Acoustics & Music Theory
why it sounds like that
Concept

Probe-tone — where it appears

A single note played after a key-establishing context and rated for fit, which is the method behind the measured tonal hierarchy. Its results correlate closely with plain counts of note frequency in the corresponding repertoire.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside 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 How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire.

The chord that did not come

A deceptive cadence is described as a surprise, and the explanation offered is that the wrong chord arrived. Measured against the tonal hierarchy already in use, the wrong chord is the second best-fitting triad in the key — and two of its three voices do exactly what they would have done in the right one. The surprise is not statistical. It is one voice, and it is the bass.

harmony · Tonal-expectation The price of a tonic. Every note of Dorian is given the same duration except its tonic, which is lengthened; the horizontal axis is the share of the total that goes to it. The key-finder answers with the parent key until 25.0 per cent of the time is spent on the modal tonic, and with D minor above it. At the left-hand edge every note has equal weight, which is the pitch-class set itself — and with every weight identical the correlation is not merely low but undefined, because a flat histogram has no variance to correlate with anything.

What a tonic costs in seconds

The standard key-finding algorithm cannot be run on a pitch-class set at all — a flat histogram has no variance and the correlation is undefined. Give it durations and it answers with the parent key for all seven modes identically, and it takes between 15.8 and 30.0 per cent of the total time spent on one note before it names that note instead.

perception · Modes 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 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 The passage built so the two statistics disagree. A body of chords diatonic to C major, of growing length, ended by a ii–V–I in G. The bag of notes says one key and the ordered pair says the other, which is the case the earlier figures never contained. The line is the cadence evidence for G, and under the axis is what each reading actually names at each length. The cadence reading holds G up to a body of 6 bars and is overturned at 8, so one explicit cadence is worth about that many bars of profile evidence — and it is overturned by the body's OWN incidental root motions rather than by the histogram at all. That is the finding the earlier essay could not have: on real material the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.

The passage built to make them disagree

A cadence count and a key-profile correlation have been put on one scale and run on a passage where the two agree, which tells nobody anything. Building one where they disagree — the notes of one key and the cadences of another — measures the exchange rate at about six bars per cadence, and finds something the agreement case concealed: the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.

harmony · Progression The twelve keys a key-finder has never had. Every scheme the key-finder reads, read twice: over the twelve major collections the model has always used, and over twenty-four with the harmonic minor added. The pale bar is the first and the dark one the second. On 5 of the 6 the extra twelve states change nothing a reader would see — the largest loss of certainty is 3.2 per cent, on the rondo — and no bar of any of them is renamed. The exception is the ostinato, every bar of which the twelve-collection model calls E♭ and the twenty-four-collection model calls C minor — the same seven notes, the right name. So the missing states were not costing the key-finder its answers. What they were costing is the ability to say which of a collection's seven degrees is home, and that is a different repair.

The key-finder with no tonic

Thirteen essays of key-finding have run over twelve major collections and not twenty-four keys, so a passage in A minor is read as C. Adding the missing twelve costs almost nothing and fixes almost nothing — because the model has no tonic in it at all, and below three raised sevenths in a passage the extra states are worth exactly zero.

harmony · Key-relations 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 The same interval, mistuned by the same amount, at each of its two ends. A C to G in the major key, played 25 cents wrong, with the departure carried by the lower note, split between the two, and carried by the upper note. All three are the same interval size; what differs is which note is off the scale. The share of the departure that survives into what a listener hears is 32 per cent when the lower note carries it and 56 when the upper does. The middle bar is the mean of the other two to within a hundredth, so the averaging is linear and the asymmetry is the whole of the effect. Two things produce it: the prior is 9.0 cents wide at the C and 10.0 at the G, and the likelihood is 13.2 cents wide at the lower pitch and 8.8 at the higher.

An interval is two posteriors subtracted

Treating a key as a prior over one note predicts that an interval's pull is not the single-note pull doubled, because the two degrees are not equally weighted. Half of that is wrong: splitting a mistuning between the two notes gives exactly the mean of what each end gives alone, to a thousandth, at every one of the twenty-one intervals in the scale. What is not the mean is which end carries it — and the pull turns out to be largest not on the shortest notes but on notes of about an eighth of a second, where the likelihood is a quarter of a semitone wide.

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 The setting a listener reports is not the interval they preferred. Five published preferences for an interval size, and the value a listener would have to play in a key context for that preference to be what they hear. The key pulls a heard interval back toward the scale, so a listener adjusting until it sounds right has to overshoot — and the reported setting, which is what they played, exaggerates the preference. The corrections run from 1.6 cents to 12.8 on notes of 0.25 seconds. The largest is nearly a syntonic comma, on a preference of thirteen cents, which is to say that the correction is bigger than the effect it is a correction to.

The setting is not the preference

Every published number for an interval listeners prefer — the pure third a quartet is said to find, the raised leading note, the harmonic seventh — is a value somebody adjusted until it sounded right. A listener adjusting inside a key is adjusting through the posterior computed just before, so the value they stopped at is not the one they preferred: it is the one whose heard size equals it. On quarter-second notes the correction for a pure major third is 12.8 cents, which is nearly a syntonic comma and is larger than the 13.7-cent preference it corrects.

intervals · Pitch-acuity One noise everywhere, and the noise each boundary actually gets. The seven boundaries of the tempered diatonic scale, with the noise the earlier model gives each of them — 11 cents, the same everywhere — and the noise the harmonicity model gives them instead, which runs from 11.0 cents to 35.6. The error rate is the sum of these over the octave rather than seven copies of one, so it rises from 5.1 per cent to 11.8. And the moment the boundaries differ, where they are put matters: a scale that moved its degrees would move its boundaries onto different intervals and would pay a different sum. That is the earlier null broken by the assumption its own last section named.

A boundary beside a fifth

A closed form established earlier says an n-category division costs n·σ·√(2/π) whatever the widths are, so a boundary costs the same wherever it is put and the step pattern cannot matter. Its own last section named the assumption that produces the null: one σ, applied to every boundary in the octave. Let σ follow how securely each interval is held and the formula becomes a sum over boundaries rather than n copies of one — the diatonic's error rises from 5.1 per cent to 11.8, and where the degrees are put matters again.

scales · Categorical-hearing Each tradition's own steps against the equal division of the same size. Six scales, each drawn against the equal division into the same number of degrees, under both models of how securely an interval is held. On the harmonicity model the tempered diatonic is 11 per cent better than seven equal steps; on the profile model the same comparison is 0.9 per cent, which is nothing. The two models disagree about the one comparison anybody would want the measure for, and only one of them is free of circularity: the probe-tone profile was measured on listeners raised inside the diatonic tradition, so using it to explain why the diatonic is well chosen assumes the answer. The harmonicity model assumes only that a simple ratio is easier to hold than a complicated one.

The unequal scale that is easier to name

The whole of the earlier result was that a scale's step pattern cannot matter, and every tradition it drew agreed with the equal division of its own size to three decimal places. With one noise per boundary the comparison is live again, and the tempered diatonic beats seven equal steps by eleven per cent — while the pentatonics gain nothing and the maqam scales gain two tenths of one per cent. Only one of the two security models produces the effect, and it is the one that was not measured on listeners raised inside the tradition it is being used to explain.

scales · Categorical-hearing

Named alongside it

The objects these essays reach for when they reach for this one.

Scale degreeKey-findingTonal hierarchyDifference limenCentsExpectationCadenceJust intonationModulationTonicBayesian inferenceCategorical perception

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