The setting is not the preference

Everything on the last two rungs is a forward map: a value is played, and the arithmetic says what a listener hears. Almost every published fact about interval preference is the map run backwards.

A preference is measured by asking somebody to adjust an interval until it sounds right, and recording where they stopped. What they were adjusting toward is a property of their hearing; what was written down is a property of the stimulus. Those are the same number only if a listener hears what is played, and the last two rungs are about the fact that inside a key they do not.

The corrections run from 1.6 cents to 12.8 on notes of a quarter of a second, and the largest of them is bigger than the preference it corrects.

The direction is always the same, and it exaggerates #

A key pulls a heard interval toward the scale interval. A listener adjusting until an interval sounds a particular amount off the scale therefore has to overshoot, because part of what they play is absorbed on the way in.

So the correction has a sign and the sign never changes: the reported setting is always further from the tempered value than the preference behind it. Every preferred-interval measurement made in a key context exaggerates, and by an amount that is a fixed fraction of the preference rather than a fixed number of cents.

That fraction is the reciprocal of the survival, which the previous rung tabulated degree by degree. A major third is a tonic against a third degree, whose survivals are 0.32 and 0.52; the interval’s departure sits on the upper note, so 52 per cent of it arrives, so the setting has to be 1/0.52 of the preference. Thirteen point seven cents becomes twenty-six and a half, and the correction is 12.8.

The pure fifth is the case where it barely matters: two cents becomes 3.6, a correction of 1.6 cents, which is inside the scatter of any measurement of it. The reason is not that the fifth is special but that its preference is small; a correction that multiplies has nothing to work on.

The correction is largest on notes nobody would design an experiment with #

The size of the correction depends on how long the notes were, and the dependence is not the obvious one.

The corrections peak at about an eighth of a second — 23.7 cents on the major third, nearly twice the preference — and are smaller at a sixteenth. That is the same non-monotonicity the tenth rung found: a note too short to place is not pulled to the scale, because a likelihood much wider than the spacing between degrees sees a nearly flat prior.

At the long end they settle rather than vanishing. At two seconds a note the corrections are between 0.4 and 2.4 cents, which is small against every preference except the fifth’s. That is the floor the steady-tone limen leaves, and no experimental design removes it.

So an experiment measuring a preference has a length it should use and it is a long one. A design using notes of an eighth of a second would report a preference nearly twice the size of the one its listeners had, and would report it with perfectly good precision, because the bias is in the mean rather than in the scatter.

What the octave says about whether any of this is real #

There is one interval whose preference has been measured many times, whose measurement is famously larger than any theory predicts, and which this rung’s correction cannot touch.

Listeners set the octave wide, with pure tones, by fifteen to twenty cents in the middle of the range — and they do it with tones that have no partials, so nothing about a piano’s stiffness explains it.

The correction here would make that number larger rather than smaller, and it cannot be applied for a reason worth stating. An octave in this ladder’s prior is not an interval between two degrees; the prior repeats at the octave by construction, so both notes of an octave sit on the same component and are pulled identically. An identical pull on both ends leaves the interval exactly where it was.

That is a genuine null and it is the best evidence available that the machinery is not producing corrections everywhere by construction. The one interval the model says it cannot bias is the one whose measured preference is largest and least explained, and the model says nothing about it.

Where the correction has already been made without being named #

The place this arithmetic most nearly meets the record is performance intonation rather than the laboratory.

Measurements of what performers actually play — as against what listeners set — are made on recordings, with notes of whatever length the music has, and they are made on the acoustic signal. Nothing about a listener’s prior enters them at all.

So there are two literatures reporting numbers in cents for the same intervals, one of them biased by this and one not, and they disagree. The melody ladder’s fifth rung records that performers play a leading note about twenty-two cents above the vertical answer; the setting literature reports smaller melodic deviations. This rung predicts the disagreement has the wrong sign to be explained by the bias — the bias makes settings larger, so it should make the setting literature report the larger numbers — which means whatever separates the two is something else.

