The two parameters turn out to have a ceiling between them

A sharper cue is worth nothing to a reading that moves swept what a change of key costs the reading and found that the bass cue’s whole advantage is borrowed from it. It ended by naming the experiment that would say whether the two are one thing:

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. 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.

The contour is drawn and the answer is no. What the figure shows instead is the reason no, and the reason is a ceiling: a place in the plane where one of the two parameters stops doing anything and the other has to do all of it.

The product is not constant, by a factor of five #

The test is arithmetic rather than a matter of looking. If the two parameters multiply, then any pair $(b, k)$ giving the same degree share satisfies $b \cdot k = \text{constant}$, so the product read along a contour should not move.

It moves by a factor of five on every contour drawn. Along the fifty per cent line the product runs from 0.10 to 0.50; along sixty, from 0.43 to 1.84; along seventy, from 0.66 to 2.56; along eighty, from 1.85 to 4.91; and along ninety, from 2.71 to 15.20.

The fifty per cent line is the one to be careful with, because a reading at fifty per cent is barely reading at all and a modulation and a borrowing are one number apart there. The ninety is the one that says what is happening. A factor of five and a half along a single contour is not a bent hyperbola; it is a curve that has stopped being a function of one product at all, and it does so at the top of the plane rather than in the middle of it. Every contour runs up and to the left, turns, and then goes vertical. Vertical means the bass weight has stopped mattering: above it, the only way to buy another point of degree share is reluctance.

Where the bass cue stops buying anything #

The ceiling can be read off the grid directly, and it moves with the key cost in a way that explains the shape of every contour.

At a key cost of 0.5 the reading saturates at a bass weight of 0.75: giving the cue more than three quarters of a unit changes nothing, and the reading names 46 per cent of degrees however loudly the bass shouts. At a cost of 1.2 the ceiling is at 1.25 and the share is 64 per cent. At 2.2 it is at 1.5 and 92 per cent. At 4 it has moved to 3.

The ceiling rises with the reluctance, which is the whole of why the two parameters do not commute. A reading that will change key cheaply cannot use a loud bass cue, because the cue points at whatever chord the bar is flattered by and the reading follows it out of the key — which is the margin the dynamic program already had spent on the wrong bar. A reading that is reluctant can use as much cue as it is given, up to a point that rises with how reluctant it is. The bass weight and the key cost are not two ways of spending one budget; the second sets how much of the first is spendable.

That is a different kind of relationship from either of the two the earlier essay could name, and it has a name in the shape of the surface rather than in the vocabulary of the model. It is not a product and it is not independence. It is a constraint: the cue is usable only under the reluctance that will hold it.

The key cost has a best value, and nothing had found it #

Read the plane the other way and something appears that no sweep of a single parameter could have produced.

Every figure on this account has swept one parameter with the other held, and every one of them found the same monotone thing: more reluctance is better. The earlier essay states it outright: every rule reads more degrees right at a key cost of 4 than at 0.5. Swept across the whole plane, that stops being true at the top.

The best column is at a key cost of 3, where the reading names 93.1 per cent of degrees. At 4 it names 93.1 as well but needs twice the bass weight to get there. At 6 the best it can do is 91.5, and it is worse at every bass weight than it was at 3.

A reading too reluctant to change key reads worse than one that will. The reason is not subtle once it is stated: the passages being read modulate, and a reading that charges too much for a key change stays in the old key through a real modulation and gets every degree after it wrong. The cost of being wrong about a key change runs both ways, and only one of the two directions had ever been swept far enough to see.

It is worth being precise about how large that effect is, because it is easy to overstate. Between a key cost of 3 and one of 6 the best reading loses 1.6 points of degree share, which is about one bar in sixty over the schemes read here. That is a real loss and it is not a collapse: a reading that never changes key still names nine degrees in ten correctly, because most bars of most of these schemes are in the key the piece started in. What the interior optimum establishes is the sign of the curve past three rather than a dramatic size, and the sign is the part that was assumed.

This matters for a number the account has been quoting. The key cost has been 2.2 since the reading was built with no tonic in it, and 2.2 was chosen rather than fitted. On this grid it is just below the best — 92.0 against 93.1 — which is a good place for an unfitted constant to have landed and is not where anybody put it deliberately.

Where the ceiling is, in the units the reading actually has #

A weight is an awkward thing to have a ceiling in, because nothing else on this account is measured in the same units. It is worth converting it into something that is.

The bass weight is how much the reading adds to a key’s score for each bar whose bass it explains, against a profile correlation that runs between about zero and one. So a weight of 1.5 is the bass cue counting for rather more than the whole of the pitch-class evidence in a bar, and the ceiling says that past that point the bass is already deciding every bar it can decide and the profile is deciding the rest. Adding more cannot change a decision the cue has already won.

That has a consequence for a claim the earlier essay made and could not test. It found the sharp rule reading 92 per cent against the published bass rule’s 68, and attributed the gap to the sharpness of the cue. The plane says the gap is a gap between a rule that reaches the ceiling and one that does not: at a key cost of 2.2 the sharp rule saturates at a bass weight of 1.5, and the published rule is still climbing at three. The sharp cue is not better because it is sharper; it is better because it runs out of things to get wrong sooner. Those are different claims and only the second is about the cue.

