Harmony and voice leading

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.

Assumes: The chords are a weak witness to the barline · How often the chord changes

The chords are a weak witness to the barline put a harmonic reading and a metrical one on the same footing, each in standard deviations above its own chance, and measured how strongly each speaks for the barline a passage was written with. The metre stood up to 5.9 standard deviations above chance there. The chords, with every note a tone of its bar’s chord, stood 1.55 at best. They broke the one tie the metre always has — between the barline and the barline half a bar later — the right way two times in three, and the essay’s last section named the reason and did not test it.

The reason was the question the reading asked. It read each bar by its best chord and averaged how well those chords fitted, which is the question of which notes are the chord asked of a bar already given. Nothing in it compares one bar with the next. And what a student taking dictation is taught to listen for, and what how often the chord changes measured as harmonic rhythm, is exactly a comparison: the chord changes at the barline and holds inside the bar.

So the debt was a reading that hears the change, put against the same passage-wide null and added to the same metre. Here it is, on the same fifteen hundred constructed passages.

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. — Fig. 1 The passage both earlier essays drew — chords C, F, E minor and D minor, every note a chord tone — with each candidate barline scored four ways in standard deviations above chance: the metre, the chords read bar by bar, the chords read by where they change, and the metre plus the change. The metre ties the written barline with the half-bar at 3.97. The bar-by-bar reading prefers a barline two quavers late. The change reading stands 4.30 above chance at the written barline and exactly 4.30 below it at the half-bar, and the sum takes the written barline alone, at 8.27.

A reading that compares neighbours

The change reading cuts the passage into half-bars and reads each by its best chord — the same best-chord fit the bar-by-bar reading uses, applied to four slots instead of eight, and read on a half-bar’s own metrical grid so that the same notes count the same whichever half of a bar a candidate puts them in.

Every boundary between two half-bars is then one of two kinds under a given candidate. Either it is one of that candidate’s barlines, or it is the middle of one of its bars. For each boundary the reading asks how different the chords on its two sides are, as the share of the first chord’s tones the second does not hold: nought for the same triad, a third for two triads sharing two tones, one for triads with nothing in common. The score is the mean difference across the barlines minus the mean difference across the middles.

A candidate at the right barline, in a passage whose chord changes at every bar, puts every change at a barline and every hold at a middle, and scores high. The candidate half a bar away puts every change at a middle and every hold at a barline, and scores low by exactly the same amount — because moving by half a bar turns every barline into a middle and every middle into a barline, and the score is the difference between the two. The change reading is antisymmetric under the one shift the metre cannot see. That is what makes it the right instrument for the tie: it cannot tie the barline with the half-bar unless it scores both at nothing.

The null is the one the previous essay built, on the principle a count and a correlation established for putting two statistics in one unit: the passage’s pitch classes shuffled among its own notes, which keeps every onset and destroys which notes sound together. Because a shuffled passage read at the half-bar is the same shuffled passage read at the barline with the sign reversed, the null is antisymmetric too, and the standard score at the half-bar is exactly the negative of the standard score at the barline — in every passage, and so in every average.

It breaks the tie

The tie is the whole of the difficulty, so it comes first.

The tie the metre always has, broken by where the chords change, 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 ties the half-bar 31%, chords, z, break it 71%, change, z, breaks it 94%; rhythm 0.25: metre ties the half-bar 79%, chords, z, break it 71%, change, z, breaks it 96%; rhythm 0.5: metre ties the half-bar 100%, chords, z, break it 64%, change, z, breaks it 84%; rhythm 0.75: metre ties the half-bar 100%, chords, z, break it 58%, change, z, breaks it 90%; rhythm 1: metre ties the half-bar 100%, chords, z, break it 48%, change, z, breaks it 81%. — Fig. 2 With every note a chord tone: how often the metre ties the written barline with the half-bar, how often the bar-by-bar chord reading then prefers the written barline in its standard units, and how often the change reading does. The bar-by-bar reading breaks the tie right on 71, 71, 64, 58 and 48 per cent of tied passages as the rhythm grows more regular; the change reading on 94, 96, 84, 90 and 81.

