159edo

159edo, or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave 2/1. Like 53edo, 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), but this time, you have access to near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, giving you consistency up to the 17-odd-limit.

Theory

159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.

Edostep interpretations

159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:

• 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
• 256/255, the difference between 16/15 and 17/16
• 289/288, the difference between 17/16 and 18/17

159edo tempers out the following commas:

• The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
• The vulture comma (the difference between four 320/243 intervals and the tritave)
• The amiton (the difference between a stack of five 10/9 intervals and 27/16)
• The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
• The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
• 1029/1024 (the difference between a stack of three 8/7 intervals and 3/2)
• 385/384 (the difference between 77/64 and 6/5)
• 4000/3993 (the difference between a stack of three 11/10 intervals and 4/3)
• 625/624 (the difference between 25/24 and 26/25)
• 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
• 1089/1088 (the difference between a stack of two 33/32 intervals and 17/16)

JI approximation

Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. 159edo is most usefully considered as a 2.3.5.11.17.23 system, though you can always add single instances of 7 or 13 for flavor.

Approximation of prime harmonics in 159edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	-0.1	-1.4	-2.8	-0.4	-2.8	+0.7	-3.2	-1.9	-3.2	+2.1
	Relative (%)	0.0	-0.9	-18.7	-36.9	-5.0	-37.0	+9.3	-42.0	-24.6	-41.9	+28.3
Steps (reduced)		159 (0)	252 (93)	369 (51)	446 (128)	550 (73)	588 (111)	650 (14)	675 (39)	719 (83)	772 (136)	788 (152)