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 10 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 33%, metre alone 50%, chord changes alone 10%; 6% held (convention strength 0.2): metre then bass 68%, bass alone 41%, metre alone 50%, chord changes alone 15%; 9% held (convention strength 0.3): metre then bass 75%, bass alone 50%, metre alone 50%, chord changes alone 18%; 15% held (convention strength 0.4): metre then bass 81%, bass alone 66%, metre alone 50%, chord changes alone 13%; 16% held (convention strength 0.5): metre then bass 85%, bass alone 71%, metre alone 50%, chord changes alone 14%; 20% held (convention strength 0.6): metre then bass 86%, bass alone 74%, metre alone 50%, chord changes alone 11%; 26% held (convention strength 0.7): metre then bass 90%, bass alone 82%, metre alone 50%, chord changes alone 13%; 27% held (convention strength 0.8): metre then bass 91%, bass alone 82%, metre alone 50%, chord changes alone 13%; 28% held (convention strength 0.9): metre then bass 95%, bass alone 89%, metre alone 50%, chord changes alone 10%; 33% held (convention strength 1): metre then bass 96%, bass alone 90%, 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 As the bass stops standing still, what marks the barline stops being its motion and becomes its note. Passages of eight bars at one chord a bar, and how often each reading places the barline, against how often the bass moves to another tone of the chord on a crotchet. Every reading is the metre's own candidates with one witness breaking its tie. bass moving on 0% of crotchets: moves 100%, root 50%, surer of the two 100%, metre alone 50%; bass moving on 13% of crotchets: moves 100%, root 94%, surer of the two 100%, metre alone 50%; bass moving on 25% of crotchets: moves 100%, root 99%, surer of the two 100%, metre alone 50%; bass moving on 38% of crotchets: moves 99%, root 100%, surer of the two 100%, metre alone 50%; bass moving on 50% of crotchets: moves 95%, root 100%, surer of the two 100%, metre alone 50%; bass moving on 63% of crotchets: moves 88%, root 100%, surer of the two 100%, metre alone 50%; bass moving on 75% of crotchets: moves 77%, root 100%, surer of the two 100%, metre alone 50%; bass moving on 88% of crotchets: moves 54%, root 100%, surer of the two 100%, metre alone 50%; bass moving on 100% of crotchets: moves 20%, root 100%, surer of the two 94%, metre alone 50%. A bass that never moves inside the chord moves only at chord changes, which at this rate are barlines, so the move reading is certain and the root sounds throughout the bar and says nothing; a bass that moves on every crotchet moves everywhere and sounds the root only where the chord has just arrived.

A walking bass marks the bar by the note it is on

The reading that finds a barline from where the bass moves collapses to twenty per cent when the bass moves on every crotchet, and the bass the last essay called busiest moved twice a bar, because its beat was the half-bar. Given one that really walks, the same computation with sounds the chord's root in place of moves rises from nothing to a whole note a bar exactly as the other falls, and the two are equal at a bass moving on three crotchets in eight. Neither places a barline alone: the root is still sounding on the quaver after it. The metre, which cannot tell a barline from a half-bar, tells it from the quaver after it at once.

harmony · Progression Two chords a bar over a moving bass: every witness there is, at chance together. Passages of eight bars at two chords a bar, with the chords always voiced over their own roots, and how often each reading places the barline against how often the bass moves inside the chord. 0%: metre alone 50%, with the moves 50%, with the root 50%, with both 50%; 13%: metre alone 50%, with the moves 49%, with the root 50%, with both 49%; 25%: metre alone 50%, with the moves 49%, with the root 50%, with both 49%; 38%: metre alone 50%, with the moves 49%, with the root 50%, with both 49%; 50%: metre alone 49%, with the moves 53%, with the root 49%, with both 54%; 63%: metre alone 50%, with the moves 54%, with the root 50%, with both 54%; 75%: metre alone 50%, with the moves 48%, with the root 50%, with both 48%; 88%: metre alone 50%, with the moves 51%, with the root 50%, with both 51%; 100%: metre alone 50%, with the moves 51%, with the root 50%, with both 51%. The metre ties the barline with the half-bar because both carry the same strong slots; the chords change at both; the bass states a root at both and moves between both. Nothing in the passage distinguishes them.

A texture in which nothing says where the bar begins

Two chords a bar over a bass that moves inside the chord is the one arrangement in which every witness this account has stands together at the metre's coin toss. The metre ties the barline with the half-bar, the chords change at both, and the bass states a root at both and walks between both — at every busyness, and whichever question the bass is asked. The only thing that breaks the symmetry is the voicing convention, and what it is worth falls from 98 per cent to 90 as the bass fills in, not because fewer changes can be held but because the motion around them is evidence for nothing. Chasing five points of residue above the toss found a defect that had been flattering the reading since it was written.

harmony · Progression The leap at the barline is not in the harmony: it is in which octave the root is played in. How often the size of the bass's motion places the barline, at one chord a bar over eight bars, against how often the bass moves inside the chord, under two realisations of the same pitch classes in a register. Roots at the bottom of the compass: 100% at 0%, 100% at 13%, 100% at 25%, 99% at 38%, 96% at 50%, 95% at 63%, 93% at 75%, 92% at 88%, 90% at 100%. Least motion: 100%, 100%, 96%, 91%, 83%, 69%, 52%, 32%, 20%. Also drawn, the reading of whether the bass moved at all: 100%, 100%, 99%, 99%, 93%, 89%, 70%, 54%, 20%. The two realisations use the same notes in the same order and differ only in which octave each is played in.

The leap at the barline is a fact about register

A bass leaps to a root at the barline and steps inside the bar, and no reading of pitch classes can see it — so the essay that named it recorded it as a limit. Given octaves and the size of every motion, it is worth nothing: realised at least motion the reading tracks whether the bass moved at all and ends 0.87 semitones on the wrong side of zero, because a line that always takes the nearest octave has no leaps to take. Play a stated root at the bottom of the compass instead, and the same sixteen notes in the same order turn a margin of minus 1.67 semitones into plus 1.67 and hold at 90 per cent where the move reading has fallen to 20. The barline is not where the bass moves furthest. It is where the bass goes back down.

harmony · Progression

Named alongside it

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

Harmonic rhythmMetreRegisterDownbeatEvidenceInferenceKey-findingModulationScale degreeInversionProgressionNull model

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