112edo

← 111edo

112edo

113edo →

Prime factorization	24 × 7
Step size	10.7143¢
Fifth	66\112 (707.143¢) (→33\56)
Semitones (A1:m2)	14:6 (150¢ : 64.29¢)
Dual sharp fifth	66\112 (707.143¢) (→33\56)
Dual flat fifth	65\112 (696.429¢)
Dual major 2nd	19\112 (203.571¢)
Consistency limit	3
Distinct consistency limit	3

112 equal divisions of the octave (abbreviated 112edo or 112ed2), also called 112-tone equal temperament (112tet) or 112 equal temperament (112et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 112 equal parts of about 10.7 ¢ each. Each step represents a frequency ratio of 2^1/112, or the 112th root of 2.

Theory

112edo has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the patent fifth- is identical to the perfect fifth of 56edo, a great inverse gentle fifth where +5 fifths gives a near-just 28:27 while -8 fifths gives a near-just 32:39 (identical to 2 degrees of 7edo) and +9 fifths gives a close approximation to 17:21.

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from 17edo, but sharing a similar structure.

Odd harmonics

Approximation of odd harmonics in 112edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	+5.19	-0.60	-4.54	-0.34	-4.89	-4.81	+4.59	+2.19	+2.49	+0.65	+3.87
	relative (%)	+48	-6	-42	-3	-46	-45	+43	+20	+23	+6	+36
Steps (reduced)		178 (66)	260 (36)	314 (90)	355 (19)	387 (51)	414 (78)	438 (102)	458 (10)	476 (28)	492 (44)	507 (59)

