Harmony and voice leading

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.

Assumes: A fourth decision, and two that were never made · Three decisions that constrain each other

Three decisions that constrain each other resolved a passage’s key, metre and chords together, by scoring every combination and keeping the best. A fourth decision, and two that were never made added a fourth and, reading the search in order to add it, found that two of the three were not being decided at all. The key was the same at every hypothesis. And the metre scored all eight of its candidate barlines identically, because the passages were one bar of eight quavers with a note on every slot, and a pattern with a note everywhere looks the same wherever the bar is said to begin.

That essay named the repair. A passage of four bars with rests in it puts a different pattern of notes under the metrical grid at each candidate barline, and it also puts different notes into each bar, so the chords read in each bar change with the barline too. The barline becomes a decision the metre and the chords both have an opinion about. The prediction recorded with it was that the chords would move the barline more often than the barline moves the chords, because the chord reading has a hundred and forty-four candidates to choose among and the metre score only a handful of values.

Four bars, eight barlines, three readings

The passages are constructed, so that the right answer is known. Each is four bars of eight quavers with the barline written at the first slot. Each bar has a triad of C major, drawn at random and never the same as the bar before. Two dials decide what the notes look like. Rhythm regularity is how much likelier a note is on one of the metre’s strong slots — the downbeat and the half-bar — than on any other: at nought every slot is equally likely to hold a note, at one only the strong slots do. Chord regularity is how likely a note is to be a tone of its bar’s chord: at nought every note is a random scale tone, at one every note is a chord tone.

Each passage is then read under each of the eight places the barline could fall. The metre reading is the one the joint search uses: the onset pattern rotated to that barline and scored against the strong positions, then scaled by its own range across the eight so that the best candidate scores one and the worst nought. The chord reading is the best chord’s score in each of the four bars that barline makes, averaged. The search’s reading is the product of the two.

Four bars and eight barlines: the metre ties the barline with the half-bar, and the chords decide. 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 score 15, metre, scaled 1.00, chords 0.833, product 0.833; 1 quaver late metre score -21, metre, scaled 0.00, chords 0.862, product 0.000; 2 quavers late metre score -15, metre, scaled 0.17, chords 0.842, product 0.140; 3 quavers late metre score -9, metre, scaled 0.33, chords 0.695, product 0.232; 4 quavers late metre score 15, metre, scaled 1.00, chords 0.634, product 0.634; 5 quavers late metre score -21, metre, scaled 0.00, chords 0.813, product 0.000; 6 quavers late metre score -15, metre, scaled 0.17, chords 0.813, product 0.135; 7 quavers late metre score -9, metre, scaled 0.33, chords 0.813, product 0.271. Best metre score: as written and 4 late. Best metre, scaled: as written and 4 late. Best chords: 1 late. Best product: as written. — Fig. 1 One constructed passage of four bars — chords C, F, E minor and D minor, every note a tone of its bar’s chord — read under its eight candidate barlines. The metre scores the written barline and the one four quavers late identically, at 15. The chords prefer the barline one quaver late, at 0.862 against 0.833. The product takes the written barline.

The passage above shows every result in this essay at once. The metre scores the written barline 15 and the barline four quavers late 15 as well; it cannot tell them apart. The chords prefer neither: their best barline is one quaver late, at 0.862, with the written one at 0.833 and the half-bar at 0.634. The product multiplies a metre factor of one at the written barline and at the half-bar by the chords’ 0.833 and 0.634, and takes the written barline. So the chords did decide something — which of the metre’s two equal candidates wins — and the barline they would have chosen on their own was never available to them, because the metre’s scaled score there is nought.

The metre cannot see the half-bar

The tie at four quavers late is not a feature of this passage. The metre’s strong positions in a bar of eight are the downbeat and the half-bar. Move the barline by half a bar and the downbeat lands where the half-bar was and the half-bar where the downbeat was, so exactly the same slots sit under the strong positions and the metre score is identical. It is identical for every passage, and every figure here checks that from the onsets before drawing anything.

