The part of the error a key cannot touch

Four earlier essays turn one dial. An interval is two errors establishes that judging an interval is judging two pitches and combining them; how much an anchor would have to be worth prices the correlation between those two errors, which is what a shared reference would buy; the notes in between asks what intervening material does to it; and the quantity a rival account says is not there asks whether the whole quantity describes anything real.

All four apply the correlation to the whole of a note’s limen. That limen is not one thing.

Two errors, and only one of them belongs to the listener #

How long a note has to be is the third rung of this ladder and it is where the limen stopped being a single number. A tone of duration $T$ occupies a band about $1/2T$ wide whatever the ear does with it, so its frequency is not specified more finely than that; the effective limen is the larger of the listener’s steady-tone figure and that Fourier bound.

At A4 the steady figure is 4.04 cents and the two are equal at 486 milliseconds. Below that the note is the limit and above it the listener is. Every rung since has taken the effective limen from that function and got on with the arithmetic.

The two halves are not the same kind of quantity, and the difference is exactly the difference the correlation is about.

It is worth being careful about why the second of them is not negotiable, because the ladder has a rung where a listener beats a physical expectation and it is the octave that is not two to one. Listeners set an octave several cents wide of a 2:1 and do it reliably, which shows that a pitch judgement is not a read-out of a frequency. But a stretched octave is a systematic displacement of a judgement, not extra resolution: the listener is answering a different question consistently, and the spread around that answer is still bounded by what the signal delivers. The Fourier bound is a bound on the spread and says nothing about where the middle of it sits.

The steady-tone limen is internal noise. It is a property of a listener judging a stimulus, it is the thing a key, a drone, a preceding note or a memory for the note itself could plausibly reduce, and a shared reference reduces it in a correlated way for two notes judged against the same reference. That is the whole content of the anchor arithmetic and it is a reasonable thing to hope for.

The Fourier bound is not the listener’s at all. It is a statement about the signal: a note of that length does not contain the information, and there is no context, no key and no reference that puts it there. Two short notes judged against a perfect reference still have independent errors of that size, because the errors come from two different notes’ own finite lengths.

So the correlation cannot be a free parameter. Writing the two components in quadrature, the largest correlation two notes’ errors can have is

$$\rho_{\max} \;=\; \frac{\sigma_{\text{listener}}^{2}}{\sigma_{\text{listener}}^{2}+\sigma_{\text{duration}}^{2}}$$

and it is a function of note length and register rather than a dial.

What the ceiling actually is #

At A4 with a quarter-second note — a quaver at a hundred and twenty, which is an ordinary note in ordinary music — the listener’s part is 4.04 cents and the duration’s is 7.85, so the ceiling is 0.21. At A2 the same note gives 0.07. At A6 it gives 0.77.

Put the other way round, a correlation of one half needs a note of 486 milliseconds at A4, 702 at A3, 910 at A2 and 138 at A6. A correlation of 0.75 needs 842 milliseconds at A4, and 0.9 needs 1,459.

There is a neat coincidence in the middle of that, and it is not a coincidence. A correlation of exactly one half is the two components being equal, and the two components being equal is the crossover the duration rung computed by a different route three rungs ago. The note length at which a shared reference could halve the interval’s variance is the note length at which the listener stops being the limit, and those are two descriptions of one event.

The computation that refused #

The consequence for what is already written is uncomfortable and it should be stated first rather than last.

The figure that prices the anchor draws the interval limen against the correlation, from zero to 0.95, and it is drawn at a note length of a quarter of a second. At that length the ceiling is 0.21, so four-fifths of that figure’s horizontal axis is a region the arithmetic cannot reach. The number it reports most prominently — that halving the interval limen needs a correlation of 0.75 — is true and unreachable at the note length it is drawn at.

The same reading applies to the rung about intervening material. The notes in between draws two published accounts of what happens to a shared reference when other notes come between the two being judged, one of which refreshes it and one of which decays it, and both are drawn as movements of the correlation from a starting value of 0.5 or 0.2. Both movements are movements within a range the note length has to make available first, and at a quarter-second note at A4 the whole of that range is 0.21 wide. Whether a shared reference decays or refreshes is a real question and a listener does forget; what the arithmetic adds is that the answer is unmeasurable on short notes, because there is almost nothing there to decay.

