How much of the reading arrives late

The margin the dynamic program already had found a quantity that was free. A dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key was already computed and had never been printed.

It also had a term in it that a listener cannot have, and its last paragraph said so:

Every margin above uses the whole passage, forward and backward, which is a reading a listener cannot have.

Dropping that term is one line, because the program already keeps the two halves separately. What comes out is the size of the retrospective element in tonal hearing, which this ladder has assumed at every rung and never measured.

Two halves of one program #

The dynamic program keeps two tables and only one of them is available in real time.

The forward table holds, for every key and degree at every bar, the best score of any path into that state. It is built left to right, and its entry at bar n uses bars one to n.

The backward table holds the best score of any path onward from that state. It is built right to left, and its entry at bar n uses bars n+1 to the end.

The best path through a state is the sum of the two, and that is what the tenth rung’s margin is computed from. It is the right quantity for an analyst, who has the score in front of them and is asking what the best reading of the whole passage is.

It is not a quantity a listener has at bar n, because half of it is made of bars they have not heard.

Dropping the backward term gives the causal margin: how sure a listener could be at the moment, given everything so far and nothing after. Everything else in the calculation is unchanged.

More than half of the certainty is hindsight #

On the thirty-two-bar song the ladder uses throughout:

The mean two-sided margin is 3.90 bits. That is a comfortable reading — a rival key would have to be nearly sixteen times less likely to be inside a bit of it.

The mean causal margin is 1.79 bits.

So 54 per cent of this passage’s certainty arrives after the bar it is about. A listener at bar four has a margin of one bit where an analyst has 3.9; a listener at bar twelve has 2.9 where an analyst has 5.0.

The pattern through the passage is worth reading. The causal margin starts at zero — the first bar of a passage has no evidence at all and every key that contains its chord is equally good — and climbs in steps as evidence accumulates. The two-sided margin starts near its maximum, because the backward term at bar one contains the whole piece.

That difference at the opening is the honest shape of the thing. An analyst knows the key of bar one; a listener does not, and the gap between them is largest at the beginning and closes as the passage goes on.

On an ambiguous passage the readings differ #

The 32-bar song is a clear case and both passes name the same key at every bar. The interesting case is the passage this ladder built to be ambiguous.

Two keys at once constructs a progression whose chords are shared between two keys and which resolves toward one of them only at the end. Running both passes over it:

The mean two-sided margin is 0.69 bits and the causal margin is 0.18.

And at nine of the eleven bars the two passes name a different key.

That is the result that makes the rung worth writing. On a genuinely ambiguous passage, the analyst’s reading and the listener’s are not the same reading with different confidence — they are different readings, most of the time.

The causal margin at the first eight bars is exactly zero: a listener has no basis whatever for preferring one of the two candidate keys, because the evidence that distinguishes them has not arrived. The two-sided margin at those same bars is 0.6, because the ending is in the backward table.

So the analyst says the passage was in the second key all along. The listener says nothing for eight bars and then agrees. Both are correct about different questions.

Where the hindsight is concentrated #

The retrospective fraction is a mean and it is not evenly spread, which is where the musical content is.

At the opening it is total. Bar one’s causal margin is zero in every passage tried, because a single chord belongs to several keys and there is nothing before it. Its two-sided margin is near the passage’s maximum. So the first bar of a piece is the bar an analyst is most sure about and a listener is least.

In the middle it is small. By the twelfth bar of the song the causal margin has climbed to 2.9 against a two-sided 5.0, so the listener has 58 per cent of the analyst’s certainty. Evidence has accumulated and the two readings have converged.

At a modulation it opens again. The causal margin drops sharply at the bar where the key changes and takes several bars to recover, while the two-sided margin barely moves — because the backward pass already knows the new key is established.

That last pattern is the one worth naming. A modulation is a local collapse of a listener’s certainty that an analyst does not experience, and its width in bars is how much evidence a modulation needs measured from the listener’s side rather than from the analyst’s.

