A sharper cue is worth nothing to a reading that moves

The essay that gave the cue the bar in octaves found that a cue given the bar in octaves — which names the chord rather than guessing at it from the bass — reads the right scale degree in 92 per cent of bars where the published cue reads 68. It ended on a suspicion: that the margin depends on a parameter the account has held fixed for five essays.

The parameter is what a change of key costs the reading. It is the term that stops a dynamic program hopping to a new key on every bar that would be flattered by one, and every figure from the essay that built the reading with no tonic in it onward has it at 2.2.

The suspicion was right and the size of it is larger than expected.

The margin is not the cue’s #

At a key cost of 0.5 the rule that names the chord reads 46 per cent and the rule that names three chords reads 47. The sharp cue is worth nothing — slightly less than nothing — against the vaguest one available.

At 2.2 the same two rules read 92 and 61. At 4, 91 and 59.

So the twenty-four-point margin the previous essay reported is not a property of the cue. It is a property of the cue and the key model together, and the model contributes most of it. A reading that will change key for one bar’s evidence cannot use a cue that names a chord, because the chord it names is consistent with several keys and the cheap model simply goes to whichever is locally best.

Why a vague cue survives a cheap model #

The mechanism is worth stating because it inverts the usual intuition about evidence.

A cue that names one chord a bar says: this bar is one of the three or four keys in which that chord occurs. A cue that names three chords says: this bar is one of rather more keys. Both are constraints, and the first is tighter.

A tight constraint is only useful to a reader who is going to stay somewhere. The dynamic program’s job is to find a path through keys, and the bass term enters as a per-bar preference. With a cheap key change the path is free to take each bar’s preference at face value, and a tight per-bar preference produces a path that jumps — right about each bar in isolation and wrong about the passage. With an expensive change the path has to satisfy a run of bars at once, and a tight preference on each of them narrows the run sharply.

So the sharp cue’s evidence is only cashed when the model is forced to reconcile several bars. The reluctance is what turns per-bar evidence into a reading, and without it a sharper cue is just a louder voice in a room where everybody is being listened to equally.

Where the rules change places #

The crossing is not a single point, because the four series do not cross at once.

Below a key cost of about 1.5 the published rule on a real line is the worst thing in the figure by a long way — 20 per cent at a cost of 0.5, against 47 for the member rule and 49 for the published rule on a line of roots. A cue that is actively wrong on a third of bars, fed to a model that will follow it, is much worse than no cue at all.

Between 1.5 and 2.2 the register rule pulls away and the member rule stalls. Above 2.2 nothing much changes: the register rule and the root-on-roots rule sit together above 90 and the member rule stays at 60.

So the account’s chosen 2.2 is just past the point where the sharp cue’s advantage has arrived, which is the setting at which the previous five essays’ comparisons are most flattering to the sharpest cue. That is not a criticism of the choice — 2.2 was fixed long before any of this and for other reasons — but it is the kind of coincidence worth reporting rather than discovering later.

The published cue is the one the cost ruins #

There is a second result in the figure and it is sharper than the first, because it concerns the rule anybody would actually implement.

The published cue on a real bass line reads 20 per cent of degrees right at a key cost of 0.5 — against 47 for the member rule, 46 for the register rule and 49 for the published cue on a line of roots. It is not merely the worst of the four; it is worse than a quarter of the range between chance and the best.

That is the combination of two errors reinforcing each other. The cue is confidently wrong on the third of bars where the bass is not a root, and the cheap model acts on each bar’s confidence. A wrong answer delivered sharply into a reader with no inertia is worse than no answer at all, and the arithmetic puts a number on how much worse.

Which is the practical warning this essay leaves. A key-finder that uses the published bass cue on real music, with a permissive key model, is not merely leaving accuracy on the table — it is performing worse than one with no bass term. The failure is invisible at the key cost used here, where the same cue reads 68 per cent and looks like a useful if imperfect contribution.

What the key cost actually is #

Naming what has been swept is worth a paragraph, because “what a change of key costs” is a modelling term with a listener’s quantity behind it.

It is how much evidence a listener demands before deciding they have modulated. How much evidence a modulation needs is the essay that made it a quantity, and it found the answer to be several bars rather than one. A cost of 0.5 is a listener who changes key at the first flattering chord; a cost of 4 is one who will not change without a cadence, which is nearly the distinction a modulation and a borrowing are one number apart puts a number on.

Put that way the finding is about listening rather than about dynamic programs. A listener who hears every borrowed chord as a modulation cannot use their knowledge of chords, because chord knowledge tells them where they could be and the decision about where they are has already been made locally. A listener who holds a key against pressure converts the same chord knowledge into a reading.

That makes the two quantities complementary in a specific way: the sharper the local evidence, the more a reader needs the reluctance to use it, and the sharpest possible local evidence is useless without any.

What the schemes say about it #

the previous essay reported the oddity and left it: at a key cost of 2.2 the register rule reads the rondo best and the thirty-two-bar song worst, and the member rule reads the song perfectly.

The key-cost result explains it. The song modulates and returns within a short span, so the reading’s correct path changes key several times — and a cue sharp enough to pin each bar’s chord, fed into a model reluctant enough to hold a key, resists the genuine modulations along with the spurious ones. The rondo’s difficulty is the opposite: inverted dominants whose bass misleads the root rule, on a scheme whose key plan is stable.

So the two schemes want different settings and no single pair of parameters serves both. A reading tuned to follow the song’s modulations reads the rondo’s inversions wrongly, and one tuned to hold through the rondo’s inversions misses the song’s modulations — which is the corner two cues meet in seen along a third axis. The account has been reporting an average over six schemes and calling it an accuracy.

