88edo

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88edo

89edo →

Prime factorization	23 × 11
Step size	13.6364¢
Fifth	51\88 (695.455¢)
Semitones (A1:m2)	5:9 (68.18¢ : 122.7¢)
Dual sharp fifth	52\88 (709.091¢) (→13\22)
Dual flat fifth	51\88 (695.455¢)
Dual major 2nd	15\88 (204.545¢)
Consistency limit	7
Distinct consistency limit	7

88 equal divisions of the octave (abbreviated 88edo or 88ed2), also called 88-tone equal temperament (88tet) or 88 equal temperament (88et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 88 equal parts of about 13.6 ¢ each. Each step represents a frequency ratio of 2^1/88, or the 88th root of 2.

Theory

Using two different approximations to the perfect fifth (one of 51 steps and one of 52 steps), it is compatible with both meantone temperament and the particular variety of superpyth temperament(s) supported by 22 equal temperament, respectively. The meantone fifth is 0.0384 cents flatter than that of Lucy Tuning and, thus, audibly indistinguishable from it. It also gives the optimal patent val for the 11-limit mothra and euterpe temperaments.

Odd harmonics

Approximation of odd harmonics in 88edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	absolute (¢)	-6.50	-4.50	-0.64	+0.64	-5.86	+4.93	+2.64	+4.14	+2.49	+6.49	-1.00
	relative (%)	-48	-33	-5	+5	-43	+36	+19	+30	+18	+48	-7
Steps (reduced)		139 (51)	204 (28)	247 (71)	279 (15)	304 (40)	326 (62)	344 (80)	360 (8)	374 (22)	387 (35)	398 (46)