It also says something about why modulations are prepared. A pivot chord, a dominant of the new key, a sequence approaching it — all of the standard preparations are devices for shortening exactly this gap, by giving the forward pass evidence before the change rather than after it.

Fifty-four per cent, run on the other five #

Both figures above are one passage each, and 54 per cent is stated as though it were a property of tonal hearing. The collection holds six chord schemes and the same two passes cost nothing to run on all of them.

The retrospective fraction is between 48 and 54 per cent on every scheme that has a margin at all, on passages of 16 to 40 bars, with and without modulations, in three centuries of idiom. So the number generalises across this corpus in a way the essay had no right to assume and now has some right to state — about half of a reading arrives late, and the half does not depend much on what is being read.

The ostinato is the control and it does the correct thing, which is to refuse. Thirty-two bars of one chord give a margin of zero under both passes, so the retrospective fraction is nought over nought and is not a number. A passage with no reading has no retrospective share of one.

The revision count behaves quite differently from the fraction, and it is the more musical of the two. Four of the six schemes revise nowhere: the two passes name the same key at every bar, and hindsight buys confidence rather than a different answer. The blues revises at bar one only, which is the opening, where a causal pass has nothing to go on. The rondo revises at bars 15 to 19 — a five-bar window straddling its return from the dominant to the tonic at bar 17 — and at none of its other three key changes. So disagreement between an analyst and a listener is not spread through a passage in proportion to its difficulty; it is concentrated at one modulation, and the other three are heard as they happen.

What the ladder has been assuming #

Ten rungs of this ladder produce readings, and every one of them is two-sided. It is worth naming which of their claims this touches.

The key plan is the form reads a whole movement’s key scheme, which is an analyst’s object by construction and is unaffected.

How much evidence a modulation needs counts the chords a modulation takes to establish itself. That is a causal question asked with a two-sided instrument, and the causal version of it would give a larger number — a modulation takes longer to establish for a listener than for an analyst, because the analyst has already seen it succeed.

A modulation and a borrowing are one number apart turns on a cost parameter, and the distinction it draws is exactly the kind that hindsight settles: a borrowed chord is one that turned out not to be a modulation, which is a statement about what came after.

So the ladder’s most listener-facing claims are the ones most affected, and the ones about form are not affected at all. That is a clean division and it is not one the ladder had drawn.

Is a listener causal? #

The obvious objection to the whole rung is that a listener may not be causal either, and it deserves an answer.

A listener certainly revises. Hearing a passage and realising in retrospect that the opening was in a different key is an ordinary musical experience, and the whole rhetoric of a deceptive resolution or a modulation revealed at a cadence depends on it. So a listener is not a strictly forward machine.

But a listener’s revision is not the backward table. The backward table is optimal: it knows the best continuation from every state, using the whole rest of the passage, computed with perfect memory and no cost. A listener revising a reading is doing something much weaker — holding one or two alternatives, reweighting them when something arrives, and forgetting most of the detail.

So the two computations bracket the truth rather than one of them being right. The causal margin is a lower bound on what a listener has and the two-sided margin is an upper bound, and the interesting quantity — how much revision a real listener does — is somewhere between them and is a listening experiment.

Reporting both, rather than picking one, is what this rung can do. The gap between them is 2.1 bits on an ordinary passage, and that is the size of the space a theory of musical revision has to fit inside.

One line, and why it was free #

It is worth being explicit about how little work this rung was, because that is a fact about the method rather than about the result.

The eighth rung built a dynamic program to find a best key path. A dynamic program computes a forward table and, if it wants the best path through each state rather than merely the best path overall, a backward one. The tenth rung noticed that having both tables makes the margin free. This rung notices that having them separately makes the causal margin free too.

