Harmony and voice leading

A listener who knows the style needs no introduction

A listener who weights each witness to the barline by its record needed eight bars of walking bass before a syncopated tune was heard against the bar nine times in ten, and with no introduction at all it learned the syncopation's lesson instead. Finer records — one for each side a witness takes, one for each section — change nothing. What changes everything is arriving with the style's habits: a listener who already gives the bass's motion and the melody's long notes no vote hears the tune against the bar 93 per cent of the time from its first bar, and 94 per cent for a tune that syncopates every bar. The root need not be trusted more. The others need to be trusted less.

Assumes: A listener who weights its witnesses needs eight bars of bass, not twelve · Two bars of drum hold the barline; a dozen bars of bass do not

A listener who weights its witnesses needs eight bars of bass, not twelve gave the barline listener a record for each witness — the bass’s root, the bass’s motion, the melody, a drum — and weighted each vote by how often that witness had agreed with the reading the listener held. Over a walking bass the root earned a second vote, the motion lost half of one, and a syncopated melody lost almost all of its, so a tune was heard against the bar nine times in ten after eight bars of ground rather than twelve. With no ground at all, the same learning went wrong: the melody’s long notes on the half-bar set the first reading, and the listener learned to distrust the root.

That essay left two ways to repair the no-introduction case. One was finer records — a listener that trusts the bass’s motion in an introduction and discounts it under a tune, or trusts a witness when it says one thing and not when it says the other. The other it named as the largest influence it had not measured: the credit a listener brings from earlier pieces, which it had set by hand to be the same for every witness.

The first repair does nothing, and the second does everything. Records kept for each section, or for each side a witness takes, leave every result within a point of the single record’s. A listener who arrives already giving the bass’s motion and the melody’s long notes no vote — the habits of a style in which the bass walks and the tune syncopates — hears the tune against the bar 93 per cent of the time from its very first bar.

Fig. 1 A syncopated melody over one chord a bar and a walking bass, after a stated number of bars of the harmony and bass alone: the share of its bars heard against the barline by a listener counting every vote alike, by one learning each witness’s reliability from nothing, and by one arriving with the bass’s motion and the melody’s long notes credited at a coin toss and learning from there. With no introduction: 40, 39 and 93 per cent. After two bars: 57, 71 and 99.

Habits as a starting credit

The listener is the earlier one exactly: it reads the passage a bar at a time, each witness votes for the barline or the half-bar, the votes are weighted by the log-odds of each witness’s estimated reliability, and the listener keeps a running total and hears whichever reading leads. What changes is where the reliabilities start. In the earlier essay every witness was credited with a reliability of three in four, worth four bars of evidence, before a note was heard. Here each witness can be credited separately.

A coin toss is the natural credit for a witness a style has taught the listener to ignore. In swing, in the popular song over a walking bass, in bossa nova, the bass walks on every beat, so whether it moves at the barline or the half-bar says nothing about which is which; and the tune’s long notes are placed early, against the bar, as a matter of style. A listener who has heard a great deal of such music can be expected to have learned both facts before the first bar of a new piece. Credited at a coin toss, a witness’s weight starts at zero: its votes are ignored until the record in this piece earns it something.

The root keeps the default credit. The drum, when there is one, too. Nothing else about the listener changes, and the passages are the same passages, generated from the same seeds, as in both earlier essays.

What a coin toss means in a record

A credit of a coin toss is not a fixed decision to ignore a witness. It is a starting point that the record in this piece can move, and the arithmetic of the log-odds weight says which way it will. A witness credited at one half starts at no vote; if it then agrees with the reading the listener holds more often than not, its estimated reliability rises above one half and it earns a vote; if it agrees less often, its estimate falls below one half and — since this listener ignores witnesses below a coin toss rather than reading them backwards — it stays at none.

Over a walking bass under a syncopated tune, both discounted witnesses stay at none. The bass’s motion votes for the half-bar a little more often than for the barline, as a walking bass marks the bar by the note it is on found, and the melody’s long notes vote for it far more often. Neither ever earns a vote back, so the habits are confirmed by the piece rather than revised by it. Meanwhile the root, the only witness with a vote, is credited for agreeing with the reading it set, and its weight rises from one to about two over eight bars. Under the melody each bar then banks nine-tenths of a vote for the barline on average, where the earlier listener with no habits banked seven-tenths after a long ground and lost ground with none.

