108edo

← 107edo

108edo

109edo →

Prime factorization	22 × 33
Step size	11.1111 ¢
Fifth	63\108 (700 ¢) (→ 7\12)
Semitones (A1:m2)	9:9 (100 ¢ : 100 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine temperament, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is 27edo's superpyth fifth.

Prime harmonics

Approximation of prime harmonics in 108edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-1.96	+2.58	-2.16	+4.24	+3.92	-4.96	+2.49	+5.06	+3.76	-0.59
	Relative (%)	+0.0	-17.6	+23.2	-19.4	+38.1	+35.3	-44.6	+22.4	+45.5	+33.8	-5.3
Steps (reduced)		108 (0)	171 (63)	251 (35)	303 (87)	374 (50)	400 (76)	441 (9)	459 (27)	489 (57)	525 (93)	535 (103)

Miscellany

Aside from tuning, 108 is the smallest number with a prime factorization of the form [math]\displaystyle{ p^p \cdot q^q }[/math]. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments.

Circulating temperaments in 108edo

Tones	Pattern	L:s
5	3L 2s	22:21
6	6edo	equal
7	3L 4s	16:15
8	4L 4s	14:13
9	9edo	equal
10	8L 2s	11:10
11	9L 2s	10:9
12	12edo	equal
13	3L 10s	9:8
14	10L 4s	8:7
15	3L 12s
16	12L 4s	7:6
17	6L 11s
18	18edo	equal
19	13L 6s	6:5
20	8L 12s
21	3L 18s
22	20L 2s	5:4
23	16L 7s
24	12L 12s
25	8L 17s
26	4L 22s
27	27edo	equal
28	24L 4s	4:3
29	21L 8s
30	18L 12s
31	15L 16s
32	12L 20s
33	9L 24s
34	6L 28s
35	3L 32s
36	36edo	equal
37	34L 3s	3:2
38	32L 6s
39	30L 9s
40	28L 12s
41	26L 15s
42	24L 18s
43	22L 21s
44	20L 24s
45	18L 27s
46	16L 30s
47	14L 33s
48	12L 36s
49	10L 39s
50	8L 42s
51	6L 45s
52	4L 48s
53	2L 51s
54	54edo	equal
55	53L 2s	2:1
56	52L 4s
57	51L 6s
58	50L 8s
59	49L 10s
60	48L 12s
61	47L 14s
62	46L 16s
63	45L 18s
64	44L 20s
65	43L 22s
66	42L 24s
67	41L 26s
68	40L 28s
69	39L 30s
70	38L 32s
71	37L 34s
72	36L 36s
73	35L 38s
74	34L 40s
75	33L 42s
76	32L 44s
77	31L 46s
78	30L 48s
79	29L 50s
80	28L 52s
81	27L 54s
82	26L 56s
83	25L 58s
84	24L 60s
85	23L 62s
86	22L 64s