88edo

← 87edo

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

Theory

Using two different approximations to the perfect fifth (one of 51 steps and one of 52 steps), 88edo is compatible with both meantone and the particular variety of superpyth 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 (%)	-47.7