A walking bass marks the bar by the note it is on

A bass that holds through a change marks the barline ended on a failure it had produced deliberately. Its reading scores a candidate barline by the share of that candidate’s barlines on which the bass moves, less the share of its mid-bars, and at one chord a bar that reading is perfect — until the bass starts moving inside the chord as well. Turned up to a bass that took every opportunity to move, the same reading placed the barline in sixteen per cent of passages, which is worse than guessing, and it was worse than guessing for a reason that had nothing to do with chance. A bass that arpeggiates always moves at the half-bar, because it moves to a tone other than the one it is on. At the barline it states the new root, which is sometimes the note the arpeggio has just reached, and then it stands still. So the barline becomes the one beat on which the bass sometimes does nothing, and a reading of motion points half a bar late.

The repair the essay named was to ask a different question of the same line. A bass that moves all the time marks its barline by arriving — by sounding the root of the chord that has just come in — and between barlines by passing through the chord’s other tones. That is the same computation with sounds the root in place of moves, and it is the whole of what follows.

The dial that was turned and the one that was not #

Before any of that, a correction, because the busiest bass the earlier essays could draw was not busy.

arp is the probability that the bass, inside a chord, moves to another of the chord’s tones at an opportunity, and an opportunity was any slot whose metrical weight cleared a threshold. That threshold was set once, at the level the metre model here calls the half-bar. A bar of eight quavers has exactly one such slot inside it. So a bass with arp at one moved once a bar and sounded twice — and the sentence “a bass that moves on every beat”, which the earlier essay used five times, described a line of minims.

Dropping the threshold one level puts the opportunity on the crotchet, and the same dial at full strength then produces a bass sounding four notes in a bar of four — which is what the words have always meant. Every figure below uses that level. It is not a repair of a bug: the old setting produced exactly the bass it was asked for, and the number it produced was reported honestly. It is a repair of a description, and the description was doing work, because the failure that reading suffered is a failure specifically about a line with no gaps in it, and the line that produced it had gaps.

The correction makes the earlier failure milder and the question sharper. At the half-bar level the move reading fell to sixteen per cent. At the crotchet it falls to twenty, and the four points are not a reprieve: the reading is now failing on a line that has no gaps in it, which is the line the failure was always supposed to be about.

Sounding, not attacking #

The reading has one decision in it and everything turns on that decision.

A bass note can be counted at the slot it is struck, or at every slot through which it is sounding. The first is what every reading of these passages has used, because a passage here is a list of onsets and an onset is what a list of onsets contains. The second requires filling in: a bass that states a root on the first quaver of a bar and then says nothing for three beats has one entry in the list and four slots of sound, and until now nothing asked what it was doing in between.

Take the second. A candidate barline scores the share of its barlines at which the note the bass is sounding is the current chord’s root, less the share of its mid-bars. And the choice decides the whole character of the result, in the direction that is easy to get backwards:

• A bass that never moves inside the chord states the root at the barline and is still sounding it at the half-bar, and at every other slot of the bar. Every candidate scores zero. The root reading is worth exactly nothing — and it is worth nothing in the one texture where the move reading is perfect.
• A bass that always moves passes through the chord’s other tones between the barlines, so the root sounds only where the chord has arrived. The root reading is at its strongest — in the one texture where the move reading has collapsed.

Counted at the attack instead, the reading is not a complement but a restriction: the bass attacks the root here is a subset of the bass moves here, and it inherits the same behaviour. The complementarity is entirely a consequence of counting sound rather than counting events, and it is the reason a reading that looks like a small variant on the previous one behaves like its opposite.

The crossing #

The hero’s two curves are success rates, which are bounded and so flatten at their ends. The evidence underneath them is not bounded, and it is straighter.

The move reading’s margin runs 1.00, 0.81, 0.66, 0.55, 0.40, 0.28, 0.14, 0.01 and −0.13 across the sweep: a straight fall, through zero, to a reading that prefers the half-bar. The root reading’s runs 0.00, 0.23, 0.41, 0.55, 0.63, 0.69, 0.64, 0.60, 0.49 — a rise, a peak at about two-thirds, and then a slight decline. They cross at 0.55 of a bar each, at a bass moving on 37 per cent of its crotchets — three in eight.