On passages where the metre ties the barline with the half-bar, the bar-by-bar reading chose right on 71, 71, 64, 58 and 48 per cent as rhythm regularity rose from nought to one. The change reading chooses right on 94, 96, 84, 90 and 81 per cent.

The recorded question was whether a reading that hears the change would break the tie as reliably as a dictation class says it does. The answer is nearly, and it is worth being exact about the word. Ninety-four per cent is not certainty, and eighty-one, at the most regular rhythm, is one passage in five wrong. But it is a different kind of number from the previous essay’s, which at the most regular rhythm was a coin, and from the product of the chords never move the barline, under which the chords could not break a tie in the metre’s favour or against it. The harmony in these passages did carry the barline all along; the reading that averaged fits bar by bar was throwing the evidence away.

How far above chance each witness stands

The previous essay’s sharpest result was a ratio: the metre’s evidence for the right barline over a wrong one reached 7.85 standard deviations and the chords’ 0.87, a factor of about nine. The change reading has to be put on the same scale.

How far above chance the metre, the chords and the chord change stand at the barline. 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, chords z, barline 1.55, change z, barline 2.44, change z, half-bar -2.44, change z, elsewhere 0.00; rhythm 0.25: metre z, barline 1.40, chords z, barline 1.20, change z, barline 2.06, change z, half-bar -2.06, change z, elsewhere 0.00; rhythm 0.5: metre z, barline 2.62, chords z, barline 0.71, change z, barline 1.53, change z, half-bar -1.53, change z, elsewhere 0.00; rhythm 0.75: metre z, barline 4.03, chords z, barline 0.44, change z, barline 1.36, change z, half-bar -1.36, change z, elsewhere 0.00; rhythm 1: metre z, barline 5.90, chords z, barline -0.18, change z, barline 1.00, change z, half-bar -1.00, change z, elsewhere 0.00. — Fig. 3 With every note a chord tone, averaged over a hundred passages a setting: the metre’s standard score at the written barline, the bar-by-bar chord reading’s, and the change reading’s at the barline, at the half-bar and at the six other candidates. The change reading stands 2.44, 2.06, 1.53, 1.36 and 1.00 above chance at the barline, exactly as far below it at the half-bar, and at nought elsewhere.

At the written barline the change reading stands 2.44 standard deviations above chance with no rhythmic regularity, 2.06 at a quarter, 1.53 at a half, 1.36 at three quarters and 1.00 when every strong slot holds a note and no weak one does. The bar-by-bar reading stood 1.55, 1.20, 0.71, 0.44 and −0.18.

The half-bar column is the antisymmetry, visible: −2.44, −2.06, −1.53, −1.36, −1.00. So the change reading’s margin between the right barline and the half-bar is twice its score at the barline — 4.88 standard deviations with no rhythm, 2.00 with a fully regular one. The bar-by-bar reading’s margin there was 1.18 at best.

The six other candidates average nought, and that is also by construction rather than by luck. A candidate one or three quavers off puts some changes at barlines and some at middles, in a proportion that reverses between it and the candidate half a bar from it, so across all six the reading cancels. What the change reading knows is where in the bar the chords turn over. It does not know, and does not claim to, whether a bar begins on the first slot or the third.

Against the metre the factor of nine is gone for the question the metre cannot answer. For the barline against the half-bar, the metre’s evidence is exactly nothing and the change reading’s is between two and five standard deviations. For the barline against everything else the metre is still the far stronger witness, and it should be: it is the reading built for that.

