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 synsharp and synflat, one is left with 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) | |