None of the arithmetic in those rungs is wrong. What is wrong is the implicit claim that the dial is free to turn, and the correction is not that the anchor is worth less than the ladder said but that the anchor is worth what the ladder said, in a regime the ladder never checked it was in.

What the anchor is actually worth #

Applying the ceiling to the interval limen turns the correction into a number, and the number goes both ways.

A fifth above A4, with each note a quarter of a second long, has an interval limen of 9.44 cents if the two errors are independent. At the correlation of one half the ladder priced, it would be 6.93 — an improvement of 27 per cent. At the ceiling of 0.21, it is 8.48, an improvement of 10 per cent. Down at A2 the same interval goes from 37.54 to 36.29, an improvement of 3.3 per cent against the 27 the ladder’s figure implies.

Now lengthen the notes to a second. The ceiling at A4 rises to 0.81, and the interval limen falls from 5.40 independent to 2.40 at the ceiling — an improvement of 56 per cent, twice what a correlation of one half would have bought.

So the correction is not that the ladder overstated the anchor. It is that the ladder stated one number for a quantity that runs from three per cent to fifty-six across the notes music is made of, and the direction of the error depends on which note. A shared reference is worth almost nothing on a fast bass line and more than the ladder ever claimed on a slow melody, and the two are the same listener with the same key.

Where music is #

The regime question has an answer and it is not close.

At a quarter-second note, every pitch on a piano below about E♭6 — eighty of its eighty-eight keys — is duration-limited. Under this collection’s own convention, which takes the effective limen as the larger of the two rather than their sum, that means the listener’s noise is not the binding constraint anywhere in that range, and the shareable fraction is not merely small but exactly zero.

Under the softer quadrature reading the ceiling rises smoothly rather than stepping, and it is still 0.04 at the bottom of the compass, 0.11 at A3, 0.21 at A4 and 0.43 at A5. It first passes one half at about 1 kHz.

There is a second reason the quarter-second figure is generous rather than harsh, and it belongs here rather than in the caveats. A note in music does not present a steady frequency for the whole of its length. It has an attack during which it is not yet at its pitch, it often has vibrato, and the length a note is written to have is not the length it sounds for. Whatever fraction of the note is usable for a pitch judgement is less than the whole of it, and the bound goes as one over that fraction — so a note that is nominally 250 milliseconds and usable for 150 has a duration bound of 13.1 cents at A4 rather than 7.85, and a ceiling of 0.087 rather than 0.21. Every number above is the best case.

This is the same shape as the reason a low note cannot start on time and it has the same cause: the lower the note, the more of its own period a judgement has to wait for, and a bass line moving at the same tempo as a treble line gets proportionally less of what it needs. The bass is the register where intonation is hardest, every ensemble player knows it, and the usual explanation is that low pitches are harder to hear. The arithmetic says the bass is where a shared reference is worth least, which is a different claim and predicts something the usual one does not: giving a bass player a drone should help less than giving a violinist one, by about a factor of three at ordinary note lengths.

What it does to the experiment the ladder is owed #

The seventh rung ended by specifying an experiment: a context, twelve starting notes, four interval sizes and a threshold at each, to distinguish an account in which a listener forms an interval from one in which each note is referred to a key. It priced what the degree account predicts — a spread across starting notes of between 1.30 and 4.81 cents, depending on a stability exponent nobody can pin down.

Those numbers now come with a design constraint, and it is severe.

An effect of 1.30 cents cannot be seen through a duration bound larger than it. At A4 the bound falls below 1.30 cents when each note lasts 1.51 seconds; at A3 it takes 3.03 seconds; at A5, 756 milliseconds. If the experiment is run at the optimistic end of the predicted range — a spread of 4.81 cents — the requirement relaxes to 408 milliseconds at A4 and 817 at A3.