The two together find the barline

The chords read by where they change, beside the chords read bar by bar, 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: sum of z finds it 22%, chord change finds it 65%, change, z, finds it 55%, metre + change finds it 45%; rhythm 0.25: sum of z finds it 58%, chord change finds it 55%, change, z, finds it 56%, metre + change finds it 78%; rhythm 0.5: sum of z finds it 61%, chord change finds it 47%, change, z, finds it 50%, metre + change finds it 83%; rhythm 0.75: sum of z finds it 58%, chord change finds it 39%, change, z, finds it 39%, metre + change finds it 89%; rhythm 1: sum of z finds it 48%, chord change finds it 20%, change, z, finds it 20%, metre + change finds it 81%. — Fig. 4 With every note a chord tone: how often the metre plus the bar-by-bar reading finds the written barline, how often the change reading finds it alone, raw and in standard units, and how often the metre plus the change reading finds it. The published sum finds it on 22, 58, 61, 58 and 48 per cent of passages; the metre plus the change on 45, 78, 83, 89 and 81.

Added to the metre’s standard score, the change reading finds the written barline on 45, 78, 83, 89 and 81 per cent of passages. The sum with the bar-by-bar reading found it on 22, 58, 61, 58 and 48. At every setting of the rhythm dial the new sum is better, by between 20 and 33 points, and a hundred passages a setting resolves differences of five.

The change reading alone is a different story, and the story is the interesting part. On its own it finds the barline on 65, 55, 47, 39 and 20 per cent of passages — worse and worse as the rhythm grows more regular. It cannot choose among the six candidates that are neither the barline nor the half-bar, so on its own it is guessing across those, and the more regular the rhythm, the fewer notes a half-bar holds. At a rhythm regularity of one there are two notes a bar, one to a half-bar, and a best chord fitted to a single note is a best chord fitted to almost nothing.

That makes the two readings complementary in a way neither essay before this could see. The rhythm that makes the metre strong starves the change reading, and the rhythm that silences the metre feeds it. With no rhythmic regularity the metre finds the barline on one passage in ten and the change reading on two in three; with complete regularity the metre ties everything it finds with the half-bar and the change reading, with one note a half-bar, still breaks four ties in five. Neither is a good reading of a barline alone, and between them the barline is found on four passages in five or more wherever the rhythm has any regularity at all, and on nearly half where it has none.

A control, and what moves the barline

A reading that improves the search could be improving it for a reason that is nothing to do with harmony, so the same four rates go on passages whose notes are random scale tones.

The chords read by where they change, beside the chords read bar by bar, 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: sum of z finds it 6%, chord change finds it 21%, change, z, finds it 24%, metre + change finds it 18%; rhythm 0.25: sum of z finds it 23%, chord change finds it 17%, change, z, finds it 15%, metre + change finds it 27%; rhythm 0.5: sum of z finds it 39%, chord change finds it 11%, change, z, finds it 12%, metre + change finds it 40%; rhythm 0.75: sum of z finds it 58%, chord change finds it 19%, change, z, finds it 19%, metre + change finds it 64%; rhythm 1: sum of z finds it 53%, chord change finds it 13%, change, z, finds it 12%, metre + change finds it 49%. — Fig. 5 The same with every note a random scale tone, so neither chord reading carries information about the barline. The change reading alone finds it on 21, 17, 11, 19 and 13 per cent of passages, about chance, and the metre plus the change finds it on 18, 27, 40, 64 and 49 — within a few points of the metre alone.

With random notes the change reading finds the barline at chance, and adding it to the metre leaves the metre roughly where it was: 18, 27, 40, 64 and 49 per cent against the metre’s own 10, 31, 50, 50 and 50. The small gains and losses are the size of a hundred passages’ noise. The improvement with chord tones is the harmony’s, which is the claim the change reading has to earn and does.

