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.

Approximation of prime harmonics in 53edo

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)