That is a place the account could have been confirmed and is not. Recording it is more useful than not mentioning it: the correction predicts a specific direction of disagreement between two literatures and the disagreement runs the other way.

The correction depends on which degree the interval is measured from #

Every setting above is measured from the tonic, which is what an adjustment experiment almost always uses because a context establishes a key and the key names a reference. The previous rung’s table says that choice is not neutral.

A setting task holds one note and adjusts the other, so the survival that matters is the moving note’s. Measured from the tonic, a major third’s moving note is the third degree at 0.52 and a major sixth’s is the sixth at 0.70 — which is why the third’s correction is 12.8 cents and the sixth’s is 6.8, on preferences that are almost the same size.

Move the same interval to a different place in the scale and the correction changes with it. A major third from the fourth degree to the sixth has a moving note at 0.70 rather than 0.52, so its correction is a third smaller; a major third from the fifth degree to the leading note has a moving note at 0.79 and a correction half the size again.

So the same interval has a different correction depending on which degree it is built on, and by a factor of about one and a half across the scale. That is a testable difference between this account and every account in which an interval preference is a property of the interval: measure a pure major third at three places in the key and this says the reported settings should differ, in a direction the profile predicts, by several cents.

It is also a diagnosis of a discrepancy that already exists. Preferences for the same interval reported by different studies vary by more than their stated precisions, and the studies differ in the contexts they use. Some of that spread is a property of the design rather than of the listeners.

What a drone does, and why it is the cheapest fix #

There is one experimental arrangement in which none of this applies, and it is already used for other reasons.

If the two notes sound together, the listener is nulling beats between their coincident partials rather than comparing two pitch estimates. A tuner counts beats and a beat can be counted to a fraction of a hertz, which is a fraction of a cent at these frequencies — an order of magnitude finer than any pitch judgement and entirely outside the mechanism this ladder models.

So a harmonic adjustment task measures the preference directly, with no correction at all, and a melodic one does not. That is not a new observation about the two designs; what is new is the size of the difference it should produce. If the same listeners are asked for a pure major third melodically and harmonically, the melodic answer should be about twice as far from the tempered value on quarter-second notes, and the two should converge as the notes lengthen.

Whether they do is a measurement. It is one of the cheapest tests of this whole ladder, it needs no equipment beyond what an adjustment task already uses, and its outcome is not obvious in advance — a null would say the prior account is wrong about melodic settings, which is most of what the last four rungs have been about.

What the pictures cannot show #

The correction is computed by inverting a map whose inputs are a profile and a bound, and the inversion is exact for that map and says nothing about how well the map describes a listener. If a real listener’s prior is narrower than the profile implies, every correction here is too small; if it is wider, they are too large. The model has one free parameter — the width at the most strongly specified degree — and it is a parameter this ladder has never measured, only assumed.

Nothing here models what a listener is doing when they adjust. Adjusting is a search with a stopping rule, and a stopping rule has hysteresis in it: a listener approaching from below stops in a different place from one approaching from above, which is why the method of adjustment is usually run in both directions and averaged. This calculation assumes the average of that procedure recovers the posterior’s mean, which is plausible and is not proved.

The five preferences are taken from the literature at their nominal values and treated as though a single number stood for each. Every one of them is a distribution over listeners with a spread of several cents, and correcting a mean without correcting the spread is only half the arithmetic. The correction scales the mean by 1/survival and would scale the spread by the same factor, so a corrected preference is not merely shifted but is also less precisely known than the reported one — which is the part of the result an experimenter would care about most and which no figure here draws.

There is also an assumption buried in treating the five preferences as independent quantities to be corrected one at a time. A listener adjusting a major third in a key is not adjusting one interval in isolation: the third they are moving is also a scale degree with a place in a collection, and moving it changes what collection the passage sounds like. The prior here has no term for that — it is twelve fixed components and no dependence between them — so the correction it computes is the correction for a listener who judges each note against a fixed template. A listener who re-reads the key when the third moves far enough is doing something this model cannot represent.