Intervals

Steps	Cents	Ups and downs notation (dual flat fifth 65\112)	Ups and downs notation (dual sharp fifth 66\112)	Approximate ratios
0	0	D	D	1/1
1	10.7143	↑D, ↓3E♭♭	↑D, ↓5E♭
2	21.4286	↑↑D, ↓↓E♭♭	↑↑D, ↓4E♭
3	32.1429	↑3D, ↓E♭♭	↑3D, ↓3E♭	56/55
4	42.8571	↑4D, E♭♭	↑4D, ↓↓E♭	40/39
5	53.5714	↑5D, ↓6E♭	↑5D, ↓E♭	33/32
6	64.2857	↑6D, ↓5E♭	↑6D, E♭	26/25
7	75	D♯, ↓4E♭	↑7D, ↓13E
8	85.7143	↑D♯, ↓3E♭	↑8D, ↓12E	21/20
9	96.4286	↑↑D♯, ↓↓E♭	↑9D, ↓11E	55/52
10	107.143	↑3D♯, ↓E♭	↑10D, ↓10E	52/49
11	117.857	↑4D♯, E♭	↑11D, ↓9E
12	128.571	↑5D♯, ↓6E	↑12D, ↓8E	14/13
13	139.286	↑6D♯, ↓5E	↑13D, ↓7E
14	150	D𝄪, ↓4E	D♯, ↓6E
15	160.714	↑D𝄪, ↓3E	↑D♯, ↓5E
16	171.429	↑↑D𝄪, ↓↓E	↑↑D♯, ↓4E
17	182.143	↑3D𝄪, ↓E	↑3D♯, ↓3E	49/44
18	192.857	E	↑4D♯, ↓↓E	28/25
19	203.571	↑E, ↓3F♭	↑5D♯, ↓E	55/49
20	214.286	↑↑E, ↓↓F♭	E
21	225	↑3E, ↓F♭	↑E, ↓5F	25/22
22	235.714	↑4E, F♭	↑↑E, ↓4F
23	246.429	↑5E, ↓6F	↑3E, ↓3F
24	257.143	↑6E, ↓5F	↑4E, ↓↓F	65/56
25	267.857	E♯, ↓4F	↑5E, ↓F
26	278.571	↑E♯, ↓3F	F	75/64
27	289.286	↑↑E♯, ↓↓F	↑F, ↓5G♭	13/11, 77/65
28	300	↑3E♯, ↓F	↑↑F, ↓4G♭	25/21
29	310.714	F	↑3F, ↓3G♭
30	321.429	↑F, ↓3G♭♭	↑4F, ↓↓G♭
31	332.143	↑↑F, ↓↓G♭♭	↑5F, ↓G♭	40/33
32	342.857	↑3F, ↓G♭♭	↑6F, G♭	39/32
33	353.571	↑4F, G♭♭	↑7F, ↓13G
34	364.286	↑5F, ↓6G♭	↑8F, ↓12G
35	375	↑6F, ↓5G♭	↑9F, ↓11G
36	385.714	F♯, ↓4G♭	↑10F, ↓10G	5/4
37	396.429	↑F♯, ↓3G♭	↑11F, ↓9G	44/35
38	407.143	↑↑F♯, ↓↓G♭	↑12F, ↓8G
39	417.857	↑3F♯, ↓G♭	↑13F, ↓7G	14/11
40	428.571	↑4F♯, G♭	F♯, ↓6G	32/25, 50/39
41	439.286	↑5F♯, ↓6G	↑F♯, ↓5G
42	450	↑6F♯, ↓5G	↑↑F♯, ↓4G	13/10
43	460.714	F𝄪, ↓4G	↑3F♯, ↓3G
44	471.429	↑F𝄪, ↓3G	↑4F♯, ↓↓G	21/16
45	482.143	↑↑F𝄪, ↓↓G	↑5F♯, ↓G	33/25
46	492.857	↑3F𝄪, ↓G	G	65/49
47	503.571	G	↑G, ↓5A♭
48	514.286	↑G, ↓3A♭♭	↑↑G, ↓4A♭	35/26
49	525	↑↑G, ↓↓A♭♭	↑3G, ↓3A♭
50	535.714	↑3G, ↓A♭♭	↑4G, ↓↓A♭
51	546.429	↑4G, A♭♭	↑5G, ↓A♭
52	557.143	↑5G, ↓6A♭	↑6G, A♭
53	567.857	↑6G, ↓5A♭	↑7G, ↓13A
54	578.571	G♯, ↓4A♭	↑8G, ↓12A	7/5
55	589.286	↑G♯, ↓3A♭	↑9G, ↓11A
56	600	↑↑G♯, ↓↓A♭	↑10G, ↓10A
57	610.714	↑3G♯, ↓A♭	↑11G, ↓9A
58	621.429	↑4G♯, A♭	↑12G, ↓8A	10/7
59	632.143	↑5G♯, ↓6A	↑13G, ↓7A
60	642.857	↑6G♯, ↓5A	G♯, ↓6A
61	653.571	G𝄪, ↓4A	↑G♯, ↓5A
62	664.286	↑G𝄪, ↓3A	↑↑G♯, ↓4A
63	675	↑↑G𝄪, ↓↓A	↑3G♯, ↓3A	65/44
64	685.714	↑3G𝄪, ↓A	↑4G♯, ↓↓A	52/35
65	696.429	A	↑5G♯, ↓A
66	707.143	↑A, ↓3B♭♭	A
67	717.857	↑↑A, ↓↓B♭♭	↑A, ↓5B♭	50/33
68	728.571	↑3A, ↓B♭♭	↑↑A, ↓4B♭	32/21
69	739.286	↑4A, B♭♭	↑3A, ↓3B♭
70	750	↑5A, ↓6B♭	↑4A, ↓↓B♭	20/13
71	760.714	↑6A, ↓5B♭	↑5A, ↓B♭
72	771.429	A♯, ↓4B♭	↑6A, B♭	25/16, 39/25
73	782.143	↑A♯, ↓3B♭	↑7A, ↓13B	11/7
74	792.857	↑↑A♯, ↓↓B♭	↑8A, ↓12B
75	803.571	↑3A♯, ↓B♭	↑9A, ↓11B	35/22
76	814.286	↑4A♯, B♭	↑10A, ↓10B	8/5
77	825	↑5A♯, ↓6B	↑11A, ↓9B
78	835.714	↑6A♯, ↓5B	↑12A, ↓8B
79	846.429	A𝄪, ↓4B	↑13A, ↓7B
80	857.143	↑A𝄪, ↓3B	A♯, ↓6B	64/39
81	867.857	↑↑A𝄪, ↓↓B	↑A♯, ↓5B	33/20
82	878.571	↑3A𝄪, ↓B	↑↑A♯, ↓4B
83	889.286	B	↑3A♯, ↓3B
84	900	↑B, ↓3C♭	↑4A♯, ↓↓B	42/25
85	910.714	↑↑B, ↓↓C♭	↑5A♯, ↓B	22/13
86	921.429	↑3B, ↓C♭	B
87	932.143	↑4B, C♭	↑B, ↓5C
88	942.857	↑5B, ↓6C	↑↑B, ↓4C
89	953.571	↑6B, ↓5C	↑3B, ↓3C
90	964.286	B♯, ↓4C	↑4B, ↓↓C
91	975	↑B♯, ↓3C	↑5B, ↓C	44/25
92	985.714	↑↑B♯, ↓↓C	C
93	996.429	↑3B♯, ↓C	↑C, ↓5D♭
94	1007.14	C	↑↑C, ↓4D♭	25/14
95	1017.86	↑C, ↓3D♭♭	↑3C, ↓3D♭
96	1028.57	↑↑C, ↓↓D♭♭	↑4C, ↓↓D♭
97	1039.29	↑3C, ↓D♭♭	↑5C, ↓D♭
98	1050	↑4C, D♭♭	↑6C, D♭
99	1060.71	↑5C, ↓6D♭	↑7C, ↓13D
100	1071.43	↑6C, ↓5D♭	↑8C, ↓12D	13/7
101	1082.14	C♯, ↓4D♭	↑9C, ↓11D
102	1092.86	↑C♯, ↓3D♭	↑10C, ↓10D	49/26
103	1103.57	↑↑C♯, ↓↓D♭	↑11C, ↓9D
104	1114.29	↑3C♯, ↓D♭	↑12C, ↓8D	40/21
105	1125	↑4C♯, D♭	↑13C, ↓7D
106	1135.71	↑5C♯, ↓6D	C♯, ↓6D	25/13
107	1146.43	↑6C♯, ↓5D	↑C♯, ↓5D	64/33
108	1157.14	C𝄪, ↓4D	↑↑C♯, ↓4D	39/20
109	1167.86	↑C𝄪, ↓3D	↑3C♯, ↓3D	55/28
110	1178.57	↑↑C𝄪, ↓↓D	↑4C♯, ↓↓D
111	1189.29	↑3C𝄪, ↓D	↑5C♯, ↓D
112	1200	D	D	2/1

Music

• Circulating 2.3.7.11.13 Floaty Piano Improv by Cam Taylor