Knowing the melody is worth more than knowing the bass

Fig. 2 With no introduction and after two bars of ground, the share of the syncopated melody’s bars heard against the barline by a learning listener arriving with no habits (39 and 71 per cent), with the bass’s motion discounted (48 and 85), with the melody’s long notes discounted (86 and 94), and with both (93 and 99).

The two habits are not equal. Discounting the bass’s motion alone lifts the no-introduction case from 39 per cent to 48. Discounting the melody alone lifts it to 86. Both together, 93.

The asymmetry has a plain reason. In the first bar of a passage with no introduction, the melody’s long note on the half-bar is the vote that most often overturns the root. A syncopated melody needs its phrases to land found the syncopation outvoting the root two to one in most passages, and the earlier essay traced the whole capture to the first bars’ reading. Take away the melody’s vote and the first bar is decided by the root and the motion, which lean toward the barline; take away the motion’s as well and it is decided by the root alone, which almost always votes for the barline when it votes at all.

The root’s credit does not matter

Fig. 3 With no introduction, the share of the syncopated melody’s bars heard against the barline by a learning listener, against the credit it brings: the root’s credit swept from 0.6 to 0.95 with the motion and melody at a coin toss — 93 per cent throughout — and the motion’s and melody’s credit swept together from a coin toss to the default with the root left at three in four: 93, 84, 75, 62, 55 and 39 per cent.

It would be natural to think the habit that matters is trusting the bass’s root — that an experienced listener hears the barline because it knows the bass lands on the root at the downbeat. The figure says otherwise. With the motion and melody discounted, how much the root is trusted at the start makes no difference at all: 93 per cent whether it starts at a reliability of 0.6 or 0.95. Once the root is the only witness with a vote, its votes decide the bar whatever their size, since a running total’s sign does not depend on the size of the only thing being added.

What matters is how little the others are trusted. Swept together from a coin toss to the default credit, the motion and melody take the no-introduction case from 93 per cent down through 84, 75, 62 and 55 to 39. An experienced listener’s advantage is in what it ignores. That is a precise version of an observation about expertise that appears in many places: novices attend to everything, and experts to what is diagnostic.

How sure the habit has to be

The habit need not be certain to do most of its work. Credited at 0.55 — barely more than a coin toss — the motion and the melody each start with a fifth of a vote, and the tune is heard against the bar 84 per cent of the time with no introduction. At 0.6 they start with 0.37 of a vote and the figure is 75 per cent; at 0.65, 0.56 of a vote and 62 per cent; at the default three in four, a full vote and 39.

The shape of that decline is the arithmetic of the first bar. With no introduction the listener’s first reading is set by whichever side the first bar’s weighted votes favour, and the capture happens when the melody and the motion together outvote the root. The root casts at most one vote; the melody and the motion, each at a fifth of a vote, cannot outvote it even together, and the first bar is the root’s unless the root abstains. As their weights approach a half each, two of them opposing a lone root come closer and closer to a tie, and past it the syncopation wins the first bar whenever both vote against the barline. Most of the benefit of the style’s habits is therefore bought by a habit strong enough to keep the melody and the motion together below the root — which is a much weaker requirement than ignoring them.

A tune that syncopates every bar

The hardest case in the earlier essays was a melody with a long note on the half-bar in every bar. Two bars of drum hold the barline found that no introduction a song would use made the bass alone a ground for it, and the learning listener needed eight bars. A listener with the style’s habits hears it against the bar 94 per cent of the time from its first bar and all of the time after four. The syncopation’s strength does not matter to a listener who does not count the syncopation as evidence about the bar.

This is the result the earlier essays were circling. The syncopated tune is heard as syncopated — as placed against a bar the listener holds — not because the ground before it built up enough confidence to outlast it, nor because the listener learned within the piece to discount it, but because the listener knew, before the piece began, what kind of evidence a tune’s long notes are in this style.

What an introduction is for, then

For the listener with the style’s habits, the ground before the tune scarcely matters to the barline: 93 per cent with none, 99 after two bars, all of it after four. That does not make introductions pointless in the music; it changes what they are for. The chords never move the barline, and for this listener nothing in the harmony needs to. An introduction still sets the tempo, the key and the band’s sound, and for a listener arriving late in a performance a dancer who comes in late needs the downbeat marked found a marked downbeat is what settles the metre fastest.

What the arithmetic takes away is the introduction’s burden as a ground — the dozen bars of walking bass the earlier essays found a listener without habits would need. Those bars were needed only by a listener who had to learn, inside the piece, what the style teaches over years. A practised listener does not need them, and the styles in question characteristically do not supply them: a count-in and a bar of rhythm section before the tune is common, and on this account it is enough because the listeners it is played for already know how to listen.