And every setting here is melodic. A listener adjusting two notes sounding together is nulling beats, which is a criterion finer than any pitch estimate by an order of magnitude, and none of this reaches them.

The one correction that is already applied, under another name #

Piano tuning provides a case where an equivalent correction has been in the practice for a century and is justified differently.

A tuner stretches the octaves — more at the ends of the keyboard than in the middle — and the standard justification is the string’s inharmonicity, which makes a piano’s second partial sharp of twice its fundamental so that a beat-free octave is a wide one. That is a claim about the instrument and it is correct as far as it goes.

What the floor above says is that the audibility of a mistuning is not uniform across the keyboard either: about a quarter of a departure never reaches a listener at middle C and about half never reaches them at A2. So the bass is the register in which a tuner’s errors are least heard and the register in which the stretch is largest, and the two are usually discussed as though only the second existed.

The reading is a caution rather than a discovery. It says a tuner’s freedom in the bass is greater than the beat criterion alone implies, which is consistent with the wide spread between tuners’ bass octaves that anybody who has measured several instruments reports. It does not say the stretch is a perceptual effect: the inharmonicity account is quantitative and this is not a competitor to it.

Whose measurements, and what to do about them #

The literature this bears on is the twentieth-century psychoacoustics of interval preference — adjustment tasks with tonal contexts, run on listeners raised inside the Western tradition, mostly on short tones. It is the source of most of the cents figures this collection quotes for what listeners like.

The recommendation the arithmetic supports is narrow and cheap. Report the note length. A preference measured on two-second tones needs a correction of a cent or two and one measured on eighth-second tones needs one of twenty; the two are different measurements of different quantities and are usually cited as though they were the same. Nothing else about the design has to change, and a length is one number.

There is a third thing and it is the one that costs an experiment nothing at all. A preference measured melodically and the same preference measured harmonically bracket the answer: the harmonic setting is uncorrected and the melodic one carries the whole bias, so running both puts a bound on the correction without needing this model to be right. That is the design a sceptic of everything on this page should prefer, and it is the design that would settle it.

The second thing follows from the first and is a warning about replication. Two laboratories measuring the same preference with different note lengths will disagree by more than either one’s scatter, and will disagree in a way that looks like a difference between listener populations. The bias is in the stimulus rather than in the people, and it is the size of a comma.

Closing this anchor #

pitch-acuity closes at twelve rungs. What bounds a model is having said something about every variable it has, and this one has four.

How finely can a pitch be told apart? Rungs one and three: a limen, and the Fourier bound that floors it as a note shortens.

What does a second note do? Rungs four to eight: two errors added, the correlation a shared anchor would buy, what intervening material does to it, and how much of the error a shared anchor could ever reach.

What does a key do? Rungs nine to eleven: not a reduction of noise but a prior, producing a bias rather than a variance, applied to both notes of an interval and asymmetric between its ends.

And what does the octave do that none of that explains? Rung two, and it is still unexplained — the subjective octave is wide with pure tones, and this rung has just shown that the machinery built since cannot bias an octave at all.

Every variable the model has now has a rung, and this one is the last because it turns the model around and asks what it says about the measurements the model was built from. There is one thing on the list this rung has made worse rather than better, and closing an anchor is the place to say so. The prior’s free parameter — the width at the most strongly specified degree — was chosen at the ninth rung to make the single-note bias come out the size the ladder wanted, and every correction on this page scales with it. Nine cents was a reasonable choice and it is the one number in four rungs of arithmetic that nothing measured. Halving it halves every correction here; doubling it doubles them. The corrections’ ratios to each other are safe, because they come from the profile, and their absolute sizes rest on one assumption made three rungs ago for a different purpose.

What is not on the list belongs elsewhere: how finely a chord can be tuned is beating, because that is a beat criterion rather than a pitch one; what a listener names rather than places is categorical hearing; and what a scale does with the resolution it is given is beyond twelve. Those are different models rather than further rungs.