Interval table

Steps	Cents	Ups and downs notation (dual flat fifth 51\88)	Ups and downs notation (dual sharp fifth 52\88)	Approximate ratios
0	0	D	D	1/1
1	13.6364	↑D, ↓3E♭♭	↑D, ↓3E♭
2	27.2727	↑↑D, ↓↓E♭♭	↑↑D, ↓↓E♭	65/64, 78/77
3	40.9091	↑3D, ↓E♭♭	↑3D, ↓E♭	40/39, 49/48
4	54.5455	↑4D, E♭♭	↑4D, E♭
5	68.1818	D♯, ↓4E♭	↑5D, ↓11E	25/24, 80/77
6	81.8182	↑D♯, ↓3E♭	↑6D, ↓10E	21/20, 22/21
7	95.4545	↑↑D♯, ↓↓E♭	↑7D, ↓9E
8	109.091	↑3D♯, ↓E♭	↑8D, ↓8E
9	122.727	↑4D♯, E♭	↑9D, ↓7E
10	136.364	D𝄪, ↓4E	↑10D, ↓6E
11	150	↑D𝄪, ↓3E	↑11D, ↓5E	12/11, 35/32
12	163.636	↑↑D𝄪, ↓↓E	D♯, ↓4E	11/10
13	177.273	↑3D𝄪, ↓E	↑D♯, ↓3E
14	190.909	E	↑↑D♯, ↓↓E	39/35, 49/44
15	204.545	↑E, ↓3F♭	↑3D♯, ↓E	44/39
16	218.182	↑↑E, ↓↓F♭	E	25/22
17	231.818	↑3E, ↓F♭	↑E, ↓3F	8/7, 55/48, 63/55
18	245.455	↑4E, F♭	↑↑E, ↓↓F
19	259.091	E♯, ↓4F	↑3E, ↓F	65/56
20	272.727	↑E♯, ↓3F	F
21	286.364	↑↑E♯, ↓↓F	↑F, ↓3G♭
22	300	↑3E♯, ↓F	↑↑F, ↓↓G♭	25/21
23	313.636	F	↑3F, ↓G♭	6/5
24	327.273	↑F, ↓3G♭♭	↑4F, G♭
25	340.909	↑↑F, ↓↓G♭♭	↑5F, ↓11G	39/32
26	354.545	↑3F, ↓G♭♭	↑6F, ↓10G	16/13, 49/40
27	368.182	↑4F, G♭♭	↑7F, ↓9G
28	381.818	F♯, ↓4G♭	↑8F, ↓8G	5/4
29	395.455	↑F♯, ↓3G♭	↑9F, ↓7G	44/35, 49/39, 63/50
30	409.091	↑↑F♯, ↓↓G♭	↑10F, ↓6G
31	422.727	↑3F♯, ↓G♭	↑11F, ↓5G	14/11
32	436.364	↑4F♯, G♭	F♯, ↓4G	77/60
33	450	F𝄪, ↓4G	↑F♯, ↓3G
34	463.636	↑F𝄪, ↓3G	↑↑F♯, ↓↓G	55/42, 64/49, 72/55
35	477.273	↑↑F𝄪, ↓↓G	↑3F♯, ↓G	33/25
36	490.909	↑3F𝄪, ↓G	G	65/49
37	504.545	G	↑G, ↓3A♭
38	518.182	↑G, ↓3A♭♭	↑↑G, ↓↓A♭
39	531.818	↑↑G, ↓↓A♭♭	↑3G, ↓A♭	15/11
40	545.455	↑3G, ↓A♭♭	↑4G, A♭	48/35
41	559.091	↑4G, A♭♭	↑5G, ↓11A
42	572.727	G♯, ↓4A♭	↑6G, ↓10A	25/18, 39/28
43	586.364	↑G♯, ↓3A♭	↑7G, ↓9A	7/5
44	600	↑↑G♯, ↓↓A♭	↑8G, ↓8A
45	613.636	↑3G♯, ↓A♭	↑9G, ↓7A	10/7
46	627.273	↑4G♯, A♭	↑10G, ↓6A	36/25, 56/39
47	640.909	G𝄪, ↓4A	↑11G, ↓5A
48	654.545	↑G𝄪, ↓3A	G♯, ↓4A	35/24
49	668.182	↑↑G𝄪, ↓↓A	↑G♯, ↓3A	22/15
50	681.818	↑3G𝄪, ↓A	↑↑G♯, ↓↓A
51	695.455	A	↑3G♯, ↓A
52	709.091	↑A, ↓3B♭♭	A
53	722.727	↑↑A, ↓↓B♭♭	↑A, ↓3B♭	50/33
54	736.364	↑3A, ↓B♭♭	↑↑A, ↓↓B♭	49/32, 55/36
55	750	↑4A, B♭♭	↑3A, ↓B♭	77/50
56	763.636	A♯, ↓4B♭	↑4A, B♭
57	777.273	↑A♯, ↓3B♭	↑5A, ↓11B	11/7
58	790.909	↑↑A♯, ↓↓B♭	↑6A, ↓10B
59	804.545	↑3A♯, ↓B♭	↑7A, ↓9B	35/22, 78/49
60	818.182	↑4A♯, B♭	↑8A, ↓8B	8/5, 77/48
61	831.818	A𝄪, ↓4B	↑9A, ↓7B
62	845.455	↑A𝄪, ↓3B	↑10A, ↓6B	13/8, 80/49
63	859.091	↑↑A𝄪, ↓↓B	↑11A, ↓5B	64/39
64	872.727	↑3A𝄪, ↓B	A♯, ↓4B
65	886.364	B	↑A♯, ↓3B	5/3
66	900	↑B, ↓3C♭	↑↑A♯, ↓↓B	42/25
67	913.636	↑↑B, ↓↓C♭	↑3A♯, ↓B
68	927.273	↑3B, ↓C♭	B
69	940.909	↑4B, C♭	↑B, ↓3C
70	954.545	B♯, ↓4C	↑↑B, ↓↓C
71	968.182	↑B♯, ↓3C	↑3B, ↓C	7/4
72	981.818	↑↑B♯, ↓↓C	C	44/25
73	995.455	↑3B♯, ↓C	↑C, ↓3D♭	39/22
74	1009.09	C	↑↑C, ↓↓D♭	70/39
75	1022.73	↑C, ↓3D♭♭	↑3C, ↓D♭
76	1036.36	↑↑C, ↓↓D♭♭	↑4C, D♭	20/11
77	1050	↑3C, ↓D♭♭	↑5C, ↓11D	11/6, 64/35
78	1063.64	↑4C, D♭♭	↑6C, ↓10D
79	1077.27	C♯, ↓4D♭	↑7C, ↓9D
80	1090.91	↑C♯, ↓3D♭	↑8C, ↓8D
81	1104.55	↑↑C♯, ↓↓D♭	↑9C, ↓7D
82	1118.18	↑3C♯, ↓D♭	↑10C, ↓6D	21/11, 40/21
83	1131.82	↑4C♯, D♭	↑11C, ↓5D	48/25, 77/40
84	1145.45	C𝄪, ↓4D	C♯, ↓4D
85	1159.09	↑C𝄪, ↓3D	↑C♯, ↓3D	39/20
86	1172.73	↑↑C𝄪, ↓↓D	↑↑C♯, ↓↓D	77/39
87	1186.36	↑3C𝄪, ↓D	↑3C♯, ↓D
88	1200	D	D	2/1