Which end the mistuning is on

The previous rung found that an interval’s pull is exactly the average of its two notes’ pulls, and that the interesting quantity is therefore not the average but the difference: the same departure from the scale reaches a listener differently depending on which of the interval’s two notes carries it.

That difference can be tabulated, because a key has seven degrees and twenty-one intervals between them, and the profile that weights them is one published table.

The spread runs from 0.466 at the top to −0.002 at the bottom, and the ordering is not the one an account of intervals would produce, because it is not about intervals at all.

The table is a property of degrees, not of intervals #

Read down the ranking and the same names keep appearing in the same positions. Every interval with the tonic at the bottom is near the top of the table; every interval between two adjacent middle degrees is near the bottom.

That is because the survival is a property of each note, and the asymmetry is just a subtraction. Measured degree by degree on a quarter-second note at middle C, the share of a departure that survives runs:

Seven numbers, and the twenty-one entries of the table are their twenty-one differences. So there is no such thing as a lopsided interval; there are strongly and weakly pulled degrees, and an interval is lopsided when its two ends are far apart on that list.

The extreme case is the tonic against the leading note, at 0.32 and 0.79. A performer can be twenty-five cents wrong on the tonic and deliver eight cents of it, or twenty-five cents wrong on the leading note and deliver twenty — the same error, on two notes a major seventh apart, two and a half times as audible on one as on the other.

Two causes, and one flag separates them #

The seven numbers rise monotonically up the scale, and there are two entirely different reasons why they might.

The key’s reason is stability. The probe-tone profile rates the tonic highest and the leading note nearly lowest, and this ladder’s prior makes a degree’s component narrower in proportion to the square root of its rating — so the tonic is a 9.0-cent component and the leading note a 13.4-cent one, and a narrow component pulls harder.

The register’s reason is the Fourier bound. A quarter-second note at middle C can be measured to 13.2 cents and the same note an octave up to 6.6, because the bound is a fixed number of hertz and a fixed number of hertz is fewer cents higher up. A narrower measurement is pulled less.

Both rise up the scale, so the table cannot tell them apart — unless the first is switched off, which the machinery allows.

With every degree equally specified, the tonic-to-leading-note asymmetry falls from 0.466 to 0.305. Two thirds of it is the register and one third is the key.

That is a result worth pausing on, because the whole apparatus of this ladder is about what a key does and the answer here is that most of the effect is not about the key at all. A listener would show the same ordering with no tonal context whatever, from the Fourier bound alone, and the tonal context adds half again on top.

The intervals where the two causes cancel #

The two mechanisms both increase the survival as a note goes up the scale, but the second is a smooth function of pitch and the first is not: the profile jumps about. Where a higher degree is also a more stable one, the two disagree.

The D to E is the flattest entry in the table at −0.001. The E is two semitones higher than the D, which should make it keep more of a mistuning; the profile rates the E at 3.48 against the D’s 2.23, which makes its prior narrower and makes it keep less. The two effects are the same size to a thousandth.

The F to G is the other one, at −0.002, and for the same reason with different numbers. Both are whole tones in the middle of the scale, and both are the intervals a listener meets most often.

Nothing arranged that. The probe-tone profile is a measurement on listeners from 1982 and the Fourier bound is a fact about signals, and there is no mechanism connecting them. Two published quantities cancelling to a thousandth on the two commonest intervals in the scale is a coincidence, and it is stated here as one — but it is a coincidence with a consequence, which is that the intonation of a step in the middle of the scale is judged symmetrically and the intonation of anything involving the tonic or the leading note is not.

The ranking is not stable in time #

Everything above is computed at a quarter of a second, and the previous rung found that the pull is not monotone in note length. So the table’s entries are not either.

On notes shorter than about a tenth of a second the two curves cross, and the lower note becomes the one that delivers more. The mechanism is the one the previous rung set out: when the likelihood is much wider than the spacing between degrees, the prior stops looking like a set of attractors and starts looking flat, so the pull weakens — and it weakens first on the note whose likelihood is widest, which is the lower one.

So the table is a table about notes of ordinary length. In fast passagework the ranking inverts, and a mistuning on the tonic reaches a listener more completely than the same mistuning on the leading note. That is not a small refinement, because fast passagework is exactly the case where intonation is usually said to matter least.

Which supplies a second, cleaner test than the one at the end of this essay. The prediction is not that one end is more forgiving than the other; it is that which end is more forgiving depends on the tempo, and reverses. No account in which an interval is one quantity produces a reversal at all.

The pull is a bias and not a blur #

It is worth separating two things a key could do to a mistuned note, because this ladder spent four rungs on the first before finding the second and they behave differently.

A key that reduced noise would make a listener’s judgements more repeatable without moving their average, which is what the fifth rung and the eighth priced and could not find evidence for. A key acting as a prior moves the average and leaves the noise where it was.

The two are distinguishable by an experiment that most intonation studies already run, because a noise reduction shows up in the spread of repeated judgements and a bias shows up in their mean. That the effect on this page is a bias means it is visible in a single measurement per listener rather than needing the variance, which is the cheaper experiment by a large factor.

And it means the asymmetry is not a matter of one end being harder to judge. Both ends are judged equally well; they are simply judged wrong by different amounts, in the same direction, toward the scale.

What the profile is doing in a calculation about cents #

The probe-tone profile was measured by playing a context and then a probe tone, and asking listeners how well the probe fitted. Its numbers are ratings on a seven-point scale.

Turning that into a probability distribution over where a note is requires an identification this ladder has been making since its fifth rung and has never defended in one place: that a degree rated highly is a degree a listener expects precisely, and that “expects” means a narrow distribution over cents rather than a high probability over pitch classes.

