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 12 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 Carried across three octaves, a rise as the pitch to the power 2 leaves 4 pairs no longer beating. Every degree's pair across three octaves of slendro, the geometric mean beat rate held at 5.0 a second, when the rate rises as the pitch to the power 2 across the whole compass, and when the middle octave's rates are repeated in every octave. Carried on: low octave 0.7, 0.9, 1.2, 1.7, 2.2; middle octave 2.9, 3.8, 5.0, 6.6, 8.7; high octave 11.6, 15.1, 20.0, 26.5, 34.9, 4 of fifteen pairs above fifteen a second. Repeated: every octave 2.9, 3.8, 5.0, 6.6, 8.7. The fifteen-hertz line is where these essays stop calling a fluctuation a beat.

A gamelan's shimmer has to start again at every octave

Within one octave, a gamelan tuner who lets each pair's beat rate rise faster than its pitch names the goal tones and pays nothing in roughness for it. Carried across a real instrument's three octaves, the same rise breaks twice. The steepest rise that keeps every pair beating falls from a power of four to a power of one, and the roughness saving turns into a cost: a rise as the square of the pitch is seven per cent rougher than one rate everywhere. A tuning that gives each degree its middle-octave rate in every octave keeps both the clock and the saving at any compass. The beat rate then names the degree, not the pitch.

scales · Beyond twelve To a listener who fuses a degree's swells, a rise carried across the compass is worth nothing to the clock. Bits of position left unknown just before each goal in a sixteen-step cycle of 160 seconds, goals on degrees 1, 2, 1 and 4 of slendro, each preceded by its degree's swell of beating struck in three octaves at once, against the power the pair's beat rate rises as. fuses agreeing swells, rise carried on: 0 0.40, 0.25 0.40, 0.5 0.40, 0.75 0.40, 1 0.40, 1.5 0.40, 2 0.40, 2.5 0.40, 3 0.40; fuses agreeing swells, rates repeated: 0 0.40, 0.25 0.40, 0.5 0.40, 0.75 0.23, 1 0.15, 1.5 0.15, 2 0.15, 2.5 0.15, 3 0.15; attends to one octave, either family: 0 0.40, 0.25 0.40, 0.5 0.40, 0.75 0.23, 1 0.15, 1.5 0.15, 2 0.15, 2.5 0.15, 3 0.15. The skeleton alone leaves 0.65 bits; swells that all sound alike leave 0.40. A listener who attends to one octave hears both families identically; a listener who fuses a degree's three swells only when they agree finds, under the rise carried on, that no degree's swells agree.

A degree's three swells are one landmark only if they agree

A gamelan's goal tone is struck in several octaves at once, so the swell of beating that warns of it is three swells. A tuning whose beat rate rises with pitch across the whole compass gives each octave its own rate, and a listener who hears a degree's swells as one event only when their rates agree finds that, at any useful rise, none of them do: every goal becomes a swell of no particular rate and the clock is left exactly where swells that all sound alike leave it. No rise can both name the goals and let their swells agree, since naming needs ten times the power agreeing allows. Repeating the middle octave's rates in every octave does both.

scales · Beyond twelve The tuning that saves roughness rises only downward, and keeps the clock only if the low octave leaves the goals alone. Five ways of setting the beat rates of a gamelan's paired instruments over three octaves, the rate rising as the pitch to the power 2 where it rises: the roughness the detuned pairs add to the passage, against the bits of position a listener who fuses a goal's swells has left before each goal — filled with the goals struck in all three octaves, open with them struck only in the middle and high octaves. middle octave's rates repeated: roughness 3.06, bits 0.15 and 0.15; rise carried on: roughness 3.37, bits 0.40 and 0.40; repeated at the goals, rising elsewhere: roughness 3.42, bits 0.15 and 0.15; rising below the middle, repeated above: roughness 2.45, bits 0.40 and 0.15; the same, goals repeated in every octave: roughness 2.96, bits 0.15 and 0.15.

A gamelan's shimmer should rise only downward

A gamelan tuner who lets each pair's beat rate follow its pitch saves roughness in the low octave, where the pairs slow, and pays it in the high octave, where they race. Repeating the middle octave's rates only at the goal tones and letting the rest follow the pitch, as the lead proposed, is rougher than repeating everywhere. The tuning that saves is the one that lets the rate follow the pitch below the middle octave and repeats it above — a fifth less roughness at the square — and it keeps the clock only if the low octave does not double the goals.

scales · Beyond twelve Held to the shimmer it promised, the downward tuning is rougher than repeating the middle octave's rates. A gamelan's paired instruments over three octaves, the rate rising as the pitch to the power 2 where it rises: the roughness the detuned pairs add to the passage for each tuning at the average shimmer it actually has (open) and with every rate rescaled so the average is the one the families promised (filled). middle octave's rates repeated: average 100% of the promised, roughness 3.06, held 3.06; rise carried on: average 100% of the promised, roughness 3.37, held 3.37; repeated at the goals, rising elsewhere: average 100% of the promised, roughness 3.42, held 3.42; rising below the middle, repeated above: average 63% of the promised, roughness 2.45, held 3.26; the same, goals repeated in every octave: average 83% of the promised, roughness 2.96, held 3.32; repeated, slowed to the same average: average 63% of the promised, roughness 2.27, held 3.06.

A gamelan's shimmer has no cheaper shape, only a slower one

The tuning that let each pair's beat rate follow its pitch below the middle octave and repeated it above saved a fifth of the roughness — and slowed the ensemble's average shimmer by more than a third while doing it. Held to the shimmer it promised, it is rougher than repeating the middle octave's rates everywhere, and the middle octave's rates simply slowed to the same average are smoother than either. With the average fixed, the cheapest shape runs the other way, and a knee is never cheaper than a step.

scales · Beyond twelve

Named alongside it

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

RoughnessBeatingGamelanSlendroRegisterTriadVoicingCounterpointCritical bandwidthFrequency ratioOrchestrationCadence

All concepts