Intervals and chords

A short note is heard more in tune than it is

Four earlier essays treat a key as something that reduces the noise in a pitch judgement. Treat it instead as a prior and the prediction changes kind: not a smaller error but a systematic bias, pulling a short note toward the nearest scale degree by an amount the Fourier bound sets. Thirty cents out of tune on an eighth-of-a-second note is heard as eight. And the part the debt got wrong is the part that matters — the bias does not vanish on a long note. It stops at 17 per cent at A4 and at 48 per cent at A2, because the likelihood's width has a floor that no duration removes.

Assumes: The part of the error a key cannot touch · How long a note has to be

Every rung of this ladder that mentions a key treats it as a way of making a pitch judgement less noisy. An interval is two errors sets up the quantity; how much an anchor would have to be worth prices the correlation a shared reference would buy; the notes in between asks what intervening material does to it; and the part of the error a key cannot touch puts a ceiling on the whole idea by pointing out that half of a short note’s limen belongs to the note rather than to the listener.

A key can do something else entirely, and the machinery for it has been in this collection since the probe-tone profile arrived. Acting as a prior rather than as a filter, a key does not reduce the spread of an estimate; it moves it, toward the nearest place a note is expected to be. That is a bias rather than a variance, it is a different observable, and no rung here has computed it.

Fig. 1 How far out of tune a note sounds against how far out of tune it is, at four note lengths, at A4. The estimate is the mean of the posterior under a prior built from the probe-tone profile; the likelihood is the note’s own effective limen.

The arithmetic is a single line — the shrinkage of a Gaussian estimate toward a Gaussian prior — and everything it needs is already here. The likelihood’s width is the effective limen the third rung computed, which for a short note is the Fourier bound $1/2T$ and for a long one is the listener’s own steady-tone figure. The prior is Krumhansl and Kessler’s profile read as a place rather than as a rating.

The key drawn as a place a note is expected to be

A probe-tone profile is twelve numbers and this collection has drawn it as twelve numbers many times over.

Fig. 2 The profile as it has always been drawn here — twelve ratings of how well each pitch class fits an established key, with the nesting marked: the tonic, the rest of its triad, the rest of the scale, and everything else. Both modes are shown, because the experiment this essay predicts runs in both and the ordering differs.

To use it as a prior it has to become a density over pitch rather than over pitch class, which needs one thing the profile does not carry: a width for each degree. The width is not invented here. The rung that set two accounts of interval judgement against each other already needed exactly this quantity and gave it a value — nine cents at the most strongly specified degree, widening as the profile’s rating falls, so a leading note is specified to 13.4 cents and a raised tonic to 15.2. That parameter is used unchanged, and it is the only one in this rung.

Fig. 3 The profile as a density over pitch: one Gaussian per degree, weighted by the rating and given the width the degree account already assigns it. The pale curve is the same prior seen through a note of an eighth of a second, whose likelihood is 15.7 cents wide — nearly flat between the degrees.

The pale curve is the point. On a long note the likelihood is 4 cents wide and the twelve peaks are separate objects; on a note of an eighth of a second it is 15.7, and the structure between the degrees has almost gone. A listener judging a short note has a measurement that barely distinguishes one degree from its neighbour, and a prior that does. The estimate moves toward the prior in proportion.

Thirty cents sharp of a degree is heard as 25.0 cents on a long note, 17.1 on a note of a quarter of a second, and 7.8 on an eighth. A semiquaver at a hundred and twenty to the crotchet is 125 milliseconds. So the prediction is that intonation errors in fast passagework are heard at about a quarter of their size, and the same errors held are heard at five sixths.

Notice which way this runs against the ladder’s earlier accounting. The rungs above treat a short note as the hard case — how long a note has to be found that a quarter-second note’s limen is nineteen cents rather than five, so a listener discriminates worse there and a performer’s error is less likely to be caught. This rung adds that the error is not merely harder to catch; it is heard as smaller than it is. The two effects compound, and only the first of them has ever been priced.

They are also different kinds of statement, which is the reason for computing the second. A wider limen says a listener will sometimes fail to notice. A bias says a listener who does notice will report the wrong amount, and will do so consistently, in the same direction, on every trial.

The part the debt got wrong

The section that owed this rung stated the prediction as growing as the note gets shorter and vanishing on long notes. The first half is right. The second is not, and the reason is the same reason its own essay was about.

Fig. 4 The share of a note’s displacement from its degree that the key removes, against how long the note lasts, at four registers. Every curve falls and none reaches zero: the dashed lines are the floors, which are the listener’s own steady-tone limen read through the same formula.

The shrinkage is the likelihood’s variance over the sum of the likelihood’s and the prior’s, and the likelihood’s width is the larger of the Fourier bound and the listener’s steady-tone limen. Lengthening a note drives the first to nothing and cannot touch the second. So the pull has a floor:

17 per cent at A4, 28 at A3 and 48 per cent at A2.

