15, change, z -0.54, metre + change, z -1.93; 3 quavers late metre, z -0.31, chords, z -0.27, change, z -2.64, metre + change, z -2.95; 4 quavers late metre, z 3.97, chords, z -1.63, change, z -4.30, metre + change, z -0.33; 5 quavers late metre, z -2.45, chords, z 1.27, change, z 0.87, metre + change, z -1.58; 6 quavers late metre, z -1.38, chords, z 1.18, change, z 0.54, metre + change, z -0.84; 7 quavers late metre, z -0.31, chords, z 1.05, change, z 2.64, metre + change, z 2.32. Best metre, z: as written and 4 late. Best chords, z: 2 late. Best change, z: as written. Best metre + change, z: as written.

    The chords mark the barline by changing there

    Read bar by bar, the chords stood barely above chance at the barline and broke the metre's half-bar tie two times in three at best. Read instead by where they change — how different the chords are across a candidate's barlines against how different they are across the middle of its bars — the same notes break the tie right on 81 to 96 per cent of passages, and added to the metre they find the barline on up to 89 per cent against 61. The weakness was the question the old reading asked, not the harmony.

    part 13 · harmony
  14. 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.

    part 14 · harmony
  15. 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.

    part 15 · harmony
  16. The content errs slow and the bass errs fast. Passages of eight bars built at three harmonic rhythms, each with a bass that states every new chord's root and moves to another chord tone on a beat 50 per cent of the time. Two readings are asked the chord rate: one from how much the pitch-class content changes across a grid, one from how completely the bass's moves land on it. At two chords a bar the content reading names the rate 63 per cent of the time, too fast 0 and too slow 37; the bass reading 100, 0 and 0. At one chord a bar the content reading names the rate 67 per cent of the time, too fast 0 and too slow 33; the bass reading 18, 80 and 2. At a chord every two bars the content reading names the rate 98 per cent of the time, too fast 2 and too slow 0; the bass reading 0, 100 and 0. The two readings miss in opposite directions and at opposite ends of the tempo range: the content reading at fast harmonic rhythms, by naming a multiple, and the bass reading at slow ones, by naming its own arpeggiation.

    The bass errs fast where the content errs slow

    Asked how often the chords change, a reading built on pitch-class content names a slower multiple and never a faster rate. Give the passage a bass that states each new root and moves between chord tones inside a chord, and a reading built on the bass's moves errs the other way: it names a faster grid and never a slower one. At two chords a bar the bass is right every time; at a chord every two bars it is never right. Six ways of combining the two readings each trade one end of the range for the other.

    part 16 · harmony
  17. Holding the bass through one mid-bar change in six finds the barline five times in six. Passages of eight bars at two chords a bar, with the bass arpeggiating on 50 per cent of its beats, and each barline reading's share of passages it places correctly, against the share of mid-bar chord changes voiced over the bass already sounding. 0% held (convention strength 0): metre then bass 50%, bass alone 13%, metre alone 50%, chord changes alone 10%; 4% held (convention strength 0.1): metre then bass 62%, bass alone 34%, metre alone 50%, chord changes alone 10%; 6% held (convention strength 0.2): metre then bass 68%, bass alone 44%, metre alone 50%, chord changes alone 15%; 9% held (convention strength 0.3): metre then bass 75%, bass alone 57%, metre alone 50%, chord changes alone 18%; 15% held (convention strength 0.4): metre then bass 82%, bass alone 68%, metre alone 50%, chord changes alone 13%; 16% held (convention strength 0.5): metre then bass 85%, bass alone 74%, metre alone 50%, chord changes alone 14%; 20% held (convention strength 0.6): metre then bass 88%, bass alone 79%, metre alone 50%, chord changes alone 11%; 26% held (convention strength 0.7): metre then bass 92%, bass alone 86%, metre alone 50%, chord changes alone 13%; 27% held (convention strength 0.8): metre then bass 92%, bass alone 86%, metre alone 50%, chord changes alone 13%; 28% held (convention strength 0.9): metre then bass 95%, bass alone 92%, metre alone 50%, chord changes alone 10%; 33% held (convention strength 1): metre then bass 96%, bass alone 93%, metre alone 50%, chord changes alone 10%. The metre ties the barline with the half-bar and the chord changes are at chance, since the chords change at both; the bass's holds are the only evidence that separates them.

    A bass that holds through a change marks the barline

    At two chords a bar the chords change on the barline and on the half-bar alike, so a reading of where they change is at chance, and the metre ties the two. The bass has one more piece of evidence: a change inside the bar can be voiced over the note already sounding, and a change on the barline is voiced over its root. Hold the bass through one mid-bar change in six and the metre and bass together place the barline in 85 per cent of eight-bar passages; one in three, 97. The convention cannot be stronger than that, and the chord changes, asked first, only get in the way.

    part 17 · harmony

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