Generator

Notes on the keyboard

A piano keyboard with the notes under discussion marked. The keyboard is used throughout because it shows distance rather than name, and distance is what the theory is about.

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. 9 of its 9 placements carry sound, built from the same numbers as the drawing, offering these buttons: In turn, Together.

Called by 5 essays

the blast radius of changing it

Key character in Werckmeister III. The twelve major keys in circle-of-fifths order, each with a bar as long as its major third is sharp of a pure 5:4. The bars run from 3.9 to 21.5 cents, so the keys genuinely differ.

Keys that had characters, and could be measured

Eighteenth-century writers described E flat major as devotional and F sharp major as harsh, and modern readers file it under synaesthesia. On the instruments those writers used, the difference between the two keys was seventeen cents of third, and that is a quantity.

tuning · The comma

The major scale as a cycle. The twelve semitones drawn as a cycle, with the notes of the scale filled in. The gaps between filled positions are the step pattern, and reading them round the circle is what makes the scale's asymmetry obvious.

Seven of the twelve, chosen unevenly

A major scale is a selection of seven positions out of twelve, and the selection is lopsided on purpose. The two semitones sit where they do because an even choice would destroy the thing that makes a scale usable.

scales · The diatonic set

A memory for the note itself, and it is dated

Absolute pitch is usually described as a rare perceptual gift. It is better described as a memory for a convention — and conventions have dates. Possessors trained on A=440 mis-name Baroque pitch by a semitone, their own labels drift sharp with age, and meanwhile most listeners without it start familiar songs within a semitone of the record.

tuning · Pitch standard

Where the series stops being a chord and becomes a scale. The interval between each partial and the next, in semitones, against partial number. It is an octave, then a fifth, then a fourth, a major third, a minor third — and it crosses 2 semitones between partials 9 and 10. Each shaded band is one octave of pitch and holds twice as many partials as the band below it: 2 between 1 and 2, 3 between 2 and 4, 5 between 4 and 8, 9 between 8 and 16. Nothing about the series changes with height; what changes is the density, and a set of notes becomes usable as a melody when its neighbours are a step apart.

The register where the series becomes a scale

A melody needs steps, and a natural trumpet has only the harmonic series. The gap between one partial and the next falls as the series rises — an octave, a fifth, a fourth, a third — and does not reach a whole tone until the eighth. That single arithmetic fact decides what two centuries of brass writing could be: fanfares below, and melody only in a band one octave wide with nine notes in it, three of them badly out of tune and every one of them lipped into place by a player who had no valve to press.

instruments · Melody

Where to stop, and what stopping there gives. Each prefix of the harmonic series sounded as a chord of pure tones, with its roughness per pair and the pitch classes it contains. The smoothest prefix is the first 4, which is a bare fifth and an octave; roughness rises monotonically from there, so no roughness argument selects six. The prefixes that contain a major triad and nothing else are 5 and 6. One partial further and it is a dominant seventh; four further and there are five pitch classes. Six is the last stopping point that gives the answer the derivation wants, and nothing else in the arithmetic picks it out.

The series is not a chord

The major triad is derived from the first six partials of the harmonic series in every textbook that derives it from anything, and the derivation works: the first six contain a major triad and nothing else. Take the first four instead and there is no third at all; take seven and there is a dominant seventh with a flat seventh thirty-one cents below any keyboard note; take sixteen and there are eight pitch classes. Six is the last stopping point that gives the answer, roughness prefers three, four or ten depending on the register, and no criterion in the arithmetic selects six over any of them.

intervals · Harmonic series

All figures · What can be heard