At the bottom of the cello’s range, a note held for as long as anyone cares to hold it and thirty cents out of tune is heard as fifteen and a half cents out of tune, permanently, with no duration effect left to remove. That is the sharpest single consequence of treating the key as a prior, and it is a consequence the debt’s own wording excluded.

The register dependence is worth stating separately because it is not an artefact of the model. The prior is a prior over pitch class, so a scale degree is specified as tightly at A2 as at A5; the listener’s own limen is not, and Wier, Jesteadt and Green’s fit gives 8.6 cents at 110 hertz against 3.4 at 880. The floor is set by the ratio of those two quantities, and one of them has a register in it and the other does not.

Fig. 5 The same computation two octaves lower. A note has to last twice as long to reach the same shrinkage, and the shrinkage it reaches is twice as large: thirty cents is heard as 15.7 on a note held for a second, where at A4 it would be 25.

Which is a claim about who can play out of tune with impunity

Turn it around and it becomes a statement about the score.

The pull is toward the nearest degree, and the strength of the pull depends on how tightly the key specifies that degree. A tonic is specified to nine cents and a raised tonic to 15.2, so the same mistuning on the tonic is shrunk harder than on a chromatic neighbour. On a quarter-second note at A4, thirty cents sharp of the tonic is heard as 17.1 and thirty cents sharp of the raised tonic as 23.7.

This is a curve rather than a switch, and its shape is the ordering of the profile itself. The tonic is at nine cents of prior width, the dominant at ten, the mediant at 10.8, and the five notes outside the scale at between 14.3 and 15.2 — so the notes a listener is most sure about are exactly the notes whose mistuning is most heavily discounted. It is the same nesting the rung that found the hierarchy in the counting works from, read as a tolerance rather than as a ranking.

The most stable degree in the key is the one a player can be furthest out of tune on. That is not the direction musical intuition points — the tonic is the note everyone claims to hear most exactly — and the model is unambiguous about it: a strong prior is a prior that overrides the evidence.

The repertoire this is a claim about is the one the probe-tone profile was measured on, which is Western tonal music of roughly 1700 to 1900 and listeners raised in it. Nothing here transfers to a tradition whose stability hierarchy is different, and everything here would be a different figure with a different profile in it. What it does say inside that repertoire is specific: an out-of-tune leading note in a fast passage is more audible than an out-of-tune tonic in the same passage, by about six cents at the sizes performers actually miss by.

Where one degree stops and the next begins, and how that moves

The prior is a mixture rather than a single Gaussian, and that produces a second consequence which is not a matter of degree.

Between two adjacent degrees the pull has to change sign, and where it changes sign is a category boundary. The obvious guess is fifty cents. It is not fifty cents, because the two degrees are not equally weighted and — under the width model this ladder already uses — not equally wide.

Fig. 6 Where the pull changes sign between the tonic and its chromatic neighbour, against note length, under the two things a key might be doing to a weakly specified degree. Both run away from the midpoint as the note shortens, and they start on opposite sides of it.

Under the width model this ladder inherited — an unstable degree is specified less precisely — the boundary sits at 41.8 cents on a long note and 51.9 on an eighth-of-a-second one. Under the alternative, where every degree has the same width and only the profile’s weights differ, it sits at 51.1 and 53.9.

The two agree that the boundary runs away from the midpoint as the note shortens, and they disagree about which side of the midpoint it starts on. That disagreement is worth more than either number, because it is a question about what a key does rather than about how large an effect is. If a key specifies an unstable degree loosely, its broad component captures the middle ground and the boundary falls below fifty. If a key merely makes an unstable degree less likely without making it vaguer, the boundary falls above.

Neither has been measured, and it is exactly the kind of thing an identification experiment resolves, because a boundary is what an identification experiment measures. It is also the point where this ladder touches the one that asks how far a boundary can be pushed from the other side: that ladder asks what moves a category boundary, and this one predicts that the tempo does.

The measurement this predicts, and what it would have to resolve

The seventh rung asked for an experiment and this one asks for the same afternoon with one more condition in it.

The same 25 cents, heard 14 cents differently on a long note and a short one. A note 25 cents sharp of each of the twelve scale degrees of a major key, heard once as a note of 1 s and once as a note of 125 ms, at 440 hertz. The bar runs from the short reading to the long one, and the degrees are ordered by how strongly the key specifies them. On C, the tonic, the same mistuning is heard as 6.3 cents when short and 20.8 when long — a difference of 14.5. On C♯ the difference is 11.1. The spread between those two, 3.4 cents, is what separates a key acting as a prior from a note length acting alone, and it is the whole of what the experiment has to resolve. — Fig. 7 A note twenty-five cents sharp of each of the twelve degrees of a major key, heard once as a note of one second and once as a note of an eighth, at A4. The degrees are ordered by how strongly the key specifies them.

