105edo

105edo is the equal division of the octave into 105 equal parts of 11.429 cents each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It tempers out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the optimal patent val for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though 74edo is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

105edo close-up

C . . Dbb B## . . C# . . Db . . . C## . . D

Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments.

Circulating temperaments in 105edo

Tones	Pattern	L:s
5	5edo	equal
6	3L 3s	18:17
7	7edo	equal
8	1L 7s	14:13
9	6L 3s	12:11
10	5L 5s	11:10
11	6L 5s	10:9
12	9L 3s	9:8
13	1L 12s	8:7
14	7L 7s
15	15edo	equal
16	9L 7s	7:6
17	3L 14s
18	13L 5s	6:5
19	10L 9s
20	5L 15s
21	21edo	equal
22	17L 5s	5:4
23	13L 10s
24	9L 15s
25	5L 20s
26	1L 25s
27	24L 3s	4:3
28	21L 7s
29	18L 11s
30	15L 15s
31	12L 19s
32	9L 23s
33	6L 27s
34	3L 31s
35	35edo	equal
36	33L 3s	3:2
37	31L 6s
38	29L 9s
39	27L 12s
40	25L 15s
41	23L 18s
42	21L 21s
43	19L 24s
44	17L 27s
45	15L 30s
46	13L 33s
47	11L 36s
48	9L 39s
49	7L 42s
50	5L 45s
51	3L 48s
52	1L 51s
53	52L 1s	2:1
54	51L 3s
55	50L 5s
56	49L 7s
57	48L 9s
58	47L 11s
59	46L 13s
60	45L 15s
61	44L 17s
62	43L 19s
63	42L 21s
64	41L 23s
65	40L 25s
66	39L 27s
67	38L 29s
68	37L 31s
69	36L 33s
70	35L 35s
71	34L 37s
72	33L 39s
73	32L 41s
74	31L 43s
75	30L 45s
76	29L 47s
77	28L 49s
78	27L 51s
79	26L 53s
80	25L 55s
81	24L 57s
82	23L 59s
83	22L 61s
84	21L 63s