So a quantity that would have been a research project — how much of a listener’s harmonic certainty is retrospective — is a single if in an expression that already existed. That is the third time recently that an object built for one purpose has answered a question it was not built for, and the pattern is worth stating: a model that keeps its intermediate quantities can be asked questions its author did not have in mind, and a model that returns only its answer cannot.

The eighth rung’s own reason for using a dynamic program was that a greedy key-finder cannot keep the order — it commits to a key bar by bar and cannot revise. The irony is exact: the program was adopted because it can revise, and this rung’s contribution is to switch the revision off and measure what it was worth.

Which computation produced the numbers #

The model is the eighth rung’s: a hidden-state dynamic program over twelve keys and seven degrees, with an emission score from how well each bar’s pitch classes match each degree’s triad, a transition score from the root-motion probabilities, and a fixed cost for changing key.

The two-sided margin at a bar is the best forward-plus-backward score over any degree of the best key, minus the same for the second-best key, in bits. The causal margin is identical with the backward term set to zero.

Nothing else changes: the same emissions, the same transitions, the same key cost. The two figures differ by one term in one expression, which is what the tenth rung’s last paragraph promised.

Where the model stops #

The key cost is asserted. A modulation and a borrowing are one number apart is the rung about how much the whole reading turns on it, and this rung inherits that dependence without re-examining it.

The forward pass has perfect memory. It uses every bar so far with no decay, which is as unrealistic in its own direction as the backward pass is in its. A listener’s memory for a chord eight bars ago is not what it was for the one just heard, and a return has to be remembered is the collection’s essay about how much.

A bar is a chord. Which notes are the chord shows that segmenting a texture into harmonies is itself a decision, and both passes here take that decision as given.

And the corpus is six schemes. The retrospective fraction holds between 48 and 54 per cent across all of them, which is more than the two passages the figures draw, and it is still six encodings of conventions rather than a sample of music.

What the picture cannot show #

It cannot show what a listener revises. The gap between the two passes is a space, not a measurement, and filling it needs listeners rather than arithmetic.

Nor can it show the expectation. A listener at bar four is not only reading a key; they are expecting a continuation, and expectation is a curve is the ladder about that. A causal margin and a forward expectation are two views of the same forward pass and they have never been computed together.

And it cannot show a first hearing against a second. The form a first hearing cannot have is the essay about the difference, and a listener who knows the piece has something much closer to the two-sided reading — which suggests the right experiment is the same passage heard twice.

Whose music, and when #

The thirty-two-bar song form is a twentieth-century popular one; the root-motion probabilities and the key-distance model behind the emissions come from common-practice European tonal music. Mixing them is the ladder’s standing compromise and it is stated in the rungs that introduce each.

The ambiguous passage is constructed rather than found. That is a strength for this purpose — it isolates the case where the two passes disagree — and a weakness for any claim about how often real music is ambiguous.

There is a repertoire-shaped observation to be made and this collection cannot make it. Music of some periods is written to be read forward and music of others is written to be read backward: a classical exposition announces its key in the first bar and a late-romantic one withholds it deliberately. The retrospective fraction is exactly the quantity that would measure that difference, and measuring it needs a corpus of analysed passages from both — which is the corpus three ladders have now recorded independently.

Where this ladder goes next #

Eleven rungs. Keys are neighbours and the map is computed; the key plan is the form; three distance measures disagree; a modulation takes a measurable number of chords; the circle is a circle and the map is not; two keys at once; two keys in alternation; a model that keeps the order; the number that decides which reading it gives; what the reading is worth; and now how much of it a listener could have had at the time.

What is owed after this is memory. The causal pass has perfect recall of everything before the bar, which is as wrong as the two-sided pass’s knowledge of everything after — so the honest listener’s reading is a forward pass with a decay, in which a chord eight bars ago counts for less than the one just heard. This collection has a measured account of how a musical memory fades, on the repetition ladder, and putting a decay on the forward table is one parameter and one line. What would come out is the reading of a listener who is neither an analyst nor a machine, which is the reading this whole ladder has been standing in for.