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

108 equal divisions of the octave (abbreviated 108edo or 108ed2), also called 108-tone equal temperament (108tet) or 108 equal temperament (108et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 108 equal parts of about 11.1 ¢ each. Each step represents a frequency ratio of 2^1/108, or the 108th root of 2.

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	-18	+23	-19	+38	+35	-45	+22	+46	+34	-5
Steps (reduced)		108 (0)	171 (63)	251 (35)	303 (87)	374 (50)	400 (76)	441 (9)	459 (27)	489 (57)	525 (93)	535 (103)

Intervals

Steps	Cents	Ups and downs notation	Approximate ratios
0	0	D	1/1
1	11.1111	↑D, ↓8E♭
2	22.2222	↑↑D, ↓7E♭	66/65, 78/77
3	33.3333	↑3D, ↓6E♭	49/48
4	44.4444	↑4D, ↓5E♭	36/35, 40/39, 77/75
5	55.5556	↑5D, ↓4E♭	33/32
6	66.6667	↑6D, ↓3E♭	26/25, 28/27, 80/77
7	77.7778	↑7D, ↓↓E♭
8	88.8889	↑8D, ↓E♭
9	100	D♯, E♭	35/33, 55/52
10	111.111	↑D♯, ↓8E	16/15
11	122.222	↑↑D♯, ↓7E	15/14
12	133.333	↑3D♯, ↓6E
13	144.444	↑4D♯, ↓5E	49/45
14	155.556	↑5D♯, ↓4E	35/32
15	166.667	↑6D♯, ↓3E	11/10, 54/49
16	177.778	↑7D♯, ↓↓E
17	188.889	↑8D♯, ↓E	39/35
18	200	E	9/8
19	211.111	↑E, ↓8F	44/39
20	222.222	↑↑E, ↓7F	25/22
21	233.333	↑3E, ↓6F	8/7
22	244.444	↑4E, ↓5F	15/13
23	255.556	↑5E, ↓4F
24	266.667	↑6E, ↓3F	7/6
25	277.778	↑7E, ↓↓F	75/64
26	288.889	↑8E, ↓F	13/11, 33/28, 77/65
27	300	F
28	311.111	↑F, ↓8G♭
29	322.222	↑↑F, ↓7G♭	77/64
30	333.333	↑3F, ↓6G♭	40/33
31	344.444	↑4F, ↓5G♭	39/32
32	355.556	↑5F, ↓4G♭	16/13
33	366.667	↑6F, ↓3G♭
34	377.778	↑7F, ↓↓G♭	56/45
35	388.889	↑8F, ↓G♭	5/4
36	400	F♯, G♭	44/35
37	411.111	↑F♯, ↓8G	33/26
38	422.222	↑↑F♯, ↓7G
39	433.333	↑3F♯, ↓6G	9/7, 50/39, 77/60
40	444.444	↑4F♯, ↓5G
41	455.556	↑5F♯, ↓4G	13/10
42	466.667	↑6F♯, ↓3G	21/16, 64/49
43	477.778	↑7F♯, ↓↓G	33/25
44	488.889	↑8F♯, ↓G
45	500	G	4/3
46	511.111	↑G, ↓8A♭	35/26
47	522.222	↑↑G, ↓7A♭
48	533.333	↑3G, ↓6A♭	15/11, 49/36
49	544.444	↑4G, ↓5A♭	48/35
50	555.556	↑5G, ↓4A♭	11/8
51	566.667	↑6G, ↓3A♭
52	577.778	↑7G, ↓↓A♭	39/28
53	588.889	↑8G, ↓A♭	45/32
54	600	G♯, A♭
55	611.111	↑G♯, ↓8A	64/45
56	622.222	↑↑G♯, ↓7A	56/39
57	633.333	↑3G♯, ↓6A	75/52
58	644.444	↑4G♯, ↓5A	16/11
59	655.556	↑5G♯, ↓4A	35/24
60	666.667	↑6G♯, ↓3A	22/15, 72/49
61	677.778	↑7G♯, ↓↓A	65/44, 77/52
62	688.889	↑8G♯, ↓A	52/35
63	700	A	3/2
64	711.111	↑A, ↓8B♭
65	722.222	↑↑A, ↓7B♭	50/33
66	733.333	↑3A, ↓6B♭	32/21, 49/32
67	744.444	↑4A, ↓5B♭	20/13, 77/50
68	755.556	↑5A, ↓4B♭
69	766.667	↑6A, ↓3B♭	14/9, 39/25
70	777.778	↑7A, ↓↓B♭
71	788.889	↑8A, ↓B♭	52/33
72	800	A♯, B♭	35/22
73	811.111	↑A♯, ↓8B	8/5
74	822.222	↑↑A♯, ↓7B	45/28, 77/48
75	833.333	↑3A♯, ↓6B
76	844.444	↑4A♯, ↓5B	13/8
77	855.556	↑5A♯, ↓4B	64/39
78	866.667	↑6A♯, ↓3B	33/20, 81/49
79	877.778	↑7A♯, ↓↓B
80	888.889	↑8A♯, ↓B
81	900	B
82	911.111	↑B, ↓8C	22/13, 56/33
83	922.222	↑↑B, ↓7C	75/44
84	933.333	↑3B, ↓6C	12/7, 77/45
85	944.444	↑4B, ↓5C
86	955.556	↑5B, ↓4C	26/15
87	966.667	↑6B, ↓3C	7/4
88	977.778	↑7B, ↓↓C	44/25
89	988.889	↑8B, ↓C	39/22
90	1000	C	16/9
91	1011.11	↑C, ↓8D♭	70/39
92	1022.22	↑↑C, ↓7D♭
93	1033.33	↑3C, ↓6D♭	20/11, 49/27
94	1044.44	↑4C, ↓5D♭	64/35
95	1055.56	↑5C, ↓4D♭
96	1066.67	↑6C, ↓3D♭
97	1077.78	↑7C, ↓↓D♭	28/15
98	1088.89	↑8C, ↓D♭	15/8
99	1100	C♯, D♭	66/35
100	1111.11	↑C♯, ↓8D
101	1122.22	↑↑C♯, ↓7D
102	1133.33	↑3C♯, ↓6D	25/13, 27/14, 77/40
103	1144.44	↑4C♯, ↓5D	64/33
104	1155.56	↑5C♯, ↓4D	35/18, 39/20
105	1166.67	↑6C♯, ↓3D
106	1177.78	↑7C♯, ↓↓D	65/33, 77/39
107	1188.89	↑8C♯, ↓D
108	1200	D	2/1