Acoustics & Music Theory
why it sounds like that
Concept

Tonal hierarchy — where it appears

The measured ranking of the twelve pitch classes by how well each fits an established key. It was obtained by asking listeners to rate probe tones, and it correlates closely with how often each note occurs in the corresponding repertoire.

Named by 15 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 How many bars a key change takes to be heard. A twelve-bar progression that moves to G major at bar 6, read by the same correlation against all twenty-four profiles, with a window of 3, 4 and 8 bars. With 3 bars of history the new key is never the answer at all. With 4 bars of history the answer is G major from bar 7, one bar late, and it holds it from there. With 8 bars of history the answer is G major from bar 9, 3 bars late, and it holds it from there. The pivot bar is ambiguous by construction — it belongs to both keys, which is what makes it a pivot — so the lag is not a defect of the algorithm but a statement about how much evidence a key is.

How much evidence a modulation needs

Run a key-finder bar by bar over a progression that moves to the dominant at bar six. With four bars of history the answer becomes the new key at bar seven and holds. With three bars it never gets there at all, and reports E minor and B minor on the way. The window decides the lag as much as the music does.

perception · Key-relations 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 one number the ordered key-finder was tuned on. For each rate of alternation between two keys, the cost of changing key at which the model stops hearing two keys and starts hearing borrowed chords in one. The threshold rises with the period — 0.95 at 1 bar, 0.95 at 2 bars, 0.95 at 4 bars, 2.00 at 8 bars, 3.50 at 16 bars — so the parameter and the rate trade off against each other exactly. The value tuned earlier, 2.2, sits above every threshold on this axis, which means its verdict about fast alternation was a consequence of the tuning rather than a finding about the music. Filled means the model names two keys; hollow means it names one and calls the rest borrowings.

A modulation and a borrowing are one number apart

The key-finder that keeps the order has one tuned parameter, and said so. Sweep it and the parameter turns out to be the whole verdict: below a threshold the model hears two keys alternating, above it one key with borrowed chords. The threshold rises with how slowly the keys alternate — and the value it chose sits above every threshold in range, so its finding about fast alternation was a consequence of the tuning.

scales · Key-relations Two kinds of evidence, each measured against its own chance. Every key as a point: across, how many standard deviations its cadence count is above what the same bars resampled would give; up, the same for its pitch-class profile correlation. The winner on cadences is key C at 6.2 standard deviations and on profile C at 1.1, and they agree. Standard deviations above chance are the same unit whatever produced them, which is the commensuration this figure is for — and it costs something: it assumes the two chances are equally interesting, which is a weighting in disguise. The obvious null does not work at all for one of the two: shuffling the bars leaves a pitch-class histogram exactly as it was, so its spread is zero and every key scores nothing against it.

A count and a correlation

Cadence evidence is a count of ordered pairs and profile evidence is a correlation with a template, and the cadence essay refused to total them because they are not in the same units. Score each against its own chance and they are — standard deviations above chance are the same unit whatever produced them. Then the trouble moves: the obvious null does nothing at all to one of the two, because shuffling the bars leaves a pitch-class histogram exactly as it was.

harmony · Progression 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.

Surprise is a number

The chord that did not come was described rather than measured. Its measure is the information content of what did arrive, and a model of the probability has been to hand since the key-finding essays — eight root-motion weights, ordinal and stipulated. Reading them as a distribution prices a deceptive cadence at 2.71 bits against a perfect one's 1.85, and turns up the fact that the largest of the eight had never been read by anything.

harmony · Tonal-expectation 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 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 The mixture at which the chain acquires a tonic. Eight turns of i–iv–v–i in A minor, natural throughout, read at every mixture of the two emissions: nought is the original set overlap against the triad on each degree, one is the measured probe-tone profile rotated to each candidate key. The blocks along the top are the name the model gives, the line below is how far ahead of its best rival that name is. Below a mixture of 0.55 every bar is called C major, which is the collection and not the key; at and above it every bar is called A minor. The margin collapses to 0.33 bits at the crossing and recovers to 4.56 — higher than the 3.38 it started at, because a profile has an opinion about this passage and an overlap does not.

A tonic bought with the function

Putting the measured probe-tone profile inside the ordered key-finder is one term, and it does what was predicted: the natural-minor passage is named A minor at every bar instead of C major. It also does two things nobody predicted. It renames a scheme that has been read in the wrong key at every bar since the day it was written, and it destroys the chord's function while it is buying the key.

harmony · Key-relations Every question is answered in a corner, not on a ridge. The degree share times the minor share, over the plane of two cues: how much of the emission is the probe-tone profile, across, and how much a bass note is worth, down, with every chord's root in the bass and a bass rule that rewards the triad rooted on the bass. It runs from 0% at a profile share of 0 and a bass weight of 0 to 87% at 0.6 and 1.5. Bass 0: 0%, 0%, 0%, 0%, 41%, 32%, 46%. Bass 0.25: 0%, 0%, 0%, 0%, 52%, 55%, 44%. Bass 0.5: 0%, 0%, 0%, 0%, 68%, 62%, 51%. Bass 1: 0%, 0%, 0%, 0%, 78%, 77%, 77%. Bass 1.5: 0%, 0%, 0%, 0%, 87%, 80%, 86%. Bass 2: 0%, 0%, 0%, 0%, 87%, 87%, 87%.

Two cues meet in a corner

The profile finds a key's tonic and the bass finds a chord's degree, and until now each was swept with the other held at nothing. Swept together across 42 settings, the plane they make is not the ridge that was predicted. The tonic is a step in one direction, at a profile share of 0.55, and the bass cannot move it; the degree is a slope in the other, rising to 89 per cent as the bass is weighted, and the profile barely touches it. Every question is answered only in a corner of the plane — and the one place the two cues overlap is the one piece of music both can rescue.

scales · Key-relations

Named alongside it

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

Key-findingExpectationProbe-toneScale degreeCadenceTonicDifference limenIntervalJust-noticeable differenceLimenModulationPrior

All concepts