110edo

← 109edo

110edo

111edo →

Prime factorization	2 × 5 × 11
Step size	10.9091¢
Fifth	64\110 (698.182¢) (→32\55)
Semitones (A1:m2)	8:10 (87.27¢ : 109.1¢)
Dual sharp fifth	65\110 (709.091¢) (→13\22)
Dual flat fifth	64\110 (698.182¢) (→32\55)
Dual major 2nd	19\110 (207.273¢)
Consistency limit	5
Distinct consistency limit	5

110 equal divisions of the octave (abbreviated 110edo or 110ed2), also called 110-tone equal temperament (110tet) or 110 equal temperament (110et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 110 equal parts of about 10.9 ¢ each. Each step represents a frequency ratio of 2^1/110, or the 110th root of 2.

Theory

It tempers out 15625/15552 and 3486784401/3355443200 in the 5-limit. Using the patent val, it tempers out 1728/1715, 3125/3087, and 3645/3584 in the 7-limit.

Approximation of odd harmonics in 110edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-3.77	-4.50	+2.08	+3.36	+5.05	-0.53	+2.64	+4.14	-2.97	-1.69	+4.45
	relative (%)	-35	-41	+19	+31	+46	-5	+24	+38	-27	-15	+41
Steps (reduced)		174 (64)	255 (35)	309 (89)	349 (19)	381 (51)	407 (77)	430 (100)	450 (10)	467 (27)	483 (43)	498 (58)