How often the metre, the chords and their product find 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 · chords 0: metre finds it 10%, chords find it 12%, product finds it 8%, notes a bar 5.90; rhythm 0.25 · chords 0: metre finds it 31%, chords find it 11%, product finds it 26%, notes a bar 4.91; rhythm 0.5 · chords 0: metre finds it 50%, chords find it 12%, product finds it 41%, notes a bar 3.94; rhythm 0.75 · chords 0: metre finds it 50%, chords find it 11%, product finds it 56%, notes a bar 2.98; rhythm 1 · chords 0: metre finds it 50%, chords find it 13%, product finds it 51%, notes a bar 2.00; rhythm 0 · chords 1: metre finds it 10%, chords find it 41%, product finds it 16%, notes a bar 5.99; rhythm 0.25 · chords 1: metre finds it 34%, chords find it 35%, product finds it 56%, notes a bar 5.04; rhythm 0.5 · chords 1: metre finds it 49%, chords find it 21%, product finds it 66%, notes a bar 3.90; rhythm 0.75 · chords 1: metre finds it 50%, chords find it 17%, product finds it 56%, notes a bar 2.97; rhythm 1 · chords 1: metre finds it 50%, chords find it 11%, product finds it 45%, notes a bar 2.00. — Fig. 2 A hundred passages at each rhythm setting, with every note a random scale tone and with every note a chord tone. How often the metre, the chords and their product find the written barline — ties shared — and how many notes a bar holds. The metre never finds it more than half the time; the chords alone do best with no rhythm at all; the product does best in between.

With every note a chord tone, the metre finds the written barline on 10, 34, 49, 50 and 50 per cent of passages as rhythm regularity rises from nought to one. It can never pass fifty, and at a rhythm regularity of a half or more it sits at 49 and 50: it narrows eight candidates to two, the barline and the half-bar, and has no way to choose between them. Chance is one in eight, 12.5 per cent.

The chords alone go the other way, from 41 per cent with no rhythm to 11 per cent with complete regularity. The reason is in the last column: the more regular the rhythm, the fewer notes there are — six a bar at nought, two at one, since a perfectly regular rhythm puts notes only on the two strong slots — and two notes a bar are too few for a chord reading to say anything.

The product is best in the middle, at 66 per cent with a rhythm regularity of a half, where the metre has narrowed the choice to two and the chords still have four notes a bar to choose with. That is the joint search working as intended. At either end it is worse than one of its parts: with no rhythm, 16 per cent against the chords’ 41, because it multiplies the chords’ reading by a metre that is noise there.

The rows with random notes say what the chords are worth to the search. With random notes the product finds the barline on 8, 26, 41, 56 and 51 per cent of passages; with every note a chord tone, on 16, 56, 66, 56 and 45. So harmony that is as explicit as it can be adds thirty points at a quarter and twenty-five at a half, and adds nothing at three quarters or at one, where the bars hold three notes or two. The whole of the chords’ contribution is made where the rhythm is partly regular and the bars are still full, which is a narrow band, and a reader of real music would want to know whether most music sits in it.

When the chords miss, a quaver late is the commonest miss

The chords’ own best barline in the passage above was one quaver late. Over a hundred passages with every note a chord tone that is not their usual answer — the written barline is, on 41, 35 and 21 per cent of passages at a rhythm regularity of nought, a quarter and a half — but it is among their commonest wrong ones, at 12, 17 and 17 per cent, where any one wrong barline would get 8 to 11 if the misses were spread evenly. Moving the barline by one quaver moves one note across it, and that note lands on the last slot of the bar it joins — the lightest slot of the metrical grid, weighted an eighth of the downbeat. A bar’s best chord hardly notices a note that light. So the chord reading one quaver late is close to the reading at the written barline, and which wins is often decided by accidents of the notes rather than by where the chords change. In the passage above one of the accidents is the loop the passage is read as: its written last bar holds only two Fs, and the late barline wraps the passage’s first C into it.

