229edo

The 229 equal divisions of the octave (229edo), or the 229(-tone) equal temperament (229tet, 229et), is the equal division of the octave into 229 parts of 5.2402 cents each.

Theory

While not highly accurate for its size, 229et is the point where a few important temperaments meet, and is distinctly consistent in the 11-odd-limit. It tempers out 393216/390625 (würschmidt comma) and [39 -29 3⟩ (tricot comma) in the 5-limit; 2401/2400, 3136/3125, 6144/6125, and 14348907/14336000 in the 7-limit; 3025/3024, 3388/3375, 8019/8000, 14641/14580 and 15488/15435 in the 11-limit, and using the patent val, 351/350, 2080/2079, and 4096/4095 in the 13-limit, notably supporting hemiwürschmidt, newt, and trident.

229edo is the 50th prime EDO.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[363 -229⟩	[⟨229 363]]	-0.072	0.072	1.38
2.3.5	393216/390625, [39 -29 3⟩	[⟨229 363 532]]	-0.258	0.269	5.13
2.3.5.7	2401/2400, 3136/3125, 14348907/14336000	[⟨229 363 532 643]]	-0.247	0.233	4.46
2.3.5.7.11	2401/2400, 3025/3024, 3136/3125, 8019/8000	[⟨229 363 532 643 792]]	-0.134	0.308	5.87
2.3.5.7.11.13	351/350, 2080/2079, 3025/3024, 3136/3125, 4096/4095	[⟨229 363 532 643 792 847]]	-0.017	0.384	7.32
2.3.5.7.11.13.17	351/350, 442/441, 561/560, 715/714, 3136/3125, 4096/4095	[⟨229 363 532 643 792 847 936]]	-0.009	0.356	6.79
2.3.5.7.11.13.17.19	286/285, 351/350, 442/441, 476/475, 561/560, 1216/1215, 1729/1728	[⟨229 363 532 643 792 847 936 973]]	-0.043	0.344	6.57

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	19\229	99.56	18/17	Quintagar / quintasandra / quintasandroid
1	37\229	193.87	28/25	Didacus / hemiwürschmidt
1	67\229	351.09	49/40	Newt
1	74\229	387.77	5/4	Würschmidt
1	95\229	497.82	4/3	Gary
1	75\229	503.06	147/110	Quadrawürschmidt
1	108\229	565.94	18/13	Tricot / trident