Harmony and voice leading

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.

Assumes: The chords never move the barline · A count and a correlation

The chords never move the barline read constructed four-bar passages under each of their eight candidate barlines, by the metre, by the chords, and by the joint search’s product of the two. Whenever the metre and the chords preferred different barlines the product took the metre’s, on up to 72 per cent of passages, and in fifteen hundred passages the chords never moved it. That essay ended by saying why and what it did not yet say. The metre was scaled by its own range across the eight candidates, so its worst candidate was multiplied by nought, while the chords’ scores went in unscaled and differed across candidates by a few per cent. A product of those two numbers is decided by the metre whatever the evidence.

The repair was named two essays earlier. A fourth decision, and two that were never made noted that scaling a score by its own range is a device for putting it on a scale and not for saying whether any candidate is well attested, and that a metre scored against shuffled onsets would be a quantity the search could weigh against a chord. A count and a correlation did the same thing for a cadence count and a key profile: two statistics in different units become comparable when each is measured in standard deviations above what it gives by chance.

A null for each reading

The metre’s null is the same number of notes placed at random slots over the four bars. A random placement has no preferred barline, so one null serves all eight candidates: its mean and spread say what metre score a passage with that many notes would get by accident, and each candidate’s score is reported in standard deviations above it.

The chords’ null is the passage’s own pitch classes shuffled among its own notes, across the whole passage. Every onset stays where it is and every metrical weight is untouched; what is destroyed is which notes share a bar. That is deliberately not the null the previous essay tried, which shuffled notes within each bar and so could not credit a bar made of one chord — every shuffle of such a bar fits its chord equally well. Each candidate barline weights the notes differently, so each gets its own mean and spread.

The search then adds the two standard scores and takes the largest sum. An equal sum is itself a choice; it says one standard deviation of metrical evidence is worth one of harmonic evidence, which is the only weighting that assumes nothing.

Four bars against each axis's own chance, 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, sum of z 5.97; 1 quaver late metre, z -2.45, chords, z 2.14, sum of z -0.31; 2 quavers late metre, z -1.38, chords, z 2.15, sum of z 0.76; 3 quavers late metre, z -0.31, chords, z -0.27, sum of z -0.58; 4 quavers late metre, z 3.97, chords, z -1.63, sum of z 2.34; 5 quavers late metre, z -2.45, chords, z 1.27, sum of z -1.18; 6 quavers late metre, z -1.38, chords, z 1.18, sum of z -0.21; 7 quavers late metre, z -0.31, chords, z 1.05, sum of z 0.74. Best metre, z: as written and 4 late. Best chords, z: 2 late. Best sum of z: as written. — Fig. 1 The passage from the previous essay — chords C, F, E minor and D minor, every note a chord tone — with each candidate barline scored against each reading’s own chance. The metre stands 3.97 standard deviations above chance at the written barline and at the half-bar alike. The chords stand 2.00 above chance at the written barline and 2.14 and 2.15 at one and two quavers late. The sum takes the written barline, at 5.97.

In the passage above the null turns the metre’s tie into two equal standard scores of 3.97 and leaves the chords where they were: 2.00 standard deviations above chance at the written barline, 2.14 at one quaver late and 2.15 at two. The chords’ preference in this passage for a barline slightly late — the commonest way the chords miss, because a note moved across a barline lands on the lightest slot of the next bar, and helped here by the loop the passage is read as — survives being measured in its own units. The sum takes the written barline at 5.97, because the metre’s 3.97 there is added to the chords’ 2.00 rather than to their −1.63 at the half-bar.

The sum finds the barline where the product did

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%. — Fig. 2 A hundred passages at each rhythm setting, every note a chord tone. How often the product, the sum of standard scores, the metre alone and the chords alone in their own units find the written barline. The sum and the product are within six points of each other at every setting.

With every note a chord tone, the product found the written barline on 16, 56, 66, 56 and 45 per cent of passages as rhythm regularity rose from nought to one. The sum finds it on 22, 58, 61, 58 and 48. The two differ by six points at most, and a hundred passages a setting cannot resolve a difference that small. Putting both readings on one scale did not make the search better at the question it exists to answer.

The product and the sum of standard scores, finding the barline, chords at 0. 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 8%, sum of z finds it 6%, metre finds it 10%, chords, z 15%; rhythm 0.25: product finds it 26%, sum of z finds it 23%, metre finds it 31%, chords, z 9%; rhythm 0.5: product finds it 41%, sum of z finds it 39%, metre finds it 50%, chords, z 12%; rhythm 0.75: product finds it 56%, sum of z finds it 58%, metre finds it 50%, chords, z 8%; rhythm 1: product finds it 51%, sum of z finds it 53%, metre finds it 50%, chords, z 13%. — Fig. 3 The same with every note a random scale tone, so the chords carry no information about the barline. The sum and the product are again within a few points of each other, and the chords alone find the barline at about chance.