The root reading’s decline at the top is worth a sentence, because it is the only non-monotone thing here. A bass moving on every crotchet does not stay away from the root: it moves to a tone other than the one it is on, so having left the root at the second crotchet it may be back on it at the third. The busiest bass therefore sounds the root at a few mid-bars as well as at every barline, and gives a little of its margin back. The peak is at a bass that moves on about five crotchets in eight, which is a bass that walks and occasionally rests — and that is a fairly good description of what a bass line in this repertoire does.

What each witness knows, and what it does not #

The hero draws both readings with the metre already consulted, and that hides the most useful thing about them, which is that they answer different questions.

Asked alone, the move reading tracks its own margin: 100 per cent over a still bass, falling to 5 as the bass fills in. Asked alone, the root reading rises to 42 per cent and then stops, and no amount of extra evidence moves it past about 49.

The ceiling is not a weakness in the evidence. It is a tie, and it is the same tie in every passage. A root stated on the first quaver of a bar is still sounding on the second, so the candidate that puts the barline one quaver late scores exactly what the true candidate scores. So does the one that puts it two quavers late, if the bass has not moved by then. The root reading knows perfectly well which bar it is in and has no opinion at all about which slot within the bar the bar begins on.

That is precisely the half the metre already has. The chords are a weak witness to the barline measured the metre standing 5.9 standard deviations above chance at the written barline — it is by a long way the strongest witness in this account — and its one incapacity is the symmetry the right period at the wrong phase is about: a bar of eight quavers and its own half put the same slots under the same weights, so no count of onsets against a metrical template can separate them. Against the six candidates that are not the half-bar, the metre is not weak at all; it is decisive, because those candidates put strong material on weak slots.

The metre cannot tell a barline from a half-bar and can tell it from the quaver after it instantly; the root reading is the exact converse. That is a stronger statement than “the two are complementary”, and it is the reason the pair does better than either could by being twice as good. Two witnesses that were merely both imperfect would have to be weighed against each other, and weighing is where the chords are a weak witness to the barline found the difficulty — a reading at chance used as a filter throws out the right answer some of the time. These two never have to be weighed. The metre reduces eight candidates to two and hands them over; the bass says which of the two, and it is being asked the only question it can answer. Put together that way the eight candidates collapse to one at every busyness of bass from about one crotchet in eight upward.

The chord the reading is told #

There is an objection to all of this that has to be met before the numbers mean anything, and it is that the root reading is handed its own answer.

To ask whether the bass is sounding the chord’s root, something has to know which chord it is. In these passages the chords are given, because they were written into the passage before the melody was drawn over them — and a reader entitled to be suspicious will point out that the chords were also written in bars, so a reading that consults them may be consulting the barline it is looking for.

It is not, and the reason is the shape of the two questions rather than a reassurance about how the reading is built. Knowing the chord sequence tells a reading where the chords change. Where the chords change was the whole of the chords mark the barline by changing there, and it is drawn on every figure above as the reading that stays near chance — at two chords a bar it is at chance by construction, and at one chord a bar it is the same fact the move reading already has. What the root reading adds is not the chord but the bass’s position inside it, which the chord sequence does not contain and cannot imply: the same four chords over a still bass and over a walking one give the same changes and opposite readings, which is exactly what the two passages below show.

The honest version of the objection is different and it is not answered here. A listener does not have the chords. They have to be inferred, and which notes are the chord measured what that inference costs and found it depends on the metre — which is the thing the chord is being used to help find. Every reading of these passages since the chords never move the barline has been circular in that specific way, and the honest position is the one three decisions that constrain each other took: the circle is real, it is resolved by searching the three together rather than in sequence, and the readings here are components of that search rather than a pipeline that runs before it.

One passage, twice #

The averages hide what any of this looks like in four bars.

The still bass is the degenerate case and it is worth seeing, because it is the texture in which the older reading looks unimprovable. The bass states four roots at four barlines and holds each one for a bar. The move reading’s bars are 1.00 at the written barline and −1.00 half a bar later, which is as decided as this reading ever gets. The root reading’s bars are flat: the root is sounding at the barline and at the half-bar and at every quaver in between, so every difference it computes is zero.

Turned up, the same chords get a bass of four notes a bar, and the shaded slots are the first crotchet or two of each bar rather than all eight. The move reading’s bars are flat at zero, because the bass moves at the barline, at the half-bar and at both quarter-points alike. The root reading’s are positive at the written barline, at the quaver after it and at the two candidates that would place the barline a quaver or two early, and negative through the middle four — 0.25 against −0.25, a quarter of a slot a bar. That is a small margin and it is the same small margin in every bar, which is the kind that survives being averaged.