A reading that scores each bar by how well its best chord fits never compares one bar with the next, and the evidence the chords carry about a barline is precisely a comparison: the chord changes at the barline and not inside the bar. How often the chord changes measured that change as a rate. Nothing in the reading used here measures it at all.

When the two disagree, the search takes the metre’s side

When the metre and the chords disagree, the product takes the metre's 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 · chords 0: metre and chords disagree 49%, chords move the metre's barline 0%, metre moves the chords' barline 49%; rhythm 0.25 · chords 0: metre and chords disagree 66%, chords move the metre's barline 0%, metre moves the chords' barline 66%; rhythm 0.5 · chords 0: metre and chords disagree 72%, chords move the metre's barline 0%, metre moves the chords' barline 72%; rhythm 0.75 · chords 0: metre and chords disagree 66%, chords move the metre's barline 0%, metre moves the chords' barline 66%; rhythm 1 · chords 0: metre and chords disagree 0%, chords move the metre's barline 0%, metre moves the chords' barline 0%; rhythm 0 · chords 1: metre and chords disagree 59%, chords move the metre's barline 0%, metre moves the chords' barline 59%; rhythm 0.25 · chords 1: metre and chords disagree 47%, chords move the metre's barline 0%, metre moves the chords' barline 47%; rhythm 0.5 · chords 1: metre and chords disagree 55%, chords move the metre's barline 0%, metre moves the chords' barline 55%; rhythm 0.75 · chords 1: metre and chords disagree 44%, chords move the metre's barline 0%, metre moves the chords' barline 44%; rhythm 1 · chords 1: metre and chords disagree 0%, chords move the metre's barline 0%, metre moves the chords' barline 0%. — Fig. 3 How often the metre’s best barline and the chords’ best barline are different, how often the product then moves the metre’s choice, and how often it moves the chords’. The second column is nought in every row; the first and third are equal in every row.

The prediction was about exactly this, and the table answers it without qualification. The metre and the chords prefer different barlines on 44 to 72 per cent of passages, depending on the setting — except with a fully regular rhythm, where a bar holds only its two strong notes and every barline the chords prefer ties with one the metre prefers. On every one of those passages the product takes the metre’s barline. The chords move the metre’s choice on none of fifteen hundred passages, at any setting, with random notes or with every note a chord tone.

The mechanism is the scaling. The metre is scaled by its own range across the eight candidates, so its best candidate is multiplied by one, its worst by nought, and the rest by fractions between. The chords’ scores are not scaled at all, and across eight barlines they differ by a few per cent — from 0.634 to 0.862 in the passage above. A product of a factor that runs from nought to one and a factor that runs from 0.63 to 0.86 is decided by the first. The chords can only choose among the candidates the metre has already tied at the top.

So the number of values each score can take, which the prediction rested on, was the wrong quantity. What decides which reading moves the other is how far each one’s scores are spread across the candidates, and the scaling guarantees that the metre’s spread is the whole unit.

The half-bar is the chords’ whole job

That leaves the chords one decision, and it is the one the metre cannot make.

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 0.25 · chords 0: metre ties the half-bar 75%, chords break the tie right 41%; rhythm 0.5 · chords 0: metre ties the half-bar 100%, chords break the tie right 41%; rhythm 0.75 · chords 0: metre ties the half-bar 100%, chords break the tie right 56%; rhythm 0.25 · chords 0.5: metre ties the half-bar 78%, chords break the tie right 63%; rhythm 0.5 · chords 0.5: metre ties the half-bar 100%, chords break the tie right 54%; rhythm 0.75 · chords 0.5: metre ties the half-bar 100%, chords break the tie right 55%; rhythm 0.25 · chords 1: metre ties the half-bar 79%, chords break the tie right 71%; rhythm 0.5 · chords 1: metre ties the half-bar 100%, chords break the tie right 68%; rhythm 0.75 · chords 1: metre ties the half-bar 100%, chords break the tie right 56%. — Fig. 4 How often the metre’s best candidates include both the written barline and the half-bar, and when they do, how often the chords prefer the written one. Chance is half. Random notes break the tie at chance or worse; every note a chord tone breaks it right on 71 per cent of passages at best.