Intervals

Steps	Cents	Ups and downs notation (dual flat fifth 64\110)	Ups and downs notation (dual sharp fifth 65\110)	Approximate ratios
0	0	D	D	1/1
1	10.9091	↑D, ↓E♭♭	↑D, ↓4E♭
2	21.8182	↑↑D, E♭♭	↑↑D, ↓3E♭
3	32.7273	↑3D, ↓7E♭	↑3D, ↓↓E♭	56/55
4	43.6364	↑4D, ↓6E♭	↑4D, ↓E♭	40/39
5	54.5455	↑5D, ↓5E♭	↑5D, E♭	33/32, 65/63
6	65.4545	↑6D, ↓4E♭	↑6D, ↓14E
7	76.3636	↑7D, ↓3E♭	↑7D, ↓13E
8	87.2727	D♯, ↓↓E♭	↑8D, ↓12E	21/20
9	98.1818	↑D♯, ↓E♭	↑9D, ↓11E	35/33, 55/52
10	109.091	↑↑D♯, E♭	↑10D, ↓10E
11	120	↑3D♯, ↓7E	↑11D, ↓9E
12	130.909	↑4D♯, ↓6E	↑12D, ↓8E	14/13, 27/25
13	141.818	↑5D♯, ↓5E	↑13D, ↓7E	13/12
14	152.727	↑6D♯, ↓4E	↑14D, ↓6E	35/32
15	163.636	↑7D♯, ↓3E	D♯, ↓5E
16	174.545	D𝄪, ↓↓E	↑D♯, ↓4E	72/65
17	185.455	↑D𝄪, ↓E	↑↑D♯, ↓3E	10/9, 39/35, 49/44
18	196.364	E	↑3D♯, ↓↓E	55/49
19	207.273	↑E, ↓F♭	↑4D♯, ↓E
20	218.182	↑↑E, F♭	E
21	229.091	↑3E, ↓7F	↑E, ↓4F	8/7
22	240	↑4E, ↓6F	↑↑E, ↓3F	55/48
23	250.909	↑5E, ↓5F	↑3E, ↓↓F
24	261.818	↑6E, ↓4F	↑4E, ↓F	64/55
25	272.727	↑7E, ↓3F	F
26	283.636	E♯, ↓↓F	↑F, ↓4G♭	33/28
27	294.545	↑E♯, ↓F	↑↑F, ↓3G♭
28	305.455	F	↑3F, ↓↓G♭
29	316.364	↑F, ↓G♭♭	↑4F, ↓G♭	6/5
30	327.273	↑↑F, G♭♭	↑5F, G♭
31	338.182	↑3F, ↓7G♭	↑6F, ↓14G	39/32
32	349.091	↑4F, ↓6G♭	↑7F, ↓13G
33	360	↑5F, ↓5G♭	↑8F, ↓12G	16/13
34	370.909	↑6F, ↓4G♭	↑9F, ↓11G	26/21
35	381.818	↑7F, ↓3G♭	↑10F, ↓10G
36	392.727	F♯, ↓↓G♭	↑11F, ↓9G
37	403.636	↑F♯, ↓G♭	↑12F, ↓8G	63/50
38	414.545	↑↑F♯, G♭	↑13F, ↓7G	14/11, 33/26, 80/63
39	425.455	↑3F♯, ↓7G	↑14F, ↓6G
40	436.364	↑4F♯, ↓6G	F♯, ↓5G
41	447.273	↑5F♯, ↓5G	↑F♯, ↓4G
42	458.182	↑6F♯, ↓4G	↑↑F♯, ↓3G	13/10, 64/49
43	469.091	↑7F♯, ↓3G	↑3F♯, ↓↓G	21/16, 55/42
44	480	F𝄪, ↓↓G	↑4F♯, ↓G
45	490.909	↑F𝄪, ↓G	G
46	501.818	G	↑G, ↓4A♭	4/3
47	512.727	↑G, ↓A♭♭	↑↑G, ↓3A♭	35/26, 66/49
48	523.636	↑↑G, A♭♭	↑3G, ↓↓A♭	65/48
49	534.545	↑3G, ↓7A♭	↑4G, ↓A♭
50	545.455	↑4G, ↓6A♭	↑5G, A♭	48/35
51	556.364	↑5G, ↓5A♭	↑6G, ↓14A
52	567.273	↑6G, ↓4A♭	↑7G, ↓13A	25/18
53	578.182	↑7G, ↓3A♭	↑8G, ↓12A
54	589.091	G♯, ↓↓A♭	↑9G, ↓11A
55	600	↑G♯, ↓A♭	↑10G, ↓10A
56	610.909	↑↑G♯, A♭	↑11G, ↓9A
57	621.818	↑3G♯, ↓7A	↑12G, ↓8A
58	632.727	↑4G♯, ↓6A	↑13G, ↓7A	36/25
59	643.636	↑5G♯, ↓5A	↑14G, ↓6A
60	654.545	↑6G♯, ↓4A	G♯, ↓5A	35/24
61	665.455	↑7G♯, ↓3A	↑G♯, ↓4A
62	676.364	G𝄪, ↓↓A	↑↑G♯, ↓3A
63	687.273	↑G𝄪, ↓A	↑3G♯, ↓↓A	49/33, 52/35
64	698.182	A	↑4G♯, ↓A	3/2
65	709.091	↑A, ↓B♭♭	A
66	720	↑↑A, B♭♭	↑A, ↓4B♭
67	730.909	↑3A, ↓7B♭	↑↑A, ↓3B♭	32/21
68	741.818	↑4A, ↓6B♭	↑3A, ↓↓B♭	20/13, 49/32
69	752.727	↑5A, ↓5B♭	↑4A, ↓B♭	65/42
70	763.636	↑6A, ↓4B♭	↑5A, B♭
71	774.545	↑7A, ↓3B♭	↑6A, ↓14B
72	785.455	A♯, ↓↓B♭	↑7A, ↓13B	11/7, 52/33, 63/40
73	796.364	↑A♯, ↓B♭	↑8A, ↓12B
74	807.273	↑↑A♯, B♭	↑9A, ↓11B
75	818.182	↑3A♯, ↓7B	↑10A, ↓10B
76	829.091	↑4A♯, ↓6B	↑11A, ↓9B	21/13
77	840	↑5A♯, ↓5B	↑12A, ↓8B	13/8
78	850.909	↑6A♯, ↓4B	↑13A, ↓7B
79	861.818	↑7A♯, ↓3B	↑14A, ↓6B	64/39
80	872.727	A𝄪, ↓↓B	A♯, ↓5B
81	883.636	↑A𝄪, ↓B	↑A♯, ↓4B	5/3
82	894.545	B	↑↑A♯, ↓3B
83	905.455	↑B, ↓C♭	↑3A♯, ↓↓B
84	916.364	↑↑B, C♭	↑4A♯, ↓B	56/33
85	927.273	↑3B, ↓7C	B
86	938.182	↑4B, ↓6C	↑B, ↓4C	55/32
87	949.091	↑5B, ↓5C	↑↑B, ↓3C
88	960	↑6B, ↓4C	↑3B, ↓↓C
89	970.909	↑7B, ↓3C	↑4B, ↓C	7/4
90	981.818	B♯, ↓↓C	C
91	992.727	↑B♯, ↓C	↑C, ↓4D♭
92	1003.64	C	↑↑C, ↓3D♭
93	1014.55	↑C, ↓D♭♭	↑3C, ↓↓D♭	9/5, 70/39
94	1025.45	↑↑C, D♭♭	↑4C, ↓D♭	65/36
95	1036.36	↑3C, ↓7D♭	↑5C, D♭
96	1047.27	↑4C, ↓6D♭	↑6C, ↓14D	64/35
97	1058.18	↑5C, ↓5D♭	↑7C, ↓13D	24/13
98	1069.09	↑6C, ↓4D♭	↑8C, ↓12D	13/7, 50/27
99	1080	↑7C, ↓3D♭	↑9C, ↓11D
100	1090.91	C♯, ↓↓D♭	↑10C, ↓10D
101	1101.82	↑C♯, ↓D♭	↑11C, ↓9D	66/35
102	1112.73	↑↑C♯, D♭	↑12C, ↓8D	40/21
103	1123.64	↑3C♯, ↓7D	↑13C, ↓7D
104	1134.55	↑4C♯, ↓6D	↑14C, ↓6D
105	1145.45	↑5C♯, ↓5D	C♯, ↓5D	64/33
106	1156.36	↑6C♯, ↓4D	↑C♯, ↓4D	39/20
107	1167.27	↑7C♯, ↓3D	↑↑C♯, ↓3D	55/28
108	1178.18	C𝄪, ↓↓D	↑3C♯, ↓↓D
109	1189.09	↑C𝄪, ↓D	↑4C♯, ↓D
110	1200	D	D	2/1

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