Read as a pair, the two passages say the thing the whole essay is about in a form no summary statistic carries. The same bass line, the same chords, the same barline — and which half of the picture is informative has swapped completely, without anything in the harmony changing at all.

Choosing without being told #

All of which leaves a listener in an awkward position. Each reading is worthless in the texture where the other is perfect, and nothing tells a listener which texture they are in. Being told is not an option: how busy a bass is is exactly the kind of thing a listener would be working out from the same evidence.

They do not need to be told.

How far apart a witness spreads the candidates is a property of the passage, computable by whoever has the passage, and it requires no knowledge of what produced it. A reading that has nothing to say produces a flat set of scores; one that has something to say produces a spread one. Asking the more spread of the two picks the root reading on 0 and 2 per cent of passages at the two stillest settings, on 49 per cent at the crossing, and on 94 to 100 at the top — which is the same curve the evidence figure draws, arrived at from inside a single passage rather than from an average over two hundred.

What it costs is small and it is not nothing. At a bass moving on every crotchet the rule scores 94 per cent against the 100 that simply always asking the root would have scored, because in about one passage in sixteen the move reading happens to be the more spread and it is spread the wrong way. That is the price of a rule that does not know the answer in advance, and six points is a fair price.

Which computation produced the numbers #

The passages are the ones which notes are the chord built and the barline essays have used since: eight bars of eight quavers, a melody whose onsets are weighted toward the metrically strong slots and whose notes are tones of the current chord, and a chord sequence of diatonic triads of C major, never the same twice running, one chord to a bar.

The bass states each new chord’s root at the slot the chord begins, and inside the chord may move to another of its tones at any crotchet, with the stated probability. The move reading and the metre reading are the earlier essays’ own and unchanged. The root reading fills the bass forward from each onset to the next, marks every slot at which the sounding note is the current chord’s root, and scores each candidate by its barlines less its mid-bars. A reading’s credit for a passage is one over the number of candidates it ties for best if the written barline is among them, and zero otherwise. Every rate is over 200 passages from fixed seeds.

Where the model stops #

The chords are drawn at random. A real progression puts a cadence at a phrase end and a tonic under a downbeat, and both are barline evidence of a kind nothing here has — the cadence as evidence measured how much, on the other side of the same question. These passages give the harmony less to say than music does, and the effect runs one way: every rate here is a floor.

The bass’s motion inside a chord is unstructured. It moves to another chord tone chosen at random, which is neither a walking bass’s stepwise descent nor an Alberti figure’s fixed cycle. What that costs is a reading of pattern — a bass that repeats the same figure in every bar marks its barline by the figure’s own period, which is a cue of a different kind and is not measured here.

A passing note outside the chord is not modelled. A real walking bass spends a fair proportion of its time on notes belonging to no chord at all, which would lower the root reading’s mid-bar share further and strengthen it. The figures here are therefore a floor on that reading rather than an estimate of it.

Everything is heard perfectly. A bass line is not a list of roots found that inversions confuse a key-finder’s bass cue, and a real bass is partly masked by what is above it; every note here arrives.

What the numbers cannot say about a listener #

Whether a listener fills the bass in. The whole reading rests on treating a held note as present at every slot it covers, and that is a claim about auditory memory, not about arithmetic. It is a plausible claim — how long a note has to be found a pitch needs only tens of milliseconds to exist — but it is not tested by anything above.

How busy real bass lines are. The crossing is at three crotchets in eight, and where a repertoire sits relative to that is a count over scores with their bass lines, which is data not to hand here. A continuo bass and a bass guitar are on opposite sides of it.

Whether the choosing rule is anything a listener does. It is a demonstration that the choice can be made from the passage alone, which is the interesting half. That a listener makes it this way is not shown by finding that it works.

Still open: the two chords a bar the rate reading warned about #

Both readings work at one chord a bar and the rule chooses between them without help, which leaves the rate at which the change reading follows the chords, not the bar found the harmony at chance. At two chords a bar the chords change at the barline and at the half-bar alike, and the bass — under either reading — does the same thing at each: it states a root at both, and moves between both. The voicing convention a bass that holds through a change priced is the one thing that breaks the symmetry, and nothing has yet asked what it is worth to a bass that is also walking. The two devices may be independent, in which case a busy bass loses nothing; or the walking may dilute exactly the evidence the holding creates, in which case a bass cannot do both jobs at once and the texture with the fastest harmony and the busiest bass is the one a listener has least to go on.