A late root is believed because the record is kept by the reading

The habit worth bringing is to distrust the tune asked what a listener should assume about a piece before hearing it, when the question is where its bar begins. Four witnesses vote each bar for the barline or the half-bar: the bass’s root, the bass’s motion, the melody’s long notes, and a drum when there is one. A listener keeps a record of how often each has agreed with the reading it holds, and weighs each witness’s vote by that record, starting from a credit — a habit.

The habit that won was lopsided. Credit the bass’s root and give the bass’s motion and the melody a coin toss until they have earned more. That listener heard a syncopated tune against the bar from its first bar and lost only five bars in a hundred on a tune that marks it. A listener choosing between that habit and the opposite one did worse than one that simply kept it.

The essay ended by naming the case that could overturn it. Every passage there had a bass stating the chord’s root on the barline, so the witness the winning habit relied on was always honest. Some basses do not do that. One anticipates the next chord, arriving on its root a quaver before the barline. Another puts a different chord tone on the downbeat and the root later in the bar. In either, the habit of trusting the root and doubting the rest might be exactly the wrong way round.

It is the wrong way round for one of them. The two turn out to be different in kind, and the difference is where each bass puts its lie.

Where each bass lies #

The bass is the walking bass of the earlier essays: a note on every crotchet, one chord a bar, the root on the chord’s first beat. Two habits are added, each applied at a stated share of barlines.

An anticipated root arrives a quaver before the barline and is tied across it, so the barline itself has no new note. A late root is replaced on the barline by the chord’s fifth and arrives instead at the half-bar. Both are drawn from their own random stream after each passage is built, so that the passages, the harmony and the melodies are the same ones as before; with neither habit applied, every earlier number is reproduced, including the choosing listener’s 67 per cent on a syncopated tune with no introduction.

What each witness then says is the first figure. With a plain bass the root votes for the barline in half the bars and against it in none — in the other half it abstains, because the walking line happens to reach the root again at the half-bar — and the melody’s long notes, on a marking tune, vote for it in half. The motion votes against the barline in about one bar in eight.

Anticipated at every barline, the root votes exactly as before. The note sounding on the barline is still the new chord’s root; it simply started a quaver earlier. What changes is the motion. The bass no longer moves on the barline — the move happened a quaver before — but it still moves at the half-bar, so the motion votes for the half-bar in every bar.

With the root arriving late at every barline, the root votes for the half-bar in every bar. The motion is nearly silent, because the bass now moves both at the barline and at the half-bar.

So the anticipating bass does not make the root lie. It makes the motion lie. Only the late root lies about itself.

An anticipating bass, and the habit that already doubted the motion #

The four listeners are the earlier essay’s two, the listener with no habits and the one that trusts the root and doubts the rest, and two new ones. One arrives with the opposite habit — doubting the root and trusting the motion and the melody. The last keeps all three sets of habits and weighs them by how well each has predicted the votes so far.

As the share of anticipated barlines rises from none to all, the listener that trusts the root and doubts the rest hears the marking tune against the bar in 98 per cent of bars throughout. It was already giving the motion a coin toss, so the witness that now lies had no vote to lose. The listener with no habits, which credits the motion at three in four, falls from 98 per cent to 57. The one that doubts the root falls from 81 to nothing, because it has handed the bar to the one witness that now votes against it every time.

The choosing listener tracks the listener with no habits almost exactly: 58 per cent at the end. It cannot find out that the motion is lying any faster than the listener with no habits can, for a reason the late root makes plain.

So the earlier essay’s winning habit survives the first of the two basses completely. More than that, the reason it survives is the reason it won: it doubts the witnesses that can be moved off the barline, and keeps faith with the one that sounds on it.

A late root, and every listener that learns #

The late root is the case the earlier essay feared, and it does what was feared. As the share of barlines with the root arriving late rises, the listener that trusts the root hears the marking tune against the bar in 99, 67, 26, 6 and finally 0 per cent of bars. The listener with no habits goes 98, 79, 47, 18, 0. The choosing listener goes the same way.

The one that survives is the listener that arrived doubting the root. It hears the tune against the bar in 81 per cent of bars with a plain bass and still in 74 with every root late. It never gave the root a vote, so a lying root has nothing to lie with, and the honest melody carries the reading.

That is not a victory for doubting the root. The same listener heard the anticipating bass’s marking tune in none of its bars, and a syncopated tune over a plain bass in 4 per cent. It survives the late root because it made the opposite bet, and the opposite bet fails everywhere else.

Why learning does not rescue anyone #

The obvious hope was that a listener who learns its witnesses’ reliability would discover, a few bars in, that the root is lying, and discount it. None does. The choosing listener, which even has a set of habits that doubts the root, never switches to it. The listener with no habits never discounts the root.

The figure shows why, and the reason is built into what a record is. A witness’s record counts how often its vote agreed with the reading the listener already held. In the two bars of ground there is no melody, the motion is nearly silent, and the root votes for the half-bar: the reading forms around the half-bar. When the tune arrives and its long notes vote for the true barline, they disagree with that reading, and the record marks them down. The root agrees with the reading it made, and the record marks it up. By the end of the passage the lying root carries 2.2 votes and the honest melody none.

A record kept against the listener’s own reading can only confirm the witness that made the reading. It has no access to the truth except through the reading, so a witness that deceives the reading from the start is, by the record’s lights, the most reliable witness in the texture. The choosing listener compounds this: it scores each set of habits by how well it predicted the votes, measured against the same reading, so the habit that believes the root predicts best.

