An interval is two posteriors subtracted

The ninth rung of this ladder treats a key as a prior rather than as a reduction of noise, and finds that a note played out of tune is heard closer to the scale than it is — by three quarters of its own error on a semiquaver and by a sixth on a note of any length whatever.

It is a statement about one note. Every rung below it is about two, because a melodic interval is what a listener judges and a single pitch is not. Joining them is a subtraction: two notes each pulled toward their own nearest degree give an interval pulled toward the scale interval between them, and the arithmetic is a difference of two posteriors rather than one posterior of a difference.

Two things come out of that figure and the ninth rung predicted one of them.

The half of the prediction that is wrong #

The ninth rung’s closing section says the interval’s pull is not the single-note pull doubled, “because the two degrees are not equally weighted and the shrinkages therefore differ”. The second clause is right and the first does not follow from it.

Splitting a twenty-five-cent departure between the two notes — a quarter tone’s eighth down on the C, the same up on the G — gives a survival of 0.439. The mean of what the two ends give alone is 0.440. The two agree to a thousandth, and they agree to a thousandth at every one of the twenty-one intervals in the diatonic scale: the largest departure from linearity anywhere in the table is 0.0010.

That is worth stating plainly because it removes a complication the ladder had been carrying. The posterior mean is a nonlinear function of the measurement — it has to be, because a mixture prior has a boundary in it where the pull changes sign — but over a departure of a few tens of cents from a degree it is locally linear, and two locally linear maps subtract linearly. An interval’s pull is the average of its two notes’ pulls, and nothing about the interval as an object enters the calculation at all.

Which leaves the asymmetry as the whole of the content, and the asymmetry is large. The same twenty-five cents reaches a listener at 32 per cent of its size when the C carries it and 56 per cent when the G does.

Why the two ends differ, and it is not one reason #

Two quantities set how hard a note is pulled, and both differ between the ends of an interval.

The prior’s width at a degree is the free parameter divided by the square root of that degree’s stability in the probe-tone profile, which makes the tonic 9.0 cents wide and the raised fourth 14.3. A narrow component pulls a measurement into itself more strongly, so a mistuned tonic is dragged back further than a mistuned tritone.

The likelihood’s width is the Fourier bound at the note’s own frequency, which falls with pitch — 13.2 cents at middle C on a quarter-second note against 8.8 cents at the G above it. A narrower likelihood is a more confident measurement, and a more confident measurement is pulled less.

On a rising interval those two act against each other or together depending on the degrees, and there is no way to know which without computing it. The C to G is a case where they agree: the G is both higher and less stable than the C, so it is pulled less on both counts.

The note length at which a key does the most damage #

The ninth rung’s picture of the pull is a curve against note length that rises monotonically — a short note is pulled hard and a long one is not — and that picture is not right at the short end.

The survival falls to a minimum of 0.185 at an eighth of a second and rises again on shorter notes, reaching 0.564 at sixty milliseconds. A note too short to place is not pulled to the scale; it is left where it was.

The mechanism is in the shape of the prior rather than in anything about hearing. The prior is a mixture with its components a hundred cents apart, and how much a mixture pulls depends on how the likelihood’s width compares with that spacing. When the likelihood is very narrow the prior is nearly irrelevant and the measurement stands. When it is very wide the prior looks flat over the range the measurement could plausibly have come from, and a flat prior pulls nothing. The pull is largest in between, and the measured worst case is at a likelihood width of about twenty-five cents — a quarter of a semitone.

So there is a note length that is maximally bad for intonation, in the sense that the key hides the most of what a player did. On these figures it is between ninety and a hundred and twenty-five milliseconds at middle C, which is a demisemiquaver at a moderate tempo and a semiquaver at a fast one.

What a longer note does not buy #

Both curves flatten rather than reaching one, at 0.757 for the lower note and 0.847 for the upper.

That plateau is the ninth rung’s floor and it is a real limit rather than a numerical one. The Fourier bound falls as the note lengthens and the listener’s own steady-tone limen does not; past the crossover the likelihood stops narrowing, so the pull stops shrinking. At middle C the floor leaves about a quarter of a mistuning unhearable on a note of any length, and in the bass it leaves about half.

A quarter is not nothing. It says that a held interval, a semibreve in a slow movement, played four cents out, is heard three cents out — and four cents is inside the range that separates a pure major third from a tempered one by a factor of three.

An interval played wrong at one end is a chord played wrong at both #

There is a case this arithmetic does not cover and it is worth being explicit, because it is the case most tuning arguments are about.

Everything above is a melodic interval: two notes in succession, each measured on its own, the interval formed by subtracting one estimate from the other. A harmonic interval is not measured that way. Two notes sounding together produce beats between their coincident partials, and a beat is a far finer criterion than a pitch comparison — a tuner can null a beat to a fraction of a cent, and no pitch judgement approaches that.

So the pull this rung computes applies to the melodic case and is largely absent from the harmonic one, and that difference is the same one the melody ladder found from the other direction: a leading note wants to be 1088 cents if the argument is vertical and 1110 if it is horizontal, and the gap between them is a syntonic comma. The key prior is a horizontal mechanism. Nothing in it can survive two notes sounding at once.

Which produces a prediction with content in it. The same performer should play a melodic interval with more latitude than a harmonic one, and the extra latitude should be about a quarter of the interval’s error at ordinary note lengths. That is a measurement rather than an argument, and it has not been made here.

The same interval a third higher is a different problem #

The C to G above is one interval among twenty-one, and it is worth seeing one where the two mechanisms pull against each other rather than together.