At a rhythm regularity of a half or more the metre ties the barline with the half-bar on every passage, and at a quarter on three passages in four. Whether the search then finds the barline is entirely whether the chords prefer it to the half-bar. With random notes they prefer it on 41 to 56 per cent of those passages, which is chance. With every note a chord tone they prefer it on 71, 68 and 56 per cent as the rhythm grows more regular and the bars emptier.

That is a real effect and a modest one. A passage in which every single note belongs to its bar’s chord, and each bar’s chord differs from the last, is as harmonically explicit as music gets. The chords still place the bar on the wrong half of it three times in ten.

And it is nearly the whole of what the search achieves. Once the metre has narrowed the candidates, the search finds the barline exactly when the metre’s two include the barline and the chords then prefer it, so its success rate should be the share of passages with that tie times the share in which the chords break it right. With every note a chord tone that product is 79 per cent of 71 at a quarter, which is 56, and the search finds the barline on 56; at a half, 100 of 68 against the search’s 66; at three quarters, 56 against 56; at one, 45 against 45. With random notes at a half it is 41 against 41. The joint search is two numbers multiplied: how often the metre can get the answer down to the barline or the half-bar, and how often the chords can then tell those two apart.

The bar’s own null cannot credit a bar that is one chord

The recorded plan was to use the same scorer and the same null the cue mixture needed: each bar’s pitch classes shuffled among its own slots, and the chord score reported in standard deviations above what the shuffles give. That null made the mixture rankable. It makes the barline worse.

A bar that is one chord scores nothing against its own shuffles. 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 chords 0.833, chords, z in bar 0.00; 1 quaver late chords 0.862, chords, z in bar 0.36; 2 quavers late chords 0.842, chords, z in bar -0.05; 3 quavers late chords 0.695, chords, z in bar 0.86; 4 quavers late chords 0.634, chords, z in bar -0.74; 5 quavers late chords 0.813, chords, z in bar 0.00; 6 quavers late chords 0.813, chords, z in bar 0.00; 7 quavers late chords 0.813, chords, z in bar 0.00. Best chords: 1 late. Best chords, z in bar: 3 late. — Fig. 5 The same passage with the chords read raw and against each bar’s own shuffles. Under the written barline three of the bars are made entirely of one triad’s notes, every shuffle of them fits that triad as well, and each scores a standard score of nought. The null’s best barline is three quavers late.

Under the written barline three of the four bars in that passage are nothing but their triad — C, E and G; C, F and A; B, E and G — and a shuffle of a bar’s notes among its slots leaves them in the bar. Every shuffle fits the triad exactly as well, the spread of the shuffles is nought, and the standard score is nought. The barline three quavers late mixes notes of two chords in its bars, the shuffles scatter, and it scores 0.86. The clearer a bar’s harmony, the less its own null can say about it.

The chords read raw and against their own bar's shuffles, 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: chords find it 41%, chords, z in bar 2%, chords break the tie right 69%, chords, z in bar, break it 66%; rhythm 0.25: chords find it 35%, chords, z in bar 2%, chords break the tie right 71%, chords, z in bar, break it 52%; rhythm 0.5: chords find it 21%, chords, z in bar 4%, chords break the tie right 68%, chords, z in bar, break it 56%; rhythm 0.75: chords find it 17%, chords, z in bar 7%, chords break the tie right 56%, chords, z in bar, break it 57%; rhythm 1: chords find it 11%, chords, z in bar 13%, chords break the tie right 45%, chords, z in bar, break it 50%. — Fig. 6 With every note a chord tone: how often the chords find the barline raw and against their own bars’ shuffles, and how often each breaks the metre’s half-bar tie the right way. Against the bar’s own null the chords find the barline on 2 to 13 per cent of passages and, once the rhythm is regular, break the tie at about chance.

