Acoustics & Music Theory
why it sounds like that
Concept

Consonance — where it appears

The judgement that notes sounded together are smooth or stable, as distinct from the physical roughness between their partials. The two come apart: intervals that measure equally rough are heard differently, and what a culture treats as consonant has changed.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Two tones in the ratio 3 to 2. Two sine tones whose frequencies are in the ratio 3 to 2, and their sum. The combined pattern repeats every time both waves return to the start together, which for a simple ratio is soon — here after 2 cycles of the lower tone and 3 of the upper.

Two notes and a ratio, which is the whole of consonance

Sound two tones together and the pair either settles or does not. What decides it is the ratio of their frequencies, and the rule is that simpler ratios settle — which is two and a half thousand years old and still not quite an explanation.

intervals · Consonance The critical band, measured in semitones. The width of the ear's frequency-analysis band at each pitch, converted from hertz into semitones. Two intervals drawn as horizontal lines cross the curves: below the crossing the interval fits inside one band and its notes are not resolved from each other, and above it they are.

A third is rougher in the bass

Consonance is usually presented as a property an interval has. It is not. The same major third is muddy two octaves below middle C and clean two octaves above it, the ratio never changed, and the frequency where it stops being muddy can be solved for.

intervals · Consonance All 55 chords of 3 notes, on both counts. Every 3-note subset of the twelve pitch classes containing the root — 55 of them — with summed Plomp–Levelt roughness on a string spectrum across, and the largest whole number needed to write the chord in just intonation up. The smoothest is 0–5–10, 15:20:27, which is stacked fourths. The simplest is 0–5–9, 3:4:5, which is the major triad. Exactly one chord is in the best 18% on both axes: 3:4:5, the major triad.

Three is the largest agreeable number

Take every chord of every size that can be built from the twelve, score each one for roughness and for the size of the whole numbers it needs, and ask how many are good on both counts. At two notes the answer is one. At three notes it is still one. At four it is eight, and the question stops having an answer.

intervals · The triad Two models of consonance, and where they disagree. Every chord scored twice: horizontally by summed Plomp–Levelt roughness, vertically by the largest integer needed to write it as members of one harmonic series. Both are supposed to be measuring consonance and both are computed here from the chord itself. They correlate, but not tightly enough to be the same claim, and the chords furthest from the diagonal are the ones any experiment has to be run on.

Consonance is half learned, and this is the half

This site's founding claim is that consonance is small whole numbers. Two computable models say so and they disagree about which chords — which is already awkward. The cross-cultural evidence is worse: listeners with little exposure to Western music discriminate roughness exactly as anyone does, match octaves exactly as anyone does, and rate consonant and dissonant chords as equally pleasant.

perception · Consonance The thirteen intervals at C4, ordered by roughness, against Fux's species, 1725. Every interval within an octave above 262 Hz, ordered by the Plomp–Levelt roughness of a string spectrum — from 0.0029 for the unison to 0.2635 for the minor second — beside whether Fux's species, 1725 counts it a consonance and how many of the thirteen may follow it under that rule set. The best single threshold on roughness misclassifies 2 of the thirteen, among them the major third and the minor third, and the gap it falls in is 4.2 per cent of the roughness either side of it.

A dissonance is what has to be resolved

The perfect fourth is a consonance between two upper voices and a dissonance against the bass, and the same three notes are involved either way. Score both arrangements with the roughness model and the one the rules call a dissonance comes out thirty per cent smoother. Whatever the rule is tracking, it is not the sound.

harmony · Consonance The same roughness, before and after the window it has to be heard through. The instantaneous roughness of an interval under a vibrato, and the same quantity after a running average of 59 milliseconds — the time a dissonance has to last to be heard as one, which is 4 cycles of this interval's own 68-hertz fluctuation rather than a number chosen for the figure. The mean is identical to every digit, 0.1487 against 0.1487, because a running average cannot change an average — so the earlier Jensen factor of 1.0 survives the window untouched and its prediction that the window would shrink it is wrong. What the window destroys is the depth: 0.30 of the mean becomes 0.23, which is 77 per cent. The roughness a vibrato adds is heard; the fact that it is moving is mostly not.

The mean survives the window

A roughness that moves has a mean, a depth and a rate — all three of which a listener could only have through a temporal window. Applying the window already to hand settles which of the three survives, and the answer refutes the guess: a running average cannot change an average, so the octave's factor of nineteen stands and the movement is what goes.

intervals · The voice Both notes of a fifth, and the coincidences going out. The highest partial still above the threshold of hearing for each note of a fifth struck at 80 decibels on 130.8 hertz, against time, with a 1/n spectrum and a loss rising as the partial number to the power 1. Both curves come down from off the top of the frame — the lower note starts with 152 partials and the upper with 102, because twenty kilohertz is a ceiling in frequency and not in partial number. The rings are the interval's partial coincidences at the moment they stop existing — successive multiples of one ratio, which go out from the top down: 12:8 at 0.47 s, then 9:6 at 0.64 s, then 6:4 at 1.04 s, then 3:2 at 2.14 s. The lowest, 3:2, is the last, and after it the two notes have no partial in common that either of them can still supply.

A fifth on a piano is not a fifth a second later

Nine essays draw one note in silence, and a listener is given a texture. Put two struck notes an interval apart and consonance comes apart into two quantities that had agreed while nothing moved: the partial coincidence that names the interval outlives the strike in exactly the order common-practice theory ranks its intervals, and the roughness that scores it reorders itself inside a third of a second, with the fifth overtaken by four intervals every treatise calls harsher.

timbre · Harmonic series No seventh chord can be spaced to last as long as a triad. Every inversion and spacing within 2 octaves over C3, struck at 80 dB, for two triads and five seventh chords: the bar is the longest any spacing keeps every pair's partial coincidence, and the tick is the bound set by the chord's worst pitch-class distance — the longest any presentation of that distance lasts. major triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; minor triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; dominant seventh: 0.66 s over 32 voicings, bound 0.64 set by the tone; major seventh: 0.42 s over 32 voicings, bound 0.38 set by the semitone; minor seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; half-diminished seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; diminished seventh: 0.86 s over 32 voicings, bound 0.87 set by the tritone. Every seventh chord contains a distance worse than any a triad contains, except the diminished seventh, whose distances are only minor thirds and tritones.

A seventh chord cannot be spaced to last like a triad

A struck triad keeps the partial coincidences of all its intervals for at most 1.26 seconds, whichever way it is spaced. Run the same census over every inversion and spacing of five kinds of seventh chord and none gets near: the dominant, minor and half-diminished sevenths top out at about two thirds of a second, the major seventh at 0.42. The diminished seventh, which theory calls the least stable of them, lasts longest at 0.86 — because it is the only one with no tone or semitone among its pitch-class distances, and the worst distance a chord contains sets a ceiling no spacing can lift.

timbre · Harmonic series

Named alongside it

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

RoughnessCritical bandwidthCounterpointDecayHarmonic seriesInversionCadenceChord qualityCross-culturalDominant seventhEnculturationEnumeration

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