Those are different claims. A listener could easily be very sure that the next note is a tonic and have no particular expectation about its intonation. The identification is the assumption that makes the whole prior account produce a bias rather than only a category preference, and it is the assumption an experiment would attack first.

What supports it is that the ordering it predicts is one the practice already has a name for. The melody ladder’s fifth rung records that performers play a leading note high, by twenty-two cents against the vertical answer, and that the deviation is tolerated. This says why it can be tolerated: the leading note is the degree on which a departure is least absorbed and therefore most heard, so a performer playing it high is doing something a listener receives almost in full. The practice would be pointless on the tonic.

And what a mode change does to the table #

The prior is a major-key profile, and the same machinery has a minor one.

Reading the same argument in the minor is arithmetic and it is not done here, and the reason is a limitation rather than an omission: the minor profile’s shape differs from the major’s most at the third and the sixth and the seventh, which are exactly the degrees the modes ladder found decide a mode’s identity. So a table in the minor would differ from this one in the places where the two collections differ, and would say something about mode identification rather than about intonation. It is a genuinely different question and it belongs to a different anchor.

What can be said from here is bounded and worth saying anyway. Any prior with an uneven profile produces an asymmetric table, and every measured profile is uneven, so the existence of the effect does not depend on the major key at all. Only the ranking does.

What this says about a temperament #

There is a use for the table that has nothing to do with performance, and it is the one that connects this ladder to the tuning side of the collection.

A temperament is a decision about where to put each degree, and every temperament puts several of them wrong on purpose. A fraction of a comma distributes the Pythagorean comma over some number of fifths, and the resulting departures from just intonation are not equal across the scale — a well temperament deliberately makes some keys purer than others, which is why keys had characters.

The table says those departures are not equally audible either, and it says so in a direction that is easy to state: a temperament that puts its error on the tonic and the dominant is quieter than one that puts the same error on the leading note and the sixth degree, by a factor of about two, for a melodic listener.

That is not how any historical temperament was designed, because they were designed on the vertical criterion — beat rates between simultaneous notes — where none of this applies. But it is a criterion a temperament could be evaluated on, it needs only the twenty-one numbers above and a table of cents, and no temperament argument in this collection has ever used it.

The honest caveat is the one from two sections above. A harmonic interval is judged by beats and not by a pitch estimate, so the melodic criterion says nothing about the chords a temperament is usually argued over. What it covers is the melodic line, which is most of what is actually played and is the half of the argument the historical sources are quietest about.

What the pictures cannot show #

The table is computed at one pitch and one note length, and the register term means the numbers move if either changes. At A2 every survival is lower and the whole table compresses; at A5 every survival is higher and it compresses again from the other side. The ordering is stable because the profile does not move, and the sizes are not.

There is also a step in the argument that a figure cannot carry and a reader should be able to see. The claim that two thirds of the asymmetry is register rests on comparing the model against itself with one flag changed, and a flag is not a control group. Flattening the prior widths removes the profile’s contribution as this model represents it; if the profile’s real effect on a listener is not a width at all — if it is a weight, or a criterion shift, or something with no analogue here — then the two-thirds figure is a statement about the model rather than about hearing. What can be said without the flag is weaker and safe: the Fourier bound alone produces an ordering in the same direction, so the effect cannot be entirely tonal.

Nothing here has a decision rule in it. A survival of 0.32 means the posterior mean is pulled back to eight cents; it does not mean a listener would say “in tune”, because saying that is a criterion applied to a distribution and this ladder has computed the distribution’s mean and nothing else. Two listeners with the same posterior and different criteria disagree about every number on this page.

The seven degree survivals are computed with each degree taken at the octave it first appears in above middle C, so the register term is doing part of what looks like a scale-degree effect and part of what looks like an octave effect, and the two are not separable inside one ascending scale. A melody that put the leading note below the tonic — which is where it usually is at a cadence — would reverse the register half and leave the stability half alone. That case is not drawn and its numbers are not the ones in the table.

And the departure is a fixed twenty-five cents throughout. The previous rung showed that the arithmetic is linear over departures inside a degree’s own basin, so scaling it to ten or forty cents scales every entry; a departure of half a semitone leaves the basin and the table does not describe it.

Whose playing this describes, and the measurement it asks for #

The claim is about melodic intonation in a tradition whose scale a listener knows, judged by ear, on notes short enough for the Fourier bound to bind — which is most playing and singing, since a quarter of a second is a crotchet at a brisk tempo.

The experiment is small and has not been done. Take one interval, present it twice with the same total mistuning placed at each end in turn, and ask which of the two is further out of tune. The degree account says listeners will name the one whose departure sits on the weaker degree, and it predicts the effect size interval by interval — 0.47 on a tonic-to-leading-note pair and nothing at all on a D to an E. A design that used only middle-of-the-scale steps would find nothing and would be right to report nothing, which is the way this result is most likely to be missed.

The control is the one this rung already computed. Running the same comparison with an atonal context, or with listeners unfamiliar with the tradition, should leave two thirds of the effect standing, because two thirds of it is the Fourier bound and the bound does not care what anybody knows.

The ladder from here #

Eleven rungs, and the last two have made an interval into two estimates with a subtraction between them. The averaging is linear; the asymmetry is a property of degrees; two thirds of the asymmetry is register and one third is the key.

What follows is not another property of the model but a consequence for everything measured with it. Every published preference for an interval size — the stretched octave, the narrowed leading note, the pure third a string quartet is said to find — is a number a listener set, by adjusting until it sounded right. A listener adjusting in a key context is adjusting through the posterior these two rungs have computed, so the value they stopped at is not the value they preferred: it is the value whose heard size equals what they preferred. The correction is the inverse of the map, it is computable from the same twelve numbers, and on notes of a quarter of a second it is larger than several of the preferences it is a correction to.