The control says the same thing from the other side. With random notes the chords find the barline on 8 to 15 per cent of passages, which is chance, and the sum finds it on 6, 23, 39, 58 and 53 against the product’s 8, 26, 41, 56 and 51. Neither combination is hurt much by a reading with nothing in it, and neither is helped much by a reading with something in it. Both are doing what the metre alone does, which is 10, 31, 50, 50 and 50 with random notes and nearly the same with chord tones — and then breaking the half-bar tie about as well as a coin once the chords have nothing to say.

The chords can now move the barline, and it does not help

The one thing the null plainly changes is who is allowed to overrule whom.

Which reading moves the other's barline, as a product and as a sum of standard scores. 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 · chords 0: chords move the metre's barline 0%, chords move it, in z 12%, metre moves the chords' barline 49%, metre moves it, in z 45%; rhythm 0.25 · chords 0: chords move the metre's barline 0%, chords move it, in z 11%, metre moves the chords' barline 66%, metre moves it, in z 60%; rhythm 0.5 · chords 0: chords move the metre's barline 0%, chords move it, in z 5%, metre moves the chords' barline 72%, metre moves it, in z 70%; rhythm 0.75 · chords 0: chords move the metre's barline 0%, chords move it, in z 0%, metre moves the chords' barline 66%, metre moves it, in z 81%; rhythm 1 · chords 0: chords move the metre's barline 0%, chords move it, in z 0%, metre moves the chords' barline 0%, metre moves it, in z 0%; rhythm 0 · chords 1: chords move the metre's barline 0%, chords move it, in z 17%, metre moves the chords' barline 59%, metre moves it, in z 56%; rhythm 0.25 · chords 1: chords move the metre's barline 0%, chords move it, in z 11%, metre moves the chords' barline 47%, metre moves it, in z 43%; rhythm 0.5 · chords 1: chords move the metre's barline 0%, chords move it, in z 1%, metre moves the chords' barline 55%, metre moves it, in z 73%; rhythm 0.75 · chords 1: chords move the metre's barline 0%, chords move it, in z 0%, metre moves the chords' barline 44%, metre moves it, in z 78%; rhythm 1 · chords 1: chords move the metre's barline 0%, chords move it, in z 0%, metre moves the chords' barline 0%, metre moves it, in z 0%. — Fig. 4 How often each search moves the metre’s best barline and how often it moves the chords’, as a product and as a sum of standard scores, with random notes and with every note a chord tone. As a product the chords never move the metre’s barline. As a sum they move it on up to 17 per cent of passages — only where the rhythm is irregular, and nearly as often with random notes.

As a product the chords moved the metre’s barline on no passage. As a sum they move it on 17, 11, 1, 0 and 0 per cent with every note a chord tone. That is a change, and it is confined to passages whose rhythm says little — a rhythm regularity of nought or a quarter — because on any passage with a regular rhythm the metre’s standard score at its best candidates is so large that no chord reading can outweigh it.

The tell is the control row. With random notes, whose chords say nothing about where a bar begins, the sum moves the metre’s barline on 12, 11, 5, 0 and 0 per cent of passages — nearly as often as with every note a chord tone. On an irregular rhythm the metre’s standard scores are themselves near noise, and two noises added together take turns. The chords moving the barline there is mostly the chords’ noise beating the metre’s.

The search is still two numbers multiplied

The previous essay found that the product’s success rate is almost exactly the share of passages in which the metre ties the barline with the half-bar, times the share in which the chords then prefer the barline. The sum of standard scores can change only the second of those numbers, since a null that is the same for every candidate barline cannot separate two candidates the metre scores identically.

The tie the metre always has, and how often the chords break it the right way. 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 ties the half-bar 31%, chords break the tie right 69%, chords, z, break it 71%; rhythm 0.25: metre ties the half-bar 79%, chords break the tie right 71%, chords, z, break it 71%; rhythm 0.5: metre ties the half-bar 100%, chords break the tie right 68%, chords, z, break it 64%; rhythm 0.75: metre ties the half-bar 100%, chords break the tie right 56%, chords, z, break it 58%; rhythm 1: metre ties the half-bar 100%, chords break the tie right 45%, chords, z, break it 48%. — Fig. 5 With every note a chord tone: how often the metre ties the written barline with the half-bar, and how often the chords then prefer the written barline, raw and in their own standard units. Standardising moves the tie-break by at most four points in either direction.