Every rule improves with reluctance, which is its own result #

The four curves rise together and the fact that they all rise is worth separating from the fact that they diverge, because it is the plainer of the two findings and applies past bass cues.

Every rule reads more degrees right at a key cost of 4 than at 0.5 — the register rule from 46 to 91, the member rule from 47 to 59, the published rule on roots from 49 to 91. Reluctance is worth more than any cue in the figure. The whole spread between the best and worst cue at a fixed cost is smaller than the gain any one of them makes from being read by a reluctant model.

That is a statement about the schemes and it should be read as one. These six passages stay in a key for several bars at a time, so a reader biased toward staying is biased correctly. A passage that modulated every other bar would invert it, and the honest version of the finding is that the key model’s inertia matches this repertoire rather than that inertia is good.

But the repertoire is the one the whole account has used, and everything it has measured has been measured through a model whose largest single parameter was never swept in the same figure as the cue it was being credited to. That is the shape of the oversight and it is a common one: a baseline held constant across a comparison is a term in every number the comparison produces.

Which computation produced the numbers #

The reading, the schemes, the realisation and the three bass rules are the previous essay’s, unchanged. The single change is that the key cost is swept from 0.5 to 4 rather than held at 2.2.

The bass weight is held at 1.5 and the profile weight at nought throughout, so that the figures are about the bass axis alone. Sweeping all three at once is a volume the essays here have drawn as a plane twice and would be a solid here; what the fixed values cost is that the crossing points quoted are the crossings at this slice and not in general.

The degree share is the share of bars on which the reading names both the right key and the right degree, over all six schemes — the same measure the last three essays use.

Where the model stops #

The key cost is a constant and a listener’s is not. A listener demands more evidence to leave a key they have been in for sixteen bars than one they arrived in two bars ago, and a model with a constant cost has no term for that. The listener who forgets is the essay that gave the reading a memory, and a cost that decayed with time in key would be the natural next term.

The key cost and the bass weight are not on one scale. Both enter the same objective as log-likelihood terms, so their ratio is what matters and neither is interpretable alone — a bass weight of 1.5 against a key cost of 2.2 is a reading that will change key for about one and a half bars’ worth of bass evidence. Quoting either number without the other says nothing, and the account has quoted the bass weight alone for five essays.

And sweeping one parameter with two held is a slice. The crossing between the register and member rules is at a key cost near 1.5 at this bass weight, and there is no reason to think it stays there at another.

The four series are not four models. They differ only in what the bass term rewards, so the comparison is clean, and it is a comparison between cues rather than between accounts of listening.

What a listener would have to be like #

The arithmetic describes a pair of settings and the interesting question is whether a person has them, so it is worth saying what a person with each setting would be like to listen to music with.

A reader at a key cost of 0.5 hears every secondary dominant as a modulation and every borrowed chord as a key. They are not wrong about the chords; they are right about each bar and have no account of the passage, so asked what key a piece is in they would name several and mean it. Sharper chord hearing would not help them, and the arithmetic says exactly that: at that setting the best cue and the vaguest read alike.

A reader at a key cost of 4 holds a key through anything short of a cadence. They miss real modulations for a bar or two and read the rest correctly, and every improvement in how well they hear chords is converted into a better reading of the whole.

Neither is a better listener and the difference is not skill. It is a prior about how often music changes key, and the right value of it is a fact about a repertoire rather than about an ear. A reader tuned to the tonal repertoire these six schemes come from, dropped into music that modulates every bar, would be the one making the errors.

What the picture cannot show #

It cannot show a listener who is wrong on purpose. A passage designed to mislead — a deceptive cadence, a chromatic sequence that implies a key it never reaches — is the interesting case for a reluctant reader, and none of the six schemes contains one.

Nor a reading that revises. The dynamic program finds the best path over the whole passage at once, which no listener does. How much of the reading arrives late is the essay that priced the difference, and everything here is the offline figure.

And it cannot show what a key cost of 2.2 means in bars. The number is a log-likelihood penalty and its units are the emission’s, so it is comparable across the figures here and to nothing outside them.

Where this stands after nineteen essays #

The key-relations account has spent six essays on a bass cue and this is what the six come to.

A bass note is evidence about the key. The published form of the cue assumes the bass is a root and a real bass line is not, which costs twenty-one points — the finding a bass line is not a list of roots recorded. Giving the reading the bar in octaves recovers them and more, because what the cue was reaching for was never the bass but the chord. And the recovery is conditional on the reading being reluctant to change key — at a cheap key change, naming the chord is worth nothing at all.

The useful sentence to carry out is the last one, and it is not about basses. Local evidence and global reluctance are multiplicative rather than additive. A reader with sharp local evidence and no inertia performs like a reader with vague evidence; a reader with inertia and no evidence performs like a reader with none. The account has been measuring one of them at a time for six essays, and the number it has been reporting is the product.

Still open: whether the two parameters are one #

If local sharpness and global reluctance multiply, the pair of them may be one quantity with two names, and that is testable rather than rhetorical.

The test is a contour. Sweep the bass weight and the key cost together across the plane and draw the curves of equal degree share. If the two parameters multiply cleanly the contours are hyperbolae — a halving of one exactly compensated by a doubling of the other — and the reading has one effective parameter rather than two. If they do not, the contours bend, and where they bend is where one of the two is doing something the other cannot.

That is a single figure and it would settle what the essays here have been sweeping. It would also say something about listeners that none of the nineteen essays can: whether somebody who hears chords sharply and changes key readily is the same reader as somebody who hears them vaguely and holds on — which is a question about individual differences with an answer this model could predict before anybody measured it.