Acoustics & Music Theory
why it sounds like that
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Maqam Rast, Arabic theory

Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932.
Maqam Rast, Arabic theory. Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 21 of its 21 placements carry sound, built from the same numbers as the drawing, offering these buttons: Maqam Rast, Arabic theory — up, Maqam Rast, Turkish theory — up, Raga Bhupali — up, Raga Bhupali, ascending as the tradition does, Raga Deshkar — up, Raga Deshkar, ascending as the tradition does, Slendro, one Javanese gamelan — up, The lower tetrachord alone and 10 more.

Called by 4 essays

the blast radius of changing 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 Maqam Rast, Arabic theory. Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932.

A scale is not a set of pitches

Two ragas can have identical pitch sets and be different ragas. Two national theories of one maqam put its third degree thirty-five cents apart. Both facts are fatal to the idea that a mode is a collection of notes, and both are ordinary in the traditions concerned.

scales · Beyond twelve 3 ways to fill the same fourth. Each row is a tetrachord: a span of 498 cents — a pure fourth — with two notes inside it, drawn in cents from the lower bound. The bounding notes never move, which is what makes the tetrachord a unit; everything that varies is interior. diatonic (ditonic) puts them at 90 and 294 cents, giving steps of 90, 204, 204; intense chromatic puts them at 81 and 231 cents, giving steps of 81, 151, 267; enharmonic puts them at 63 and 112 cents, giving steps of 63, 49, 386. Against the twelve equal steps below, 1 of 3 land within twenty-five cents of a semitone at every degree; the worst miss is 37 cents, which is a quarter of a semitone and has no name on a keyboard. Nothing in this construction is a subset of an equal division, so nothing the census here does applies to any of it.

A scale built downward from a fourth

A tetrachord is a bounded span with two free notes inside it, which is a different generator from a chain of fifths and a different object from a subset of an equal division. At the maqam tradition's own quarter-tone resolution the construction admits thirty-six fillings of the fourth and 1,296 octaves built from them; the census here can see six of the first and 2.8 per cent of the second, and one of the octaves it cannot see is an ordinary mode of a living tradition.

scales · Beyond twelve Two modes, one set, and every measure of a set that cannot tell them apart. Raga Bhupali and Raga Deshkar drawn as the moves each allows: an arrow from one degree to another means the tradition's ascent or descent goes that way. Raga Deshkar's ascent omits Re, so the two graphs differ by an edge. Below, every standard measure of a scale, evaluated on both — and they are identical in every row, which the drawing checks before it is made. An ascent and a descent that between them use all 5 degrees can be chosen in 63 ways, 62 of them asymmetric. That is how many modes collapse onto one pitch set under the simplest order model there is, and a census over subsets counts the set once.

A degree is where it goes next

The first essay on scales beyond twelve said a mode is not a set and listed four things a set cannot record. This one makes the sharpest of them arithmetic. Specify a mode as an ascent and a descent — which is how a living tradition specifies one — and it becomes a directed graph on its degrees; sixty-three such graphs collapse onto a single pentatonic set, sixty-two of them with an ascent that is not the descent reversed, and every standard measure of a scale returns the same value for all of them.

scales · Beyond twelve

All figures · What can be heard