Acoustics & Music Theory
why it sounds like that
Generator

The probe-tone profile, major key

How well each of the twelve pitch classes was rated as fitting, after a context establishing the key — Krumhansl and Kessler, 1982. The shading is not part of the measurement: it is the tonic, the rest of the tonic triad, the rest of the scale and the remaining five notes, which are categories this subject had before anybody ran the experiment. The profile separates all four without overlap.
The probe-tone profile, major key. How well each of the twelve pitch classes was rated as fitting, after a context establishing the key — Krumhansl and Kessler, 1982. The shading is not part of the measurement: it is the tonic, the rest of the tonic triad, the rest of the scale and the remaining five notes, which are categories this subject had before anybody ran the experiment. The profile separates all four without overlap.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 14 of its 14 placements carry sound, built from the same numbers as the drawing, offering these buttons: The context, then the tonic, The context, then the tritone.

Called by 11 essays

the blast radius of changing it

The probe-tone profile, major key. How well each of the twelve pitch classes was rated as fitting, after a context establishing the key — Krumhansl and Kessler, 1982. The shading is not part of the measurement: it is the tonic, the rest of the tonic triad, the rest of the scale and the remaining five notes, which are categories this subject had before anybody ran the experiment. The profile separates all four without overlap.

Counting produced the hierarchy

Ask listeners how well each of the twelve notes fits after a passage in C major and the answers are not a smooth gradient. They fall into four groups with no overlap at all: the tonic, then the rest of the tonic triad, then the rest of the scale, then everything else — categories the subject had names for centuries before anybody ran the experiment.

perception · Tonal-expectation The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time.

Nothing in the census knows which note is home

All four properties six earlier essays are about are invariant under rotation and transposition — one property tuple over all eighty-four rotations and transpositions of the diatonic set. So the census that separates 349 shapes cannot separate a major scale from its own Aeolian mode, and everything that makes one note a tonic is outside it.

scales · The diatonic set How much of the rule a walk with no rule reproduces. Post-skip reversal in 20,000-note random walks with no melodic knowledge of any kind. An unbounded walk reverses after 50.0 per cent of leaps, which is the chance rate and is the check that the measurement is right. Confining it to 12 semitones raises that to 61.3 per cent. Reaching the 70 per cent that corpus studies report needs a central tendency of 0.95 — an almost deterministic pull back toward the middle at the edges of the range. A wall is not enough; there has to be a spring.

The leap that pays itself back

Every melody textbook teaches that a leap should be followed by a step in the opposite direction, and every corpus that has been counted agrees — around seven leaps in ten are answered that way. A random walk with two walls, no memory of the leap and no rule of any kind reverses after 61 per cent of them, and the residue is not a rule either. What is left when the walls are accounted for is a prediction the rule does not make, and it is the prediction that decides between them.

form · Melody A model that cannot say two keys does not say it is unsure. I – IV – V – I played in C major and in a second major key at the same time, with the second key moved round the circle of fifths. For each separation: the correlation the standard key-finder gives its single best answer, and the correlation reached by the best PAIR of key profiles — a hypothesis the finder does not have. The pair recovers both keys that are sounding at every separation, 7 of 7. The single answer names neither of them at 4 of the 7, and its confidence does not fall when it is wrong: at four steps apart it reports E minor at r = 0.886, against 0.959 for the same progression in one key.

Two keys at once

Every key figure so far assumes one key is sounding, and the standard key-finder has no value it can return that means two. Play one progression in C and the same progression a major third away at the same time, and the model does not report uncertainty: it reports E minor, at a correlation of 0.886, against 0.959 for the same progression in one key. It is as confident as it ever is, and neither key it names is being played. Give it the missing hypothesis — pairs of key profiles rather than single ones — and it recovers both keys at every separation, all seven of seven.

harmony · Key-relations What the notes in between do to the anchor the interval is measured against. How finely a 7-semitone interval can be judged when its two notes are separated by other notes rather than by silence, under the two published accounts. Confirming material restates the key and refreshes the shared reference, so the correlation climbs from 0.5 toward a ceiling and the limen falls to 4.81 cents. Overwriting material competes for the same memory, so the correlation decays to 0.04 and the limen rises to 9.28. By 8 notes the two accounts differ by 4.5 cents, which is 47 per cent of the limen with no anchor at all — and no experiment here distinguishes them.

The notes in between

Every figure until now is about two notes with nothing between them, and a melody is notes with other notes between them. Two published accounts of what the intervening material does predict opposite signs — one says the key is restated and the shared reference is refreshed, the other says each note competes for the same memory and it decays. By eight notes they differ by four and a half cents, which is nearly half the limen the interval would have with no anchor at all.

intervals · Pitch-acuity The same interval, started on each of the twelve. An interval of 7 semitones started on each pitch class of a major key, against how strongly the key specifies its two notes — the mean of the probe-tone profile at each. The interval account says the listener encodes a distance, so the key cannot enter and the prediction is a horizontal line at 5.4 cents. The degree account says the listener refers each note to the key, so its precision on a note falls as the key's specification of that note weakens; scaled to agree at the most stable start, it rises from 5.4 cents on C to 8.5 on E♭. Every earlier figure measures a quantity the second account says is not being formed at all.