Which reading moves the metre's barline, and how far apart each puts the candidates. 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 it, in z 12%, change moves it, in z 20%, chords spread, sd 2.69, change spread, sd 3.04; rhythm 0.25 · chords 0: chords move it, in z 11%, change moves it, in z 15%, chords spread, sd 2.63, change spread, sd 3.10; rhythm 0.5 · chords 0: chords move it, in z 5%, change moves it, in z 10%, chords spread, sd 2.46, change spread, sd 2.97; rhythm 0.75 · chords 0: chords move it, in z 0%, change moves it, in z 0%, chords spread, sd 2.36, change spread, sd 2.85; rhythm 1 · chords 0: chords move it, in z 0%, change moves it, in z 0%, chords spread, sd 1.11, change spread, sd 1.78; rhythm 0 · chords 1: chords move it, in z 17%, change moves it, in z 26%, chords spread, sd 3.04, change spread, sd 5.69; rhythm 0.25 · chords 1: chords move it, in z 11%, change moves it, in z 10%, chords spread, sd 2.95, change spread, sd 4.91; rhythm 0.5 · chords 1: chords move it, in z 1%, change moves it, in z 4%, chords spread, sd 2.71, change spread, sd 4.09; rhythm 0.75 · chords 1: chords move it, in z 0%, change moves it, in z 1%, chords spread, sd 2.31, change spread, sd 3.72; rhythm 1 · chords 1: chords move it, in z 0%, change moves it, in z 0%, chords spread, sd 1.43, change spread, sd 2.52. — Fig. 6 How often each sum moves the metre’s own best barline, and how far apart each chord reading puts the eight candidates in its standard units, with random notes and with every note a chord tone. With random notes the change reading spreads the candidates 3.04 to 1.78 standard deviations; with chord tones, 5.69 to 3.72 on the first four settings — well clear of the noise, where the bar-by-bar reading’s spread barely moved.

The spread column is the check the previous essay used to show the bar-by-bar reading was nearly deaf: its eight candidates spread almost as far apart with random notes as with chord tones, a difference of 0.35 of a standard deviation at most. The change reading’s spread with chord tones is 5.69, 4.91, 4.09 and 3.72 on the first four settings, against 3.04, 3.10, 2.97 and 2.85 with random notes. It separates the candidates by between 1.3 and 1.9 times what noise does, most where the rhythm says least.

The moves column says the same thing as before about irregular rhythm. Where the rhythm says nothing, the change reading moves the metre’s barline on 26 per cent of chord-tone passages and 20 per cent of random ones, and some of that is noise beating noise, exactly as it was for the bar-by-bar reading. Where the rhythm is at all regular, it moves the metre’s barline almost never, because the metre’s standard score is large everywhere except the half-bar — which is the one place the change reading is meant to act, and does.

Half the notes chord tones

The passages above are the two ends of the harmonic dial: every note a chord tone, or every note random. Real melody sits between, with passing notes and neighbour notes among the chord tones, so the middle setting is the one closest to music.

The chords read by where they change, beside the chords read bar by bar, chords at 0.5. 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: sum of z finds it 18%, metre + change finds it 19%, chords, z, break it 61%, change, z, breaks it 61%; rhythm 0.25: sum of z finds it 41%, metre + change finds it 42%, chords, z, break it 60%, change, z, breaks it 64%; rhythm 0.5: sum of z finds it 57%, metre + change finds it 64%, chords, z, break it 58%, change, z, breaks it 67%; rhythm 0.75: sum of z finds it 53%, metre + change finds it 57%, chords, z, break it 53%, change, z, breaks it 58%; rhythm 1: sum of z finds it 47%, metre + change finds it 67%, chords, z, break it 47%, change, z, breaks it 67%. — Fig. 7 With each note a chord tone half the time and a random scale tone otherwise: how often each sum finds the barline, and how often each chord reading breaks the metre’s half-bar tie. The metre plus the change finds the barline on 19, 42, 64, 57 and 67 per cent, against the bar-by-bar sum’s 18, 41, 57, 53 and 47; the change reading breaks the tie right on 61, 64, 67, 58 and 67 per cent.

At half chord tones the change reading still helps and helps less. It breaks the tie right on 61, 64, 67, 58 and 67 per cent of tied passages, against the bar-by-bar reading’s 61, 60, 58, 53 and 47, and the sum with it finds the barline on up to 67 per cent against 57. The gain is concentrated where the rhythm is regular, which is again where the metre most needs its tie broken.

The drop from the all-chord-tone passages is the cost of a best chord fitted to four slots, one or two of which are passing notes. A half-bar is a short window, and a random scale tone in it is a large share of the evidence. That is a limit of this reading’s window rather than of harmony, and it is the obvious place for a reading to improve.

