282edo

The 282 equal divisions of the octave (282edo), or the 282(-tone) equal temperament (282tet, 282et) when viewed from a regular temperament perspective, is the equal division of the octave into 282 parts of 4.2553 cents each.

Theory

282edo is the smallest equal temperament uniquely consistent through to the 23-odd-limit, and also the smallest consistent to the 29-odd-limit. It shares the same 3rd, 7th, and 13th harmonics with 94edo (282 = 3 × 94), as well as 11/10 and 20/17 (supporting the garistearn temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the optimal patent val for the jupiter temperament; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for septisuperfourth temperament. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	[32 -7 -9⟩, [-7 22 -12⟩	[⟨282 447 655]]	-0.1684	0.1671	3.93
2.3.5.7	6144/6125, 118098/117649, 250047/250000	[⟨282 447 655 792]]	-0.2498	0.2020	4.75
2.3.5.7.11	540/539, 4000/3993, 5632/5625, 137781/137500	[⟨282 447 655 792 976]]	-0.3081	0.2151	5.06
2.3.5.7.11.13	540/539, 729/728, 1575/1573, 2200/2197, 3584/3575	[⟨282 447 655 792 976 1044]]	-0.3480	0.2156	5.07
2.3.5.7.11.13.17	540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197	[⟨282 447 655 792 976 1044 1153]]	-0.3481	0.1996	4.69
2.3.5.7.11.13.17.19	456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573	[⟨282 447 655 792 976 1044 1153 1198]]	-0.3152	0.2061	4.84
2.3.5.7.11.13.17.19.23	456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287	[⟨282 447 655 792 976 1044 1153 1198 1276]]	-0.3173	0.1944	4.57