The quantity a rival account says is not there

Two earlier essays measure how much two notes' errors are correlated through a shared anchor, and price what that correlation would be worth. There is a rival account in which a listener refers each note to a key and never forms the distance at all — under which the correlation is not small, it is a description of something that is not happening. The two accounts agree on almost everything and disagree on one manipulation, and the manipulation costs an afternoon.

intervals · Pitch-acuity The surprise of each chord, against the uncertainty it arrived into. The information content of each step — minus the log of its probability under a distribution that multiplies the root-motion weight by how well the destination triad's notes fit the key — with the entropy of the moment before it drawn behind. I – IV – V – I: I→IV 1.71 bits, IV→V 2.80 bits, V→I 1.60 bits, against a mean uncertainty of 2.63; I – IV – V – vi: I→IV 1.71 bits, IV→V 2.80 bits, V→vi 2.61 bits, against a mean uncertainty of 2.63. A surprise larger than the entropy it arrived into is an outcome the model was not expecting even given how uncertain it was; one below it is an outcome the model had already mostly bet on. An earlier essay produced the first of those numbers and had no way to produce the second, because a set of preferences is not a distribution and only a distribution has an entropy.

A chord, given a key and a predecessor

A chord's improbability has been priced from its root motion alone, which left one multiplication unmade: a chord is also improbable because its notes do not fit the key, and that number has been available since the probe-tone profile. Multiplied and renormalised, the two give a conditional distribution — and a distribution has an entropy, which is the quantity a surprise has to be read against and which a list of preferences cannot supply.

harmony · Tonal-expectation Expectation as a curve, and what a change costs where it lands. Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous — it sharpens through a bar and collapses when the change arrives — and the two ingredients for it are already here, the harmonic rhythm and the metrical beat weights. The curve is the hazard: given that the chord has not changed yet, the chance that it changes on this beat. It runs from 0.043 on the weakest beat to 0.290 on the downbeat, a ratio of 6.7, against 0.125 if every beat were alike. The marked beats are where the changes actually arrive, and their timing bill is 3.6 bits against 6.0 for a listener with no metre — so these changes are 1.7 times cheaper to expect than a metreless listener would find them. That term is new: an earlier essay prices which chord arrived and this prices when, and a listener meets the sum.

Expectation is a curve, not a list

Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous, sharpening through a bar and collapsing when the change arrives — and the two ingredients for it were already here, in two other accounts. What comes out is a second surprise, for when a chord arrives rather than for which one it is.

harmony · Tonal-expectation How much of A4's pitch error a key could possibly remove. The largest correlation two notes' pitch errors can have at A4, against how long each note lasts. A note's error has two parts and only one of them is the listener's: the steady-tone limen of 4.04 cents, which context might reduce, and the bound a note of length T puts on its own frequency, which context cannot touch. Taking them in quadrature, the shareable fraction is the curve. At a quarter-second note it is 0.21, so the one-half priced earlier is not available at all until each note lasts 486 milliseconds — which is exactly the crossover found by a different route, because a correlation of a half is the two parts being equal. The step is the convention used here, which takes the larger of the two rather than their sum and therefore says the shareable fraction below the crossover is zero.

The part of the error a key cannot touch

Four earlier essays turn one dial — the correlation between two notes' pitch errors — and apply it to the whole of a note's limen. Half of that limen is not the listener's: a note of finite length does not carry its frequency more finely than 1/2T, and no context can put information into a signal that is not there. So the correlation has a ceiling, it is 0.21 at a quarter-second note at A4 and 0.07 at A2, and the figure that prices a correlation of one half is drawn where one half is unavailable.

intervals · Pitch-acuity Thirty cents out of tune is heard as 8 on a 125 ms note and 25 on a 1 s one. How far out of tune a note sounds against how far out of tune it is, at 4 note lengths, at 440 hertz. The key is treated as a prior over pitch: a mixture of Gaussians on the twelve scale degrees, weighted by Krumhansl and Kessler's probe-tone profile and given the width the degree account already uses. The likelihood is the note's own effective limen, which for a short note is the Fourier bound 1/2T. The estimate is the posterior mean, and the shrinkage toward a prior is one line of arithmetic. At 1 s a thirty-cent mistuning is heard as 25.0 cents and at 125 ms as 7.8. Every curve turns back up near the middle of the semitone, because past there the nearest degree is the other one and the pull reverses. The buttons sound at A4, which is the pitch the figure is computed at, at the shortest note length it draws.

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.

intervals · Pitch-acuity Which intervals in a key can be mistuned invisibly, and from which end. Every interval between two degrees of the major scale, ranked by how differently its two ends treat a 25-cent departure on notes of 0.25 seconds. The C to B is the most lopsided, at 47 points: a mistuning on its upper note reaches the listener nearly 2.5 times as strongly as the same mistuning on its lower one. The D to E is the most even, at -0. A negative bar is an interval whose LOWER note is the one that carries a mistuning into the listener, which happens whenever the lower degree is the less specified of the two. No account of interval perception predicts a table like this, because an interval is usually treated as one quantity rather than as a difference of two estimates.

Which end the mistuning is on

Twenty-one intervals in the major scale, each with two ends, and the same twenty-five cents reaches a listener at anywhere between 32 and 79 per cent of its size depending on which of the two notes carries it. The most lopsided is the tonic to the leading note, where a departure on the upper note arrives two and a half times as strongly as the same departure on the lower. Two mechanisms produce it and they can be separated by one flag: two thirds of the asymmetry is register and one third is the key.

intervals · Pitch-acuity

All figures · What can be heard