So the experiment is runnable, and it has to be run on long notes across a register it must state. That is not a small footnote: a design using quaver-length notes at a comfortable tempo would be measuring the Fourier bound, would find a spread that does not depend on the starting note, and would report that as evidence against the degree account when it is evidence about the stimulus.

A null result from a badly-lengthed version of that experiment would be uninformative and would look decisive, which is the worst combination an experiment can have, and the arithmetic that says so costs nothing.

What the probe profile has to do with it #

The degree account’s prediction is a pattern across starting notes rather than a size: precision should be best on the tonic, next on the triad, next on the rest of the scale. That is why the seventh rung reported a pattern and declined to quote a size.

The duration bound does something specific and unhelpful to a pattern. It is the same at every starting note, because it depends on the note’s length and its frequency and not on its scale degree, so it adds a constant to every threshold in the experiment. A constant does not destroy an ordering — but the thresholds being compared are separated by one to five cents and the constant is eight, so the relative differences shrink toward nothing and the number of trials needed to resolve them grows as the square of the ratio. Adding eight cents of irreducible noise to a five-cent effect does not halve the experiment’s power; it costs about a factor of five in trials.

Which computation produced the numbers #

The steady-tone limen is Wier, Jesteadt and Green’s 1977 fit, $\log_{10}\Delta F = 0.026\sqrt{f} - 0.533$, converted to cents — the same curve the first rung of this ladder is built on.

The duration bound is $1200\log_2(1 + 1/2Tf)$, which is the third rung’s, unchanged.

The ceiling is the ratio of the listener’s variance to the sum of the two variances. That is the largest correlation achievable if the listener’s own error were perfectly shared between the two notes, which is an upper bound and not a prediction: the real correlation is that ceiling times however much of the internal noise a shared reference actually removes.

The note length at which a correlation $\rho$ becomes available inverts the same expression, and the note length at which the duration bound falls below a stated number of cents inverts the bound directly.

Where the model stops #

Quadrature is a choice and the collection’s own convention is a different one. Taking the two components in quadrature assumes they are independent, which is right, and that they combine as variances, which is a modelling decision rather than a measurement — this collection’s max convention says the effective limen is the larger of the two, under which the ceiling is a step from zero to one at the crossover. Both are drawn; the truth is presumably between them and closer to the quadrature.

A bound on a signal is not always a bound on a judgement. The Fourier argument says a note of length $T$ does not contain a frequency specified more finely than $1/2T$. A listener with a strong prior can beat that on average, in the same way any estimator can beat its own likelihood by borrowing from a prior — and a key is exactly a prior. That is the one route by which context could touch the part of the error called untouchable here, it would show up as a bias rather than as reduced variance, and this account does not model it.

One limen for both notes. The two notes of an interval are at different frequencies with different limens, and the ceiling above uses one. The figures across the compass show how much that matters: over a fifth the two ceilings differ by about a fifth of their value.

And there is still no listener. Every number here is a bound on what could be shared, not a measurement of what is. The ladder has not measured a correlation and this rung does not measure one either; it says what range the measurement can fall in.

Where this ladder goes next #

Eight rungs. How finely two pitches can be told apart; the octave that is not two to one; the family as a function of note length; an interval as two errors; how much a shared anchor would be worth; what intervening material does to it; what the whole of it is worth if a listener forms no interval at all; and now how much of a note’s error a shared anchor could ever have reached.

What is owed after this is the bias, and it is the one thing the last section names that this collection has the machinery for and has never assembled. If a key acts as a prior rather than as a noise reduction, its effect on a short note is to pull the estimate toward the nearest scale degree — which is a bias whose size depends on how far the note is from that degree and on how weak the likelihood is, and the likelihood’s width is the Fourier bound this essay has spent itself computing. That predicts something no rung here has: on short notes, mistuned intervals should be heard as less mistuned than they are, by an amount that grows as the note gets shorter and that vanishes on long notes. This collection has the probe-tone profile, it has the duration bound, and the shrinkage of a Gaussian estimate toward a prior is one line. It needs no listener to compute and one afternoon to test, and it is the same afternoon the seventh rung asked for — run at two note lengths instead of one.