The E to B is the same interval size as the C to G and its ends differ much more: 0.518 against 0.785, an asymmetry of 0.267 against 0.242. The B is the leading note, which the probe-tone profile rates barely above the raised fourth, so its prior component is 13.4 cents wide against the tonic’s 9.0 — and it is also nearly an octave above the C, so its likelihood is narrower again.

That is the shape of the whole table the next rung draws: a mistuning survives on the leading note and disappears on the tonic, and it does so for two reasons that both point the same way at that particular pair.

Where the two reasons oppose, an interval comes out almost perfectly even. The D to E is one — 0.518 against 0.517 — because the E is higher than the D, which should make it keep more, and is also more stable than it, which makes it keep less, and the two cancel to a thousandth. Nothing chose that; it falls out of two published tables that were measured for unrelated reasons.

What a performer is actually adjusting #

There is a practical reading of the asymmetry and it inverts the way intonation is usually taught.

A player told they are out of tune on an interval adjusts, and what they can adjust is either end of it. The arithmetic says the two adjustments are not worth the same: moving the note on the strongly specified degree changes what a listener hears by half as much as moving the note on the weakly specified one, at the same number of cents.

So the efficient correction is the one applied to the less stable note — the leading note, the fourth degree, the chromatic passing note — and the correction applied to the tonic is largely absorbed. That is close to what string teaching says about the notes worth being careful with, and it is usually justified by their harmonic function rather than by anything about a listener’s estimate.

Which is a warning about the argument as much as a support for it. Two accounts predicting the same practice is weaker evidence than one, and the harmonic account was there first. What separates them is the register term: this account says the same degree is treated differently at different pitches and the harmonic account says it is not.

What the pictures cannot show #

Every number on this page is the mean of a posterior, and a listener does not report a mean. They report a category, or a judgement of in-tune-ness, or a comparison against another interval — and each of those is a different functional of the same posterior. The categorical anchor’s own machinery takes the same likelihood and asks which of several boxes a measurement falls into, which is a decision rule and not an estimate, and there is no reason the two should give the same answer.

The prior here is the probe-tone profile, which was measured by asking listeners to rate how well a note fitted a context — and is being used as a probability distribution over where a note is expected to be. Those are not the same thing and the identification is an assumption this ladder has been making since its fifth rung. It is a reasonable assumption and it is not a measurement.

And the likelihood’s width is the Fourier bound, which is a bound rather than a value: it says the measurement cannot be narrower than that, not that it is that. A listener whose pitch extraction is worse than optimal has a wider likelihood at every duration, and every survival on this page would be smaller.

The one place the linearity could have failed and does not #

The linearity result deserves one more sentence than it has had, because it is the kind of thing that is true only inside a range and the range matters.

A mixture prior is not locally linear near a boundary — the place where a measurement stops being pulled toward one degree and starts being pulled toward the next. The ninth rung located those boundaries and found they are not halfway between the degrees and that they move with the note length. A departure that crosses one is pulled the other way, and two such departures subtracted are not the average of anything.

Twenty-five cents on a quarter-second note stays well inside the basin at every degree of the scale, which is why the table is linear to a thousandth. The relevant question is where that stops being true, and the boundary figure answers it: on a quarter-second note at middle C the nearest boundary is above forty cents from every degree, so a departure has to be a fifth of a semitone or more before any of this stops being an average.

That is a real bound on the result rather than a formality. A quarter tone is not covered by anything on this page. A performer playing a note fifty cents off is not being pulled toward the degree they meant; they are being pulled somewhere between two degrees, and which one wins depends on the note’s length. That case belongs to the categorical anchor and not here.

Whose intonation this is a claim about #

The repertoire is any melodic tradition with a fixed scale that a listener knows — which is most of them, and specifically the Western tonal repertoire the probe-tone profiles were measured on, with listeners who had heard it all their lives.

For that repertoire the claim is testable in a specific place: the intonation a soloist gets away with in fast passagework against the intonation they get away with in a slow one. Teachers describe exactly that asymmetry and describe it as a matter of attention. This says it is a matter of arithmetic, and it says the effect is not monotone in tempo — a passage in demisemiquavers should hide less than one in semiquavers, because the notes have fallen below the length at which the prior can act.

That last prediction is the one worth testing, because it is the one nothing in the folk account produces and the one this rung would be embarrassed by if it failed.

There is a second repertoire where the arithmetic says something and the practice has already decided. A tradition whose scale a listener does not know supplies no prior at all: the mixture is flat, nothing is pulled anywhere, and every cent a performer plays reaches the listener. So the same passage played to an insider and to a stranger is heard with different intonation, and the stranger hears it more accurately. That is an uncomfortable sentence and it is what a prior is: information paid for with bias, and the price is charged on exactly the notes the listener knows best.

It also bounds how far any of this can be pushed as a claim about music in general. The probe-tone profiles are a measurement on listeners raised inside one tradition, and the beyond-twelve ladder’s own account of a scale is that a tradition’s grammar is a good deal more than a weight per pitch class. A prior with seven numbers in it is the smallest thing that could stand for what a listener brings, and this rung uses it because the alternative is having none.

The ladder from here #

Ten 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; how much of a note’s error a shared anchor could reach; what a key does if it is not reducing the error; and now the same question asked of two notes rather than one, where the averaging turns out to be exactly linear and the asymmetry turns out to be everything.

The asymmetry is what the next rung is. Twenty-one intervals in the scale, each with two ends, and a table of which of them can be mistuned invisibly and from which side — with a control that separates the two causes, because the key’s own unevenness and the register are both in the number and only one of them is about music.