Acoustics & Music Theory
why it sounds like that
Concept

Octave equivalence — where it appears

The treatment of two pitches a factor of two apart as the same note in a different register. It is nearly universal across traditions and it is not exact: a listener's preferred octave is measurably wider than a 2:1.

Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.

Twelve fifths do not make seven octaves. Pitch drawn as a spiral: one turn is one octave, so a note's angle is its pitch class and its radius is how high it has climbed. Twelve pure fifths wind round 7 times and a little further, finishing 23.46 cents past seven octaves — the Pythagorean comma. Nothing in the chain closes that gap; it is what a tuning has to absorb, hide or spread.

Twelve fifths and seven octaves, which are not the same thing

Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that.

tuning · The comma 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 Roughness across an octave. Sensory dissonance between two complex tones as the upper one is swept through 19.02 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 7/5, on 11/7, on 5/3, on 9/5, on 11/5, on 7/3, on 13/5, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them.

A scale without an octave, and the spectrum that asks for it

Every scale so far repeats at the 2:1. That looks like a law and it is a consequence — the octave fuses because every partial of the upper note lands on one of the lower. Take a spectrum with no even partials and the 2:1 stops doing that, while the 3:1 starts.

scales · Beyond twelve Three octaves, and none of them is 2:1. How far above an exact doubling the upper note of an octave is set, against frequency. The listener's octave is measured with pure tones, which have no partials to beat against each other, so nothing about a stiff string can account for it. The piano's stretch is a different quantity with a different cause, and the two are drawn together only so that the difference is visible.

The octave that is not two to one

The octave is the one interval nobody argues about: two to one, exact, in every tradition that has one. Asked to set an octave by ear, listeners set it wide — and they do it with pure tones, which have no partials to beat against each other. Whatever is stretching the octave, it is not the stiffness of a piano string.

perception · Pitch-acuity Every voicing of a second-inversion major triad, least rough first. All 27 arrangements of the same three pitch classes within 3 octaves from 131 Hz, scored for roughness. The best is spaced 19 then 9 semitones — wide below, close above — and the worst is the chord in close position at the bottom of the range, 7.6 times rougher with exactly the same notes in it.

The inversion that cannot end a phrase

Three positions of one triad, three measurements, and the practice contradicted at every turn. The second inversion is the smoothest of the three by roughness, the best explained of the three by a virtual-pitch model, and the one that three centuries of practice will not let a phrase end on. What forbids it is a rule about a two-voice skeleton.

intervals · The triad The same intervals, played higher and higher. Sensory roughness for four fixed intervals as the pair is transposed up five octaves, computed from the same Plomp–Levelt model as the dissonance curve. Every one of them falls as it rises, and the small intervals fall furthest — so how consonant an interval is depends on where it is played, not only on what it is.

A chord is a register

The same three pitch classes are five times rougher at the bottom of a piano than in the middle, and the arrangement that minimises the roughness over a low bass turns out to be the bass's own fifth and sixth partials. The orchestration rule about low thirds is not a convention. It falls out of the width of a critical band, exactly.

intervals · The triad How much of the writing the prohibitions forbid. I – vi – ii – V – I in C major, written in 3, 4, 5, 6 parts, with every ensemble covering the same total compass. A triad has three pitch classes, so n parts double n − 3 of them and a duet cannot state one at all. Of every ordered pair of complete voicings of V and I, the share with no parallel fifth or octave between any pair of voices: 91.2% at 3, 75.0% at 4, 44.3% at 5, 17.9% at 6.

Why the exercise is in four parts

Eight earlier essays move four voices, and nothing here ever chose four. Cut one choir's compass into three parts instead and the two most famous prohibitions in music cost exactly nothing — the cheapest realisation already obeys them. Cut it into six and they cost more than half a semitone per voice per chord change, because a parallel octave needs a doubled note and six voices sharing three notes can hardly avoid one. Four is the smallest number of parts at which the rules have a price at all, and it is the largest at which the price is small.

harmony · Voice-leading Eighty-one chords the expectation model cannot tell apart. Every voicing of a dominant seventh on G inside the three octaves above its own root, placed by how rough it is and how far its outer voices are apart, and coloured by which member of the chord is at the bottom. The roughness runs from 0.269 to 1.449, a factor of 5.4, computed from each voicing's own spectrum under Plomp and Levelt's roughness model. The identity surprise the expectation model assigns is 3.51 bits for every one of the 81, because it is a function of a scale degree and its predecessor and there is no register anywhere in it. What separates them is spacing rather than inversion: roughness falls as the outer voices spread apart, correlating -0.42 with the span, and is indifferent to which member of the chord is at the bottom at 0.02. The seventh in the bass is not what makes a chord rough; a fourth and a third packed together at the bottom of the range is.

Eighty-one chords, one number

A dominant seventh has eighty-one arrangements inside three octaves and their roughness spans a factor of five and a half. The tonal-expectation model gives every one of them the same 3.51 bits, because its states are scale degrees and there is no register anywhere in them. Conditioning the surprise on the voicing costs no corpus — and the arithmetic says the conditioning belongs beside the probability rather than inside it, for three reasons that can each be computed.

harmony · Tonal-expectation

Named alongside it

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

RegisterRoughnessTriadVoicingCounterpointCritical bandwidthFrequency ratioOrchestrationCadenceCentsConsonanceEnumeration

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