Against the bar’s own null the chords find the barline on 2, 2, 4, 7 and 13 per cent of passages across the rhythm settings, where raw they found it on 41, 35, 21, 17 and 11. At no rhythm the null’s reading is a sixth of chance. It breaks the half-bar tie right on 50 to 57 per cent of passages once the rhythm is regular, which is chance.

The two questions are different, and the null belongs to the first. The mixture asked whether the notes fit the weights — whether the long or strong notes are the chord tones — and shuffling notes among slots is exactly the test of that. The barline asks whether the notes of a bar belong together, and a shuffle that keeps every note in its bar cannot test it.

The arithmetic

The metre score is the rhythm essays’ own: three for each strong position holding a note, minus two for each one empty, minus one for each note elsewhere, over the four bars, with the onsets rotated to each candidate barline. The strong positions are the slots whose Longuet-Higgins and Lee weight is at least minus one, which in a bar of eight quavers is the downbeat and the half-bar. The chord score is the segmenter — coverage of the metrically weighted notes times the share of the chord actually sounded, over every chord in the table at every root — taken bar by bar and averaged over the four. The product and the range scaling are the joint search’s. No durations are used, so the cue mixture is at its metrical end throughout.

Each setting is a hundred passages, so a rate is good to about five points either way, and a difference of a few points between two settings is not a finding. A candidate that ties for best shares the credit equally. The bar null is sixteen shuffles.

What four constructed bars cannot show

The metre score’s strong positions are two and equal. The metrical weights it is built from do distinguish the downbeat from the half-bar — nought against minus one — and the score throws that away by calling both strong. A score that kept the grades would split the tie the chords are left to break. The tie is a property of this score, not of metre, and it is the one the bar above the bar meets one level up.

The chords are drawn independently and never repeat. Real progressions repeat chords across barlines, hold one chord for two bars and change on the half-bar, and every one of those weakens what a chord change can say about where a bar begins.

There is no bass, no duration and no register. A listener placing a barline hears which note is lowest and which is held, and the long note and the strong note found that reading notes by how long they are held names a different chord from reading them by metrical position on sixteen per cent of passages even where the two cues agree.

The passage is read as a loop. A barline one quaver late puts the passage’s first note at the end of its last bar, as the joint search always has, because the onsets are rotated rather than cut. A passage heard once has no such wrap, and its first and last bars would be read with a note fewer or more than the loop gives them.

And the passages are short. Four bars is the least over which the barline is a decision, and a listener who has heard sixteen has four times the evidence. A metre has to be able to change its mind is the model of how that evidence accumulates, and it has not been given chords.

Whose barlines

The confusion the metre is left with is a familiar one on paper. Whether a piece is barred in four or in two, whether a march is in 2/4 or 4/4 and where alla breve divides — these are exactly the choice between a barline and the barline half a bar away, and they are settled in dictation classes by harmonic rhythm: the chord changes on the downbeat. That is the rule the constructed passages obey perfectly and the chord reading used here barely hears. The repertoire in which it is most useful is the one in which a chord lasts a bar — hymn tunes, marches, the thirty-two-bar song — and the one in which it misleads is the one where harmony anticipates the bar, which syncopation is a number about the metre prices from the rhythmic side.

Still open: what each reading is worth in its own units

The metre won every disagreement because it was scaled by its own range and the chords were not scaled at all, which is a statement about two rulers and not about two kinds of evidence. The recorded repair is a null for each: the metre scored against the same number of onsets placed at random, the chords against the passage’s notes shuffled across its bars rather than within them, and the two added as standard scores. What that would settle is whether the chords lose because the search’s arithmetic makes them lose, or because a best-chord-per-bar reading carries very little evidence about where a bar begins even when every note is a chord tone.