The same thing happened under the anticipating bass, more mildly. The motion lies, but the root tells the truth, and the root outvotes it. The listener with no habits still loses some bars — the motion drags the reading in the ground — but the melody and the root agree, and together they win most of the time.

The witness that was always the problem #

The earlier essays of this sequence had already found where the danger lay. A melody marks the barline by where it lingers found the melody’s long notes the one witness that can favour one side of the bar over the other, and a syncopated melody needs its phrases to land found a syncopated melody leading a listener to the wrong bar three times in four. Syncopation is a number about the metre measured syncopation against a metre already known. Every one of those treated the melody as the witness that might lie and the bass as the one that would not. The two basses here move the lie into the bass, and the late root puts it into the one witness everything else had been checked against.

A texture in which nothing says where the bar begins found that when no witness is reliable a listener’s habits have nothing to act on. The late root is a worse case than that: one witness is reliable — the melody — and a listener who learns from agreement cannot find out which.

What an introduction does now #

Two bars of drum hold the barline, a dozen bars of bass do not and the essays after it found introductions helping: bars of ground before the tune let a listener bank confidence in the barline. That depended on the ground telling the truth.

With a plain bass, the listener with no habits hears the marking tune at 98 per cent however long the introduction. Over an anticipating bass it hears 87 per cent with no introduction, 57 after two bars, 45 after four and 42 after eight. The listener that doubts the root, over a late root, falls from 90 per cent to 38 across the same range.

In both cases the ground’s loudest witness is the liar — the motion, or the root — and the ground has nothing else in it. Every bar of introduction is a bar of the wrong reading built up and ratified before the honest tune can be heard. An introduction is only as good as the witness it lets speak.

No habit serves every bass #

Put all six combinations side by side and the earlier essay’s question — whether any single set of habits serves every style — has a plain answer. The listener that trusts the root and doubts the rest is at 98 per cent or better in the four textures with an honest root, and at nothing in the two with a late one. The listener that doubts the root reaches 74 per cent over a late root with a marking tune, nothing over an anticipating bass, and 4 per cent over a plain bass with a syncopated tune. The listener with no habits and the one choosing among all three fall to a quarter and a third over an anticipating bass with a syncopated tune, and to nothing over a late root.

So the style does have to be recognised after all, as the earlier essay suspected it might — but not by listening to the bar. A listener cannot learn from the votes which witness is lying when the lie comes first, because the votes are scored against a reading the liar made. What would tell it is something the model has no witness for: knowing, before the tune, that this is music whose bass states its root late. A walking bass marks the bar by the note it is on was right that the root is the witness a walking bass supplies. It supplies it only in the styles in which the root is where the bar begins.

The model the numbers rest on #

Passages of one chord a bar, a walking bass on every crotchet, and a melody of crotchets on chord tones with a long note in half its bars, on the downbeat for a marking tune, the half-bar for a syncopated one, and at random between the two for a tune that does both; four hundred passages at each length of introduction, seeded as in the earlier essays. The bass’s habits are applied at each barline with the stated probability: an anticipated root moves the barline’s root onset a quaver earlier and removes the barline’s own onset; a late root puts the fifth on the barline and the root at the half-bar. A listener keeps two running scores, for the barline and the half-bar, and per-witness records against the reading it held, with a starting credit worth four bars. The four habits are: every witness at three in four; the root at three in four and the motion and melody at a coin toss; the root at a coin toss and the motion and melody at three in four; and all three kept at once and weighed by how well each predicted the bar’s votes, flattened two per cent a bar towards even.

What the model leaves out #

Real basses mix their habits. An anticipating bass anticipates some chords and not others, and a bass that states a chord tone other than the root on the downbeat does so for voice-leading reasons in some bars. The figures sweep the share of barlines affected, and the conclusions about which witness lies hold at every share.

A listener has more than four witnesses. Drums, harmonic rhythm, the melody’s shape and the lyric’s stresses all mark the bar in real music, and a style whose bass states its root late usually has a drum that marks the downbeat clearly. With an honest drum in the ground, the reading would form around the barline and the record would work for the listener instead of against it. That drum is what the earlier essays found holding the bar against a doubtful bass.

And the records cannot go negative. A witness found reliably wrong is discounted to nothing, not read backwards. A listener who could read a consistent liar backwards would recover the late root’s bar from the root itself — but only if it first knew which way the reading ought to go, which is the thing it is missing.

Whose music #

The textures are the walking-bass texture this sequence has used throughout, varied in the way a bass line actually varies between styles: anticipations are characteristic of a great deal of Latin American dance music and of pop and funk bass lines that push the next chord early, and bass lines that avoid the root on the downbeat occur where a style keeps the bass off the first beat or voices it in inversion. The claim is not about any one repertoire. It is that the two ways of moving the root off the barline are different to a listener: one leaves the root honest and moves the lie to the bass’s motion, and one makes the root itself the liar, and a listener who learns from agreement with its own reading can recover from the first and not from the second.

Still open: whether one honest bar at the start is enough #

The late root defeats every learning listener because the reading forms before any honest witness has spoken. That points at a specific, computable threshold. Give the passage a single honest cue at its very start — one drum stroke on the downbeat, one long melody note on the first barline — and the reading forms around the true bar; from then on the melody agrees with it and the late root disagrees, and the record should work the right way round. The computation is this one with an honest cue in the first bar or the first few, and it would say how many bars of truth a listener needs before a lying root, heard every bar afterwards, can no longer take the reading away.