The tuning system

TUNE, from equal to just

A drawbar organ is the shortest way into what this control does, because it is the same problem answered the other way round. Every drawbar is a harmonic, but an organ has only the twelve notes of the keyboard to build one out of, so each drawbar is put on the nearest one. The second, fifth and eighth are all fifths an octave apart, which means holding a C puts three G's into the sound. A harmonic series is not spaced that way. Its intervals are whole-number ratios, 2:1, 3:2, 4:3, and apart from the octaves not one of them lands on a key. TUNE is the distance between those two answers, and it is continuous rather than a switch between them.

For harmonic n, the just interval above the fundamental is 1200 * log2(n) cents. Overtonium rounds that to the nearest semitone to get the equal-tempered position, and the remainder is the cent offset the TUNE knob dials in.

semitoneOffset(n, blend) = round(1200*log2(n)/100) + blend * (1200*log2(n) mod 100) / 100

At blend = 1 this is exactly n times the fundamental, so the stack fuses into one timbre. At blend = 0 every partial is on a semitone of the chromatic scale and the same stack reads as a chord. Nothing is written down as a constant, but rounded to whole cents the derivation reproduces the familiar table, which the test suite asserts so it cannot quietly drift.

Sitting on the line has a consequence worth knowing at the panel: on channels 1, 2, 4, 8, 16 and 32 the knob does nothing to the sound. An octave is 1200 cents in both tunings, so there is nowhere for the blend to take those six, and their readouts stay at 0.0 wherever the knob is. Hovering one of those strips says so rather than leaving you to work it out. They are not stuck, they are just not moved by that knob: STRETCH moves them, and so do DRIFT, the pitch modulator, and the oscillator character, whose rack sits every partial but the first a few cents off spec.

The two big offsets are the ones worth knowing by ear. The seventh partial lands 31 cents flat of a minor seventh and the eleventh 49 cents flat of a tritone, which is to say halfway between two keys. Those are the partials that make a just-tuned stack sound unlike anything a keyboard can play.

The eleventh has a name in Norway. Its pitch between the fourth and the tritone is the natural fourth of Norwegian folk music, which the willow flute plays because the harmonic series is all it can play. The composer Eivind Groven spent much of his life on it. From 1936 he built organs with up to three pitches behind every key, chosen automatically as the music moved, so an ordinary keyboard could play pure triads in every key and the eleventh partial came out exactly in tune. Here it is the eleventh row with TUNE at just, 551 cents above the octave below it.

Stretch

Nothing real rings at integer multiples of anything. A string with any bending stiffness has partials at n * f0 * sqrt(1 + B * n^2), sharp of the harmonic and increasingly so up the series, and it is that, not the hammer or the soundboard, that makes a piano sound like a piano rather than like a sawtooth. It is also what a tuner is matching when they stretch the octaves: tune the top of the instrument to the theory and it beats against its own overtones.

STRETCH is that stiffness, dialled by what it does to the top partial rather than by its own value, which lives between 0.00003 and 0.008 and means nothing to anyone. At +150 cents, which is a real piano:

The bottom of the series barely moves and the top of it walks away, which is the shape that matters. Push further and the partials stop agreeing on a fundamental, and the sound stops being a note with a timbre and becomes a bell. Negative pulls them inward instead, which no physical string does and which is worth having anyway.

It is a separate axis from TUNE rather than more of it, and the two combine: equal temperament with heavy stretch is a different object from just intonation with the same stretch. Zero is the plain harmonic series.

Tuning the keyboard

Everything above is about where a partial sits over the note you played. This is about where that note sits. Settings carries six temperaments.

They are derived from the circle of fifths rather than copied out as tables of cents. Almost every historical temperament is described by how much each of the twelve fifths is narrowed, so that is what the code says, and the pitch classes fall out of it. The two commas it is all built from come out at 23.460 and 21.506 cents, which are the published figures, and the tests assert the property that defines each temperament rather than the numbers that happen to result: meantone's third pure to a thousandth of a cent, Pythagorean's fifth likewise, Werckmeister at its characteristic 390.2.

Root

Picks which pitch class the temperament is built on, since an unequal temperament is only in tune in the keys near its centre, and that is the point of it. It is greyed out for equal temperament, which has no centre to have.

Reference pitch

415 through 466 Hz. A stays exactly there in every temperament, because the offsets are taken relative to A's own: a tuner tunes A first and works outwards. Only the other eleven notes move.

None of the three travels with a preset. A temperament is a property of the music you are playing rather than of any one sound in it: you set it once and work, and having a patch drag you back to equal in the middle of that would be no help. That includes Init, which clears the patch rather than the session.

Tracking

Without TRACK, a patch has the same spectrum at every pitch. Nothing real does. Play up a piano and the top of the series thins out, not because the note changed but because the body has a rolloff that stays where it is while the partials climb up through it.

TRACK is that rolloff, in dB per octave above a fixed corner of 1 kHz, which is around C6. It is measured against the fundamental rather than absolutely, and that is the part that matters: taken absolutely it would be a shelf, making high notes quieter rather than duller, which is not the thing worth having. Normalised, the fundamental keeps its level at every pitch and only the spectrum above it thins.

The consequence falls out of where the corner sits. A bass note has most of its series below 1 kHz and keeps nearly all of it. A treble note whose fundamental is already above the corner loses the full slope across every partial it has. At 6 dB per octave, the 32nd partial keeps 57% of its level at A1 and 4% at A5. Zero is off, and off is exact.

Drift, the fourth kind of detuning

Per channel rather than global, and random rather than periodic. DRIFT gives a partial a smooth random walk of up to 25 cents, redrawn as the note plays, which is the cassette warble the earlier instruments in this line got from tape. Every partial draws from its own stream, so a stack with drift on it never quite settles, and no two notes drift the same way.

It is the reason a held chord keeps moving without any modulation being dialled in, and it is why the pitch lamp on the PITCH MOD rule reads a needle rather than a pulse: what it shows is where modulation and drift together have the partial at this instant.