Acoustics & Music Theory
why it sounds like that
Concept

Pentatonic — where it appears

A five-note collection, in Western practice the longest chain of fifths that contains no semitone. It is the prefix of the diatonic set on the same chain, which is why the two share so many of their properties.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Stopping the chain of fifths at every length. The chain of fifths taken two notes at a time, three, four and so on, with each collection folded into a single octave. The smallest gap in the collection is measured at every stage: it is two semitones until the sixth note arrives, and then it is one. Five is as far as the chain goes without producing a semitone.

Five notes, and no semitones

The five black keys sound acceptable in any combination, in any order, over almost any bass note. That is not mysticism and not luck — five is the longest a chain of fifths can run before it produces a semitone, and the number is derived rather than chosen.

scales · The diatonic set Four properties, ten sizes, one survivor. Every subset of the twelve pitch classes, counted by shape, at each size from two notes to eleven, and how many shapes of each size have each property. Exactly one shape in the whole table has all four at once: the 7-note set with steps 1221222, which is the diatonic scale.

Why seven

Four properties, each shared with some other set. Run all four over every subset of the twelve, at every size from two notes to eleven — three hundred and forty-nine shapes in all — and exactly one has all four at once. It is the diatonic set, and it is the only survivor in the whole twelve-note universe.

scales · The diatonic set The sizes a chain of a pure fifth will make. Chains of a pure fifth of two to 12 notes, each folded into one octave and drawn on a 1200-cent line. A chain has exactly two step sizes at 2, 3, 5, 7, 12 notes and three or more at every other size in this range. The five-note and seven-note cases are the chain's pentatonic and its diatonic set, and they are neighbours in a series rather than two separate facts.

The only sizes a fifth will make

Stack pure fifths and fold them into an octave, and at almost every number of notes the result has three or more different step sizes. At 2, 3, 5, 7, 12, 17 and 29 it has exactly two — and at no other size below thirty. The pentatonic and the diatonic set are not two discoveries. They are neighbours in one series, and it is the comma's series.

scales · The diatonic set one measured slendro is the scale least committed to a spectrum. Each scale drawn, placed by how smooth it is against 2000 random scales of the same size, under 4 spectra. Low is smooth. Every one of them lands in the smoothest third under every spectrum, so the account that a spectrum chooses a scale survives as a statement about levels. What separates them is the length of each row's line, which is how much the answer moves when the instrument changes: one measured slendro spans 2.9 percentile points and Rast, Turkish theory spans 17.6, a factor of 6.0. The scale usually offered as the case for spectral matching is the one whose standing barely depends on the spectrum, and Rast, Turkish theory is the one that depends on it most.

The scale least committed to its own instrument

The essays on scales beyond twelve draw scales without spectra and spectra without scales, and none of them had put a tradition's own degrees through the roughness model. Doing it for six scales and five spectra says the account survives — every tradition lands between the eighth and the twenty-sixth percentile of random scales of its size, and under a pure tone not one of them does. But the scale whose standing moves least when the instrument changes is the gamelan's, at 2.9 percentile points, and the one that moves most is the five-limit diatonic at 17.4. The scale the argument is usually made about is the one it explains least.

scales · Beyond twelve Weighted by the pairs a melody sounds next to each other, every seven-note scale is rougher than most. Each scale placed by how smooth it is against 2000 random scales of the same size, under 4 spectra, with each pair of its degrees weighted by how often the plainest melody its ascent and descent permit sounds the two — here the pairs a melody sounds next to each other. Low is smooth. Raga Bhupali: 40.3 to 56.8, spread 16.5; Raga Deshkar: 25.1 to 37.5, spread 12.4; one measured slendro: 31.8 to 44.5, spread 12.7; Rast, Arabic theory: 92.0 to 98.6, spread 6.6; Rast, Turkish theory: 91.8 to 97.8, spread 6.0; the diatonic major, tempered: 90.7 to 97.0, spread 6.3; the diatonic major, five-limit just: 91.8 to 97.8, spread 6.0.

The smoothness is in the skips

Traditional scales come out smoother than random scales of their size because every pair of their degrees is counted once. Count only the pairs a melody on the scale actually sounds next to each other, and every seven-note scale here is rougher than ninety per cent of random ones, under every spectrum, and under a pure tone as well. The smoothness is carried by the pairs a melody reaches by skipping — its thirds and above all its fifths, which are smoother than all but two random scales in two thousand. Counted by their own ascents and descents, the two ragas that share one set of notes stand in different places at last: Deshkar's skipped Re buys a smoother ascent and costs it the fifths.

scales · Beyond twelve Weighted by the ring of the note before, a note every 0.6 seconds, Raga Deshkar moves least. Each scale placed by how smooth it is against 2000 random scales of its size under 4 instruments, each with its own spectrum and its own ring, with every pair of notes its plainest melody sounds weighted by how much of the earlier note is still sounding when the later one begins, a note every 0.6 seconds. Low is smooth. Raga Bhupali: plucked string, 6 s 53.6, long-ringing string, 12 s 35.8, blown note, 2 s hall 41.3, free bar, 4 s 58.5; spread 22.8; Raga Deshkar: plucked string, 6 s 39.0, long-ringing string, 12 s 25.9, blown note, 2 s hall 26.1, free bar, 4 s 42.0; spread 16.1; one measured slendro: plucked string, 6 s 42.9, long-ringing string, 12 s 27.8, blown note, 2 s hall 32.6, free bar, 4 s 46.4; spread 18.6; Rast, Arabic theory: plucked string, 6 s 92.0, long-ringing string, 12 s 65.2, blown note, 2 s hall 92.3, free bar, 4 s 96.1; spread 30.9; Rast, Turkish theory: plucked string, 6 s 88.0, long-ringing string, 12 s 58.1, blown note, 2 s hall 90.8, free bar, 4 s 93.6; spread 35.5; the diatonic major, tempered: plucked string, 6 s 84.8, long-ringing string, 12 s 54.2, blown note, 2 s hall 89.5, free bar, 4 s 91.3; spread 37.0; the diatonic major, five-limit just: plucked string, 6 s 87.7, long-ringing string, 12 s 56.7, blown note, 2 s hall 90.3, free bar, 4 s 92.9; spread 36.2.

A scale is committed to how long its instrument rings

A traditional scale's standing on the roughness model moves when the instrument changes, and that movement was read as a commitment to the instrument's spectrum. Weight every pair of notes a melody sounds by how much of the earlier note is still ringing when the later one begins, and the movement grows — the tempered diatonic's by 37 percentile points at a note every 0.6 seconds — but almost none of it is the spectrum. Put four instruments' rings on one spectrum and the scale moves 38.6 points; put four spectra under one ring and it moves 9.4. And the partials that make a fifth smooth are the first to stop sounding.

scales · Beyond twelve

Named alongside it

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

Chain of fifthsPlomp–Levelt curveRagaSensory dissonanceCritical bandwidthDiatonic scaleInterval contentMaqamMaximal evennessMelodic intervalNull modelStep pattern

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