The wrong habits

Fig. 5 A learning listener arriving crediting the melody at 0.75, 0.8 and 0.9 — the habit of a style whose tunes mark the barline — against bars of ground. With no introduction: 39, 28 and 18 per cent. After eight bars: 92, 91 and 85.

Habits cut both ways. A listener raised on music whose tunes land on the downbeat — hymns, marches, most children’s songs — would arrive crediting the melody more, not less. Credited at 0.8, the syncopated tune is heard against the bar 28 per cent of the time with no introduction; at 0.9, 18 per cent. After eight bars of ground such a listener is still behind the one with no habits, 85 per cent against 92.

So the same syncopated tune is a different experience for listeners from different traditions, on this arithmetic, and the difference is largest exactly where the music gives the least help — with no introduction. A listener who expects melodies to mark the bar hears a syncopated tune as a tune that has moved the bar, and needs a long ground to be talked out of it. Knowing every metre is slower than knowing none found that a listener who knows a whole repertoire can be slower to settle than one who knows nothing; here knowing the wrong style is worse than knowing none.

Finer records change nothing

Fig. 6 A learning listener with no habits, keeping one record per witness, a separate record for each side a witness takes, and a separate record for the bars before and after the melody enters. The three lines lie on one another: 39, 71, 83 and 92 per cent after none, two, four and eight bars.

The first repair the earlier essay proposed was finer bookkeeping, and it does nothing measurable. A listener that keeps a separate record for each side a witness takes — trusting the root when it votes for the barline and not when it votes for the half-bar — hears exactly what the single-record listener hears in every passage. So does one that keeps separate records for the ground and for the bars under the melody, to within a point.

The reason is that every record is kept against the listener’s own reading. Once a reading is settled, a side-by-side record trusts every vote that agrees with it and discounts every vote that does not, which is the single record’s behaviour made stronger; and a section-by-section record starts the melody’s section with no evidence at all, so it cannot help the listener whose trouble is the first bars. Records measure agreement with what the listener already believes, and no division of them supplies the independent information the first bars lack. Only something the listener brings from outside the piece can.

Whose habits

The styles in question are those the earlier essays named — ragtime, swing, the popular song over a walking bass, bossa nova — where the bass walks and the tune arrives early. The claim is about a listener who has absorbed two of their habits and the difference the habits make in the first bars of a new piece. How much evidence a modulation needs found a key-finder’s answer set by how much history it keeps; here the answer is set by history the listener brought, not history it kept. It fits the familiar observation that listeners used to a style hear its syncopations “correctly” at once while newcomers are turned around — a syncopation’s size, which syncopation is a number about the metre prices against a metre, is only felt against a metre the listener already holds — and it gives that observation a mechanism: not superior perception of the downbeat, but indifference to the witnesses that point away from it.

What the picture cannot show

Whether listeners’ habits take this form. The model represents a style’s habits as starting credits for four witnesses. Whether experienced listeners actually discount a walking bass’s motion and a tune’s placement, or attend more strongly to the root and the drum, or do something else that has the same effect, is a question for experiments with listeners from different traditions.

Graded habits. The credits are swept, but each listener has one set. A real listener’s habits are uncertain too, and a listener only moderately sure of a style’s habits falls between the curves drawn here.

The drum and the rest of the band. The passages here have no drum under the tune, and the habits concern only the bass and the melody. In the styles in question the drummer’s ride pattern, the comping chords and the bass all carry their own conventions about the bar, and a practised listener presumably has habits about each. Every added witness a style has taught the listener to trust would add to the root’s vote; every one it has taught the listener to ignore would simply drop out, which on this arithmetic is the cheaper kind of knowledge to have.

A style change inside a piece. A piece that begins in one style and changes — a ballad introduction before a swing chorus — would give the habitual listener the wrong credits for part of it, and the arithmetic says those are the passages where it would be captured.

Still open: a listener who has to find out which style it is in

A listener arriving at a piece does not know which style it is, and so which habits to bring. The natural next listener keeps two or three sets of credits — one for a style whose tunes mark the bar, one for a style whose tunes syncopate — and weighs them by how well each has predicted the bars heard so far. A metre has to be able to change its mind found a tracker’s stickiness is what lets it hold a beat and what stops it following a change; the same trade appears here one level up. The computation would say how many bars such a listener needs to find the right habits, and whether that is fewer than the ground the listener with no habits needs — which is the question of whether style can be recognised faster than a barline can be learned.