It barely changes it. With every note a chord tone the chords break the tie right on 69, 71, 68, 56 and 45 per cent of tied passages raw, and on 71, 71, 64, 58 and 48 per cent in standard units. Multiplied by the tie shares of 31, 79, 100, 100 and 100 per cent, those give the sum’s success where the tie is most of the story: 56 at a quarter against the sum’s 58, 64 at a half against 61, 58 against 58, 48 against 48. A null cannot make a reading know something it did not know; it can only report the reading in different units, and when the decision is a comparison between two candidates the units cancel.

Where the rhythm is irregular the tie is not most of the story — the metre narrows to the barline and the half-bar on only 31 per cent of passages with no rhythm — and that is the one place the sum and the product differ by six points. It is also the place where the chords can move the barline at all, which is the next table.

How strong each witness is

The null’s real value is not the search it produces but the quantity it makes available: how far above chance each reading stands at the right answer.

Each reading's standard score at the barline, at the half-bar and elsewhere. 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 z, barline -0.09, metre z, elsewhere 0.08, chords z, barline 1.55, chords z, half-bar 0.37, chords z, elsewhere 0.73; rhythm 0.25: metre z, barline 1.40, metre z, elsewhere -0.47, chords z, barline 1.20, chords z, half-bar 0.07, chords z, elsewhere 0.33; rhythm 0.5: metre z, barline 2.62, metre z, elsewhere -0.85, chords z, barline 0.71, chords z, half-bar 0.01, chords z, elsewhere 0.39; rhythm 0.75: metre z, barline 4.03, metre z, elsewhere -1.32, chords z, barline 0.44, chords z, half-bar 0.12, chords z, elsewhere 0.34; rhythm 1: metre z, barline 5.90, metre z, elsewhere -1.95, chords z, barline -0.18, chords z, half-bar 0.08, chords z, elsewhere -0.05. — Fig. 6 With every note a chord tone, each reading’s standard score averaged over a hundred passages: the metre at the written barline and at the six candidates other than the barline and the half-bar, and the chords at the written barline, at the half-bar and at the other six. The metre stands 5.90 above chance at the barline with a fully regular rhythm; the chords stand 1.55 above it at best.

At the written barline the metre stands −0.09, 1.40, 2.62, 4.03 and 5.90 standard deviations above chance as the rhythm grows more regular, and at the other candidates it falls correspondingly below. The chords, with every note a tone of its bar’s chord, stand 1.55 above chance at the written barline when the rhythm is irregular and the bars are full, 1.20 at a quarter, 0.71 at a half, 0.44 at three quarters and −0.18 when there are only two notes a bar. At the six other candidates they stand 0.73, 0.33, 0.39, 0.34 and −0.05 above it.

So the chords’ evidence for the right barline over a wrong one is at most 0.87 of a standard deviation, at a rhythm regularity of a quarter. The metre’s is up to 7.85. In their own units the two witnesses differ by a factor of about nine, and the product’s arithmetic, which looked like a thumb on the scale, was giving roughly the verdict the evidence gives.

The half-bar column is the one that matters most and says least. The chords’ standard score at the half-bar is 0.37 at most and near nought everywhere else. The gap between it and their score at the written barline is what breaks the tie the metre always has, and it is 1.18 standard deviations with no rhythm, 1.13 at a quarter, 0.70 at a half and 0.32 at three quarters — about one standard deviation at best, which is why the chords break the tie right only two times in three.

The spread is noise-sized for the chords

How far apart the eight barlines are in each reading's own standard units. 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 · chords 0: metre spread, sd 2.46, chords spread, sd 2.69; rhythm 0.25 · chords 0: metre spread, sd 2.95, chords spread, sd 2.63; rhythm 0.5 · chords 0: metre spread, sd 4.39, chords spread, sd 2.46; rhythm 0.75 · chords 0: metre spread, sd 6.24, chords spread, sd 2.36; rhythm 1 · chords 0: metre spread, sd 7.85, chords spread, sd 1.11; rhythm 0 · chords 1: metre spread, sd 2.41, chords spread, sd 3.04; rhythm 0.25 · chords 1: metre spread, sd 3.20, chords spread, sd 2.95; rhythm 0.5 · chords 1: metre spread, sd 4.40, chords spread, sd 2.71; rhythm 0.75 · chords 1: metre spread, sd 6.18, chords spread, sd 2.31; rhythm 1 · chords 1: metre spread, sd 7.85, chords spread, sd 1.43. — Fig. 7 How far apart the eight candidate barlines are in each reading’s own standard units — the largest standard score minus the smallest, averaged — with random notes and with every note a chord tone. The metre’s spread grows threefold with rhythm regularity. The chords’ is nearly the same whether their notes are random or all chord tones.

The last check is whether a reading can tell the candidates apart at all. Eight standard scores drawn from noise alone spread over a couple of standard deviations just by chance, so the question is whether a reading spreads its candidates wider than that.