Play a note twenty-five cents sharp of a scale degree, in an established key, at two durations, and ask a listener how far out of tune it is. The model predicts a difference between the long note and the short one of 14.5 cents on the tonic and 11.1 on the raised tonic — both large, both easy to see in an adjustment task, and both far above anything a difference limen has to resolve.

The discriminating part is not the difference itself. A duration effect with no key in it would predict a difference too, and would predict the same one on all twelve degrees. The measurement is the 3.4 cents of spread between the twelve, ordered by the profile: it is the whole content of the claim that a key is a prior rather than a note length acting alone, and it is the smallest quantity in the design. In a minor key the same computation gives 14.5 on the tonic and 11.8 at the bottom, and the ordering follows the minor profile instead — so the two keys are a second comparison inside the same session.

Three and a half cents is not comfortable. It is above the four-cent limen only in the sense that an adjustment task averages over trials, so it is a question of how many trials rather than of whether it is visible in one. That is a design constraint and it is worth stating as one rather than as a difficulty.

Which computation produced the numbers

The likelihood is the effective limen from this collection’s own duration function: the larger of Wier, Jesteadt and Green’s steady-tone fit converted to cents and the Fourier bound $1200\log_2(1 + 1/2Tf)$ , which at A4 are equal at 486 milliseconds. Nothing about that function is new here and every rung above it uses the same one.

The prior is Krumhansl and Kessler’s probe-tone profile as a mixture of twelve Gaussians on the scale degrees, weighted by the profile’s own ratings, with the width of each component taken from the degree account’s parameter of nine cents at maximal stability scaled by the inverse square root of the profile’s rating. The mixture is repeated an octave above and below, so a note near the top of an octave is not pulled downward by a prior that stops.

The estimate is the posterior mean, which for one Gaussian component is the measurement pulled toward that component’s centre by the ratio of the likelihood’s variance to the sum of the two, and for the mixture is the weighted mean of those, with the weights being each component’s own likelihood of having produced the measurement. That is the whole of the model, and every quantity in it was already published on this site.

The boundary is found by bisection on the point where the posterior mean crosses the measurement between two degrees, which is the unstable fixed point of the map and therefore what an identification experiment finds. Each figure asserts that a note played exactly on a degree is heard on it — to a tenth of a cent, since the two neighbours a semitone either side do not have equal weight and the heavier one wins by a very little.

Where the model stops

A posterior mean is not a report. The model above says where the centre of a listener’s belief is, and an experiment asks a listener to press a button or move a slider. Those coincide only if the listener’s response is unbiased in the belief, which is a large assumption in a task with an obvious “in tune” answer available: response bias toward the category is a well-known confound and would add to the effect measured here without being it. The design has to separate them, which usually means an adjustment task rather than an identification one.

And the width parameter is a fitted one. The nine cents came into this collection as a free parameter chosen so that two accounts of interval judgement agreed on a stable in-key interval. It is used unchanged here, which is the right discipline, and it is still a number nobody measured. Every shrinkage in this essay scales with it: doubling it would halve the pull at long note lengths and barely touch it at short ones, since the short-note likelihood dominates the sum.

The very short notes are past the model’s edge. At a sixteenth of a second at A4 the likelihood is 31 cents wide, which is wider than the spacing of the prior’s components, and a mixture prior read through a likelihood that broad is not distinguishing degrees any more — it is reporting the profile’s centre of mass. The figures are drawn to an eighth of a second for that reason, and the boundary figure’s last points, where it runs to 73 cents, should be read as the model breaking rather than as a prediction.

Nothing here is about intervals. Every rung below this one is about the interval between two notes and this one is about one note against a key. Two notes both pulled toward their own nearest degrees give an interval pulled toward the scale interval between them, which is the sentence the debt actually wanted, but the two shrinkages are not equal unless the two degrees are equally weighted — so the interval’s bias is not the note’s bias doubled and has not been computed.

Where this ladder goes next

Nine 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; how much of a note’s error a shared anchor could ever have reached; and now what a key does if it is not reducing the error at all, which is to move it, by three quarters of itself on a semiquaver and by a sixth on anything.

What is owed now is the interval, and the previous section names it: this rung is about one note and every rung below it is about two. Two notes each pulled toward their own nearest degree give an interval pulled toward the scale interval between them, and the size of that pull is not the single-note pull doubled, because the two degrees are not equally weighted and the shrinkages therefore differ. A fifth from the tonic to the dominant has both notes strongly specified and should shrink almost symmetrically; a tritone from the fourth degree to the leading note has one note at 11.2 cents of prior width and the other at 13.4, so a mistuning placed on the upper note survives better than the same mistuning placed on the lower — and the interval’s apparent size therefore depends on which of its two notes carries the error, which no account of interval perception predicts and which this collection can compute from the twelve numbers it already has. It needs no corpus and no listener; it is the same posterior applied twice with a difference taken, and what would come out is a table of which intervals in a key can be mistuned invisibly and from which end.