19-TET
19-TET divides the octave into nineteen equal steps of 63.16 cents. It buys you thirds and sixths that are almost exactly just, and it pays for them with a fifth that is 7 cents flat. It is the cheapest good answer to 12-TET's worst problem.
| Property | Value |
|---|---|
| Steps per octave | 19 |
| Step ratio | 2 ^ (1/19) ≈ 1.037155 |
| Step size | 63.16 cents |
| Octave | 2/1, exact |
What it fixes
12-TET's major third is 13.7 cents sharp. 19-TET's is 7.4 cents flat — half the error, in the other direction. But the real prize is elsewhere:
| Interval | Steps | 19-TET cents | Just ratio | Just cents | Error |
|---|---|---|---|---|---|
| Minor third | 5 | 315.8 | 6/5 | 315.6 | +0.1¢ |
| Major sixth | 14 | 884.2 | 5/3 | 884.4 | −0.1¢ |
| Major third | 6 | 378.9 | 5/4 | 386.3 | −7.4¢ |
| Minor sixth | 13 | 821.1 | 8/5 | 813.7 | +7.4¢ |
| Perfect fifth | 11 | 694.7 | 3/2 | 702.0 | −7.2¢ |
| Perfect fourth | 8 | 505.3 | 4/3 | 498.0 | +7.2¢ |
The minor third and major sixth are essentially pure — a tenth of a cent off, which is nothing. They have to be: 5 + 14 = 19 steps and 6/5 × 5/3 = 2, so the two errors are equal and opposite by construction.
The fifth is the price. Seven cents flat is audible as a slight slackness in open fifths, which is the standard complaint about 19-TET.
Every interval
| Interval | Steps | Cents | Nearest just ratio | Error |
|---|---|---|---|---|
| Chromatic semitone (C→C♯) | 1 | 63.2 | 25/24 | −7.5¢ |
| Minor second (diatonic) | 2 | 126.3 | 16/15 | +14.6¢ |
| Major second | 3 | 189.5 | 9/8 | −14.4¢ |
| Minor third | 5 | 315.8 | 6/5 | +0.1¢ |
| Major third | 6 | 378.9 | 5/4 | −7.4¢ |
| Perfect fourth | 8 | 505.3 | 4/3 | +7.2¢ |
| Augmented fourth | 9 | 568.4 | 45/32 | −21.8¢ |
| Diminished fifth | 10 | 631.6 | 64/45 | +21.8¢ |
| Perfect fifth | 11 | 694.7 | 3/2 | −7.2¢ |
| Minor sixth | 13 | 821.1 | 8/5 | +7.4¢ |
| Major sixth | 14 | 884.2 | 5/3 | −0.1¢ |
| Minor seventh | 16 | 1010.5 | 9/5 | −7.1¢ |
| Major seventh | 17 | 1073.7 | 15/8 | −14.6¢ |
| Octave | 19 | 1200 | 2/1 | 0 |
The minor second is larger than a 12-TET semitone
Two 19-TET steps is 126.3 cents — a quarter of a semitone wider than 12-TET's 100. The small semitone in 19-TET is the one-step chromatic semitone at 63.2 cents. Nineteen has two different semitones, and that is the whole point (see below).
Split accidentals
In 12-TET, C♯ and D♭ are the same key on the piano. In 19-TET they are not:
- C → C♯ is one step (63.2¢).
- D♭ → D is also one step.
- C♯ to D♭ is one step apart — they are different pitches, and C♯ is below D♭.
The nineteen steps are the seven naturals, plus a sharp and a flat for each of the five that take them, plus E♯/F♭ and B♯/C♭. That is 7 + 10 + 2 = 19. Nothing collapses.
This is what makes 19-TET a real meantone temperament rather than a curiosity: it notates the way Renaissance and early Baroque music was actually meant, where a D♯ and an E♭ were different notes with different functions.
Writing 19-TET expressions
2 ^ (1/19) # one step
2 ^ (6/19) # major third (6 steps)
2 ^ (11/19) # perfect fifth (11 steps)
base.f * 2 ^ (6/19) # a major third above the BaseNote
[1].f * 2 ^ (1/19) # one step above note 1A 19-TET major triad:
base.f # root
base.f * 2 ^ (6/19) # major third
base.f * 2 ^ (11/19) # fifthLegacy JavaScript syntax
module.baseNote.getVariable('frequency')
module.baseNote.getVariable('frequency').mul(new Fraction(2).pow(new Fraction(6, 19)))
module.baseNote.getVariable('frequency').mul(new Fraction(2).pow(new Fraction(11, 19)))The shipped module
The library's Scale Systems section ships a 19-TET module (file scale-systems/TET-19.json; the tile reads 19-TET).
| Notes | 20 — nineteen steps plus the note you start on |
| Base frequency | 440 Hz |
| Tempo | 100 |
| Beats per measure | 4 |
| Instrument | sine-wave |
| Note duration | beat(base) * (1/2) |
| Step | [n].f * 2 ^ (1/19) |
Note 1 is base.f; every later note is the previous one times 2 ^ (1/19). It is a chain, so moving note 1 moves the whole scale.
To load it, drag the tile from Scale Systems (or search 19 or the microtonal tag with the library magnifier) onto a note or onto the BaseNote circle — a drop on empty canvas is refused. The full loading workflow is on Equal Temperament.
All nineteen steps after the first show a ≈ prefix and cross-hatching, because 2^(k/19) is irrational. The fraction beside the ≈ is the note's approximate ratio to the BaseNote.
Working in 19-TET
Start with triads. Play a 12-TET major triad and a 19-TET one back to back. The 19-TET minor third is the one that stops beating.
Minor keys are the sweet spot. With a pure minor third and a pure major sixth, 19-TET flatters minor-mode harmony more than major.
The fifth wants company. A bare open fifth exposes the −7.2¢ error. Fill it in — the third covers it.
The arrows cannot walk the scale. The ▲/▼ arrows apply a rational interval you set in Settings → Arrows — the octave by default, but any ratio of positive integers in [1/16, 16]. A 19-TET step is not one of those. To move a note by one degree, edit the exponent.
Challenges
Notation. Standard staff notation handles 19 pitches better than you would expect — the split accidentals map onto ♯ and ♭ directly — but there is no standard for what to do beyond that.
Instruments. Almost nothing acoustic is built for 19-TET, which is precisely why you would compose it here.
Retraining. Interval sizes in steps are all different. Six steps is a major third, not four.
Next steps
- 31-TET — better fifths, plus septimal intervals
- 12-TET — the comparison case
- Pure Ratios — what 19-TET's thirds are chasing
- Custom TET — build your own division