The metre does, and by a growing margin: from 2.41 standard deviations with no rhythm to 7.85 with a fully regular one. The chords do not. With random notes their eight candidates spread 2.69, 2.63, 2.46, 2.36 and 1.11 standard deviations; with every note a chord tone, 3.04, 2.95, 2.71, 2.31 and 1.43. The difference between a passage of random notes and a passage made entirely of its chords is at most 0.35 of a standard deviation in how far apart the chord reading puts the candidate barlines — the width of the noise it is measured against.

Why the chords say so little

The reading scores each bar by how well its best chord fits, and averages. Moving the barline by one quaver moves one note across it onto the lightest slot of the next bar, so the bars one quaver late fit nearly as well as the written bars. Moving it by half a bar splits every chord between two bars, and the best chord of a bar made of half of one triad and half of another still covers a good share of its notes. Nothing in the reading asks whether the chord changes at the barline and holds inside the bar, which is the evidence a listener taking dictation uses and the evidence how often the chord changes measured as harmonic rhythm.

The passage drawn above shows what a half-bar does to the reading and why it is still not enough. At the written barline its four bars read C major, F major and E minor, each scoring a perfect 1.00, and a last bar of two Fs that no triad covers parsimoniously, at 0.33 — a mean of 0.833. Half a bar late every bar straddles two chords, and the bars read F major seventh at 0.75, A minor at 0.63, F major seventh at 0.46 and F major seventh at 0.70, a mean of 0.634. In the chords’ own units that is 2.00 standard deviations above chance against −1.63 below it, a gap of more than three and a half. That passage is unusually clean. Averaged over a hundred like it, with every note a chord tone and a half-regular rhythm, the gap between the barline and the half-bar is 0.70, because most bars hold three or four notes rather than five, and a best chord can cover three notes drawn from two triads nearly as well as three drawn from one.

So the conclusion is not that harmony is a weak cue to the barline. It is that a harmonic reading built for a different question — which notes are the chord, asked of a bar already given — is a weak cue, and that a null was needed to see it, because the product’s scaling made the same outcome look like an arithmetic accident.

The arithmetic

The metre and chord scores are the previous essay’s unchanged, over the same fifteen hundred constructed passages: four bars of eight quavers, one triad of C major a bar never repeated, the rhythm and chord dials at nought, a quarter, a half, three quarters and one, and a hundred passages at each setting. Each null is twenty-four draws for each passage, seeded so that the figures redraw identically. A standard score is the reading minus the null’s mean, over the null’s standard deviation, and is set to nought where every draw scores alike. The sum is the metre’s standard score plus the chords’. A rate is good to about five points either way.

What a standard score cannot show

A standard score is not a probability. It says how unusual a candidate is against one chosen null, and two nulls that destroy different things would give different numbers. The chord null here destroys which notes share a bar and keeps everything else; a null that also moved the onsets would be asking a different question.

Twenty-four draws is few. It is enough to estimate a mean and a spread over a bounded score, which is all a standard score needs, and not enough for the tail of the null distribution. That is why nothing here is reported as a significance level.

The equal weighting is an assumption. Any other weight on the two standard scores is a claim about how much a listener trusts rhythm over harmony, and the constructed passages cannot supply one. What they can say is that no weighting would help much, because the chords’ margin at the barline is under one standard deviation.

The passage is a loop. Every candidate barline rotates the notes around the passage rather than cutting it, so a barline one quaver late moves the first note to the end of the last bar. The chords’ preference for slightly late barlines in the example above owes part of its size to that: the written last bar holds two Fs and nothing else, and the loop hands the late last bar a C to make a chord with.

And the chord reading is the only one tried. The weakness measured is the weakness of a best-chord-per-bar score. A reading given the bass, the durations or the cadences might be a much stronger witness, and the passage built to make them disagree is a reminder that the strongest harmonic evidence about structure has so far come from ordered pairs of chords rather than from single ones.

Whose evidence

A student placing barlines in dictation is taught to listen for the chord change, and a listener tapping along to a march or a hymn tune is plainly using harmony when the rhythm is ambiguous. The measurement here does not contradict either. It says the thing they are using is a relation between neighbouring bars, and that a reading which cannot see that relation is nearly deaf to the barline however explicit the harmony — a result about an analysis, measured in its own units, and one the bar above the bar will meet again wherever a chord reading is asked where a larger unit begins.

Still open: a chord reading that looks across the barline

The evidence a chord carries about a barline is that it changes there and holds elsewhere. A reading that scored each candidate barline by how different the best chords are on either side of it, and how alike they are on either side of every slot inside a bar, would be measuring that directly, and it can be put against the same passage-wide null and added to the same metre. What it would settle is whether the chords’ factor of nine is a property of harmony as a cue to metre or of the question the segmenter was built to answer — and whether, given a reading that hears the change, the chords would break the metre’s half-bar tie as reliably as a dictation class says they do.