And it reframes what two cues meeting in a corner found. That essay saw two kinds of evidence agreeing only in a small region of a plane and treated the region as the interesting object. Here the interesting object is the boundary of the region, because on one side of it a parameter is live and on the other it is inert — and a model whose parameters go inert is a model with fewer degrees of freedom than its author thinks, wherever it happens to be standing.

What a listener would have to be like for this to be about them #

The earlier essay ended by saying the contour would answer a question about individual differences: whether somebody who hears chords sharply and changes key readily is the same reader as somebody who hears them vaguely and holds on.

The plane says they are not, and it says something sharper. Those two readers are not on one contour — they are on opposite sides of a ceiling. The reader who hears chords sharply and changes key readily is wasting the sharpness: their reading sits on the flat left-hand part of the surface, where a loud bass cue and a quiet one give the same answer, and more acuity would buy them nothing at all. The reader who hears vaguely and holds on is on the rising part, where every extra scrap of cue is worth something.

So acuity and reluctance are not exchangeable in a listener either. A listener with a sharp ear and a loose grip on the key is not equivalent to one with a dull ear and a firm grip; the first is throwing away a cue they possess and the second is using every bit of one they barely have. That is a prediction about which pairs of listeners should read the same passage the same way, and it is not the prediction that essay expected the figure to support.

Which computation produced the numbers #

The grid is nine bass weights from zero to three against eight key costs from 0.5 to 6, at every combination of which the whole reading is run over the account’s four schemes and its minor test passage. The cell is the share of bars in which the reading names both the right key and the right scale degree, which is the metric the earlier essay used and is the stricter of the two available.

The bass rule throughout is the one that names the chord outright — the bar given in octaves — because it is the sharp cue whose value was in question, and the bass line is a real one rather than a line of roots, for the reason a bass line is not a list of roots gives.

A contour is found along each column rather than by marching the grid. Within one key cost the degree share rises with the bass weight and then flattens, so the crossing of a level is a single interpolation between two adjacent cells; a column whose best never reaches the level contributes no point, which is why the lower contours stop partway across the plane instead of running off its edge.

The grid is coarse and that is a real limitation on one claim. The ceiling’s position is read to the nearest grid point, so “1.5 at a key cost of 2.2” means between 1.25 and 2. The claim that there is a ceiling does not depend on the resolution, because the cells past it are identical rather than nearly so; the claim about where it sits does.

Where the model stops #

A saturated reading is not a reading with nothing left to get right. Ninety-three per cent is not a hundred, and how much of the reading arrives late accounts for a good deal of the remainder: the bars immediately after a modulation are read in the old key by a reading that has not yet paid to leave it. Those bars are wrong at every point on this plane and no setting of either parameter recovers them.

Four schemes are not a repertoire. Every number here is a mean over the same small set of harmonic grounds this account has used throughout, and the interior optimum in the key cost is a statement about how often those modulate. A repertoire that modulates less would put the best key cost higher and might put it at infinity, which is to say the optimum is a property of the corpus and not of the reading.

The two parameters are not the only two. The account carries a segmentation, a metre and a profile exponent as well, and this plane holds all three fixed. A ceiling in one plane of a five-dimensional surface is a ceiling in that plane.

And a cost is not a probability. The key cost is a penalty in a dynamic program rather than a prior over modulation rates, and the interior optimum would be better stated as a prior if anybody had one. What the sweep says is that the penalty behaves the way a prior would, which is evidence that it is standing in for one.

What the picture cannot show #

Whether any listener has a ceiling. The saturation is a property of a reading that takes its cue as a weight in a scoring function. A listener who used a sharp bass cue differently — as a hard constraint rather than as evidence to be weighed — would have no ceiling at all, and nothing here distinguishes the two.

Nor what the flat region sounds like. Two readings on the flat part of the surface name the same degrees and are the same reading by every measure this account has. Whether they would differ on a passage nobody has run them on is exactly the kind of question a saturated metric cannot answer, and it is the reason a ceiling in an accuracy figure is not the same thing as a ceiling in a model.

Whose music, and which parameter it belongs to #

The schemes are twentieth-century popular grounds with stated modulations, which is what this account has used from its first essay because they are short and their keys are written down. They modulate more often than a chorale and less often than a development section, and the interior optimum sits where it does because of that.

Which gives the finding its honest scope. A reading built for music that modulates every eight bars wants a key cost of about three; one built for a Bach chorale would want more and one for a development section less. The number is a property of the repertoire and the model has been carrying it as a property of the listener, which is the same confusion met here before, on the tempo of a closing gesture and on the width of a perceptual category.

Still open: whether the ceiling is the segmentation’s #

The one thing the plane cannot say is what the ceiling is made of, and there is a candidate the account already has.

A bass cue can only be as sharp as the bar it is reading. The reading takes each bar’s chord from a segmentation, and the segmentation names a different chord on sixteen per cent of passages depending on which cue it is given — so on some share of bars the sharp cue is confidently naming the wrong chord. Above some weight, more confidence in a cue that is wrong on one bar in six cannot buy anything, and that is exactly the shape of a ceiling.

The test is a substitution rather than a sweep. Run the plane again with the segmentation given the right answer — the chord the scheme was built from, rather than the chord a cue infers — and if the ceiling rises or disappears, it was the segmentation’s all along. If it stays where it is, the ceiling belongs to the key reading itself and the account has found a limit on how much evidence a profile can absorb. Both outcomes are worth having, the computation is the one above with one input replaced, and it is the first question here whose answer would change what comes next rather than adding to what this essay says.