53edo
53edo, or 53 equal divisions of the octave, is the equal tuning featuring steps of (1200/53) ~= 22.64 cents, 53 of which stack to the perfect octave 2/1. Theoretical interest in this tuning system goes back to antiquity.
Theory
Unless one has a set of accidentals for the syntonic comma, referred to in this article as up and down (see the Notation section) one is left in the unenviable position of having to label a Ptolemaic major third the same way as the Pythagorean diminished fourth, for example. Apart from that issue, 53edo is very useful for 5-limit music.
JI approximation
53edo is most accurately a 2.3.5.13 tuning. Because it is not a Meantone system, there are actually multiple potential diatonic scales to use for 5-limit harmony, one of which is the Zarlino diatonic scale (LMsLMLs), tuned in 53edo as 9-8-5-9-8-9-5, though this particular scale is arguably best used for Lydian or Locrian modes. There's also the Didymic diatonic scale, tuned in 53edo as 9-8-5-9-9-8-5, which is better suited for Ionian mode and Major tonality in general. However, 53edo also features a MOS diatonic of 9-9-4-9-9-9-4, which is basically the Pythagorean diatonic scale.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -0.1 | -1.4 | +4.8 | -7.9 | -2.8 | +8.3 | -3.2 | +5.7 | -10.7 | +9.7 |
| Relative (%) | 0.0 | -0.3 | -6.2 | +21.0 | -35.0 | -12.3 | +36.4 | -14.0 | +25.1 | -47.3 | +42.8 | |
| Steps
(reduced) |
53
(0) |
84
(31) |
123
(17) |
149
(43) |
183
(24) |
196
(37) |
217
(5) |
225
(13) |
240
(28) |
257
(45) |
263
(51) | |
| Quality | Inframinor | Subminor | Novaminor | Pentaminor | Supraminor | Submajor | Pentamajor | Novamajor | Supermajor | Ultramajor |
|---|---|---|---|---|---|---|---|---|---|---|
| Cents | 249 | 272 | 294 | 317 | 340 | 362 | 385 | 408 | 430 | 453 |
| Just interpretation | 15/13 | 7/6, 75/64 | 32/27 | 6/5 | 39/32 | 16/13 | 5/4 | 81/64 | 9/7, 32/25 | 13/10 |
Diatonic thirds are bolded.
Chords
53edo has four different flavors of minor and major intervals as well as supraminor and submajor intervals. Its inframinor and ultramajor thirds approximate 15/13 and 13/10 respectively. At the same time, 53edo's subminor and supermajor intervals approximate 7/6 and 9/7. Then there's the extremely close approximations of Pythagorean minor and major thirds, as well as very close approximations of the Ptolemaic minor and major thirds. Finally, the supraminor and submajor thirds approximate 39/32 and 16/13. For fourth-bounded triads, there's only really five options. The first two, which involve the approximations of 9/8 and 32/27, have a marked propensity to cause crowding, and thus are dissonant. Then there's the next two, which involve the approximations of 8/7 and 7/6, which, due to their tuning are markedly less dissonant, but still dissonant. Finally, the fifth, which splits the perfect fourth cleanly in half, is an ambisonance- that is, an interval that is halfway between the extremes of consonance and dissonance.