The passage the figures are built on

Every rate here is over constructed passages of four bars of eight quavers, one diatonic triad of C major a bar and never the same triad twice running, so that a chord change falls at every barline by construction. The rhythm dial is how much likelier a strong slot is than a weak one to hold a note; the harmonic dial is how likely a note is to be a tone of its bar’s chord rather than a random scale tone. There are a hundred passages at each setting, and each reading’s null is twenty-four shuffles of the passage’s own pitch classes, seeded so that every figure redraws identically.

The metre is the onset score the joint search of three decisions that constrain each other has used since it was written, and the metre and the bar-by-bar chord reading are exactly as the previous essay computed them, and their columns reproduce its numbers to the last digit, which is the check that the change reading was added beside them rather than into them. The change reading’s shuffles are the same shuffles, read a second way, so the two chord readings are measured against the same chance.

What the construction hands the change reading

A chord change at every barline. Every constructed passage changes chord at each bar and never inside one. That is the case the change reading is built for and the case where it is strongest, and it is not how music always behaves: a bar can hold two chords, and two bars can hold one. A passage with a harmonic rhythm of one chord every two bars puts a hold at every other barline, and the change reading would score half its barlines as middles.

Half a bar as the unit. The reading compares half-bars because the metre’s tie is a half-bar shift. A passage in three, or one whose chords change on the third beat of four, would need the window cut differently, and a reading that chose its own window would be a reading with a free parameter.

Triads only. Dissimilarity is measured between best triads. A seventh chord over the same root shares three tones with its triad and would register as nearly no change, which is right for a V to V7 and wrong for anything else.

And a loop. The passage wraps, so the boundary after the last half-bar is the boundary before the first. That is what makes the antisymmetry exact, and in a passage that does not wrap it would be very nearly exact rather than exactly so.

What a standard score still cannot settle

Which of these readings a listener uses. The figures show that harmony, read as change, is a strong cue to where a bar begins in these passages. Whether a listener’s sense of the barline is built from something like it, from the bar-by-bar fit, or from neither, is not in a rate.

How much the change reading should be trusted beside the metre. The sum weights one standard deviation of each equally. That is the weighting that assumes nothing, and it is not a measurement of what a listener trusts.

And the other harmonic evidence. The cadence as evidence found ordered pairs of chords carrying a key that single chords could not, and the passage built to make them disagree put that evidence against a profile. A cadence is also a statement about where a phrase, and so a bar, ends. None of that is in this reading, which compares neighbouring chords without asking which chords they are.

Whose ears, and whose barlines

Harmony as a cue to metre is textbook knowledge in two traditions that rarely cite each other. Dictation teaching treats the chord change as the barline’s most reliable witness where the rhythm is ambiguous, and music psychology has measured listeners using harmonic change to infer metre — in Temperley’s preference rules for metrical structure, among others, where a strong beat prefers to fall where the harmony changes. The constructed passages here are a test of that rule’s arithmetic on the one decision the onsets alone cannot make, and the rule passes it by a wide margin once it is written as a comparison rather than a fit.

What the numbers add is the complementarity, which the rule as usually stated does not mention. A listener who relies on harmonic change for the barline relies on it most where the rhythm is least regular, and has least of it where the rhythm is most regular. The two cues arrive in opposite amounts, and the bar above the bar will meet the same pair at the next level up, where the unit is a bar and the question is which bar begins a phrase.

Still open: a harmonic rhythm that is not one chord a bar

Every passage here changes chord exactly at the barline. Real harmonic rhythm does not: a bar of two chords puts a change at the middle, and a chord held across two bars puts a hold at the barline. A change reading on passages whose chord rate is itself a dial — one chord every two bars, one a bar, two a bar — would say whether the reading finds the barline or finds the harmonic rhythm, and whether the half-bar antisymmetry, which is exact here, turns into a preference for whichever unit the chords actually change at. That is the same arithmetic on passages with a third dial, and it would settle whether harmony marks the bar or only the chord.