Acoustics & Music Theory
why it sounds like that
Concept

Chord segmentation — where it appears

Deciding which notes of a passage belong to one chord, which has to be settled before a harmony can be named at all. It is not given by the notes: the same bar admits several readings, and which one a listener takes depends on the metre as much as on the pitches.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Given the bar in octaves, the cue gets the degree back. How often the reading names the right scale degree, against how much the bass cue is worth, under three rules for what a bass note rewards. the bass is the root: 68 per cent at best; the bass is some chord tone: 67 per cent at best; the bar in octaves: 92 per cent at best; the bass is the root, on a root line: 89 per cent at best. The published cue rewards the triad rooted on the bass, which is right only when the chord is in root position; rewarding any triad containing the bass is right always and rewards three chords a bar instead of one. The third rule is not a bass cue at all — given the bar voiced in octaves a listener knows which three pitch classes are sounding, and that names the chord outright. It reaches 92 per cent on a real line against the published cue's 68 on the same line, and the dashed curve is what that cue manages on a line made entirely of roots — 89 per cent.

Given the bar in octaves, the degree comes back

A bass note is a bare pitch class in the usual figure, and a register in any realisation anybody plays. Voice each bar in octaves and a listener knows which three pitch classes are sounding and which is at the bottom, which names the chord outright — and the rule that uses it reads the right scale degree in 92 per cent of bars on a real bass line, against 68 for the published cue on the same line and 89 for that cue on a line made entirely of roots. The question was whether the register recovers the 89. It recovers it and passes it.

scales · Key-relations A sharper cue is worth nothing to a reading that will follow it anywhere. How often the reading names the right scale degree, against how much it costs to change key, at a bass worth 1.5 on a real bass line. the bass is the root: 20 per cent at a key cost of 0.5 and 68 at 4; the bass is some chord tone: 47 per cent at a key cost of 0.5 and 59 at 4; the bar in octaves: 46 per cent at a key cost of 0.5 and 91 at 4; roots, and a line of roots: 49 per cent at a key cost of 0.5 and 91 at 4. Where a key change is cheap the rule that names the chord outright reads no better than the rule that names three — 46 against 47 per cent — because a reading that will move key for one bar's evidence follows a sharp cue wherever it points. The sharper cue's whole advantage appears only once the reading is reluctant enough to stay put, and by a key cost of 2.2 it is 31 points ahead.

A sharper cue is worth nothing to a reading that moves

The rule that names the chord outright reads 92 per cent of scale degrees right where the published bass cue reads 68 — at the key cost these readings have always been run at. Sweep that cost and the advantage is not a property of the cue. Where a change of key is cheap the sharp rule reads 46 per cent and the vaguest rule 47, because a reading that will move key on one bar's evidence follows a sharp cue wherever it points. The cue's whole value is borrowed from the model's reluctance to be moved.

scales · Key-relations The two parameters are not one, and the reason is a ceiling. The plane of the two parameters these readings have been swept one at a time: how much weight the bass cue carries, against what a change of key costs. Every cell is how often the reading names the right scale degree, and the lines are the contours of equal share. along the 50 per cent contour the product of the two coordinates runs from 0.10 to 0.50; along the 60 per cent contour the product of the two coordinates runs from 0.43 to 1.84; along the 70 per cent contour the product of the two coordinates runs from 0.66 to 2.56; along the 80 per cent contour the product of the two coordinates runs from 1.85 to 4.91; along the 90 per cent contour the product of the two coordinates runs from 2.71 to 15.20. If the two multiplied cleanly those products would be constant and the contours would be hyperbolae. They are not: every contour turns upward and then vertical, because past a bass weight of about 3 more of the cue buys nothing at all and only reluctance is left to buy anything with. The key cost has an interior best, at 3 on this grid, where the reading names 93 per cent of degrees — so a reading that will not change key at all is worse than one that will, which no sweep of a single parameter had found.

The two parameters turn out to have a ceiling between them

The essay before this one asked whether the bass cue's weight and the cost of changing key are one quantity with two names, and said the test was a contour: if they multiply, the curves of equal degree share are hyperbolae. They are not. Along the ninety per cent contour the product of the two runs from 2.7 to 15.2, because past a bass weight of about one and a half the reading saturates and more cue buys nothing. And the sweep finds something no single-parameter sweep here could: the key cost has a best value, and a reading that will never change key is worse than one that will.

scales · Key-relations 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.

harmony · Progression 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.

harmony · Progression

Named alongside it

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

Harmonic rhythmMetreRegisterKey-findingModulationScale degreeDownbeatEvidenceInferenceProgressionInversion

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