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.
Edostep interpretations
53edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:
- 65/64, the difference between the 13-limit tendoneutral third 16/13 and the classical major third 5/4
- 81/80 (the syntonic comma), the difference between 5/4 and the diatonic major third
- The Pythagorean comma, the difference between the Pythagorean diatonic and chromatic semitones
- 91/90, the difference between the 13-limit ultramajor third 13/10 and the septimal supermajor third 9/7
- 64/63, the difference between the diatonic major third and 9/7
- 512/507, the difference between the 13-limit neutral thirds
JI approximation
53edo is most usefully seen as a 2.3.5.7.13 tuning, but the 2.3.5.13 restriction is more accurate and shared with its multiples like 159edo. 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 novaminor and novamajor thirds, which are extremely close approximations of Pythagorean minor and major thirds and can be referred to as such. There are also the pentaminor and pentamajor thirds, which are very close approximations of the Ptolemaic minor and major thirds and can also be referred to as such. 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, the latal triads, which involve the approximations of 8/7 and 7/6, and which, due to their tuning are markedly less dissonant, but still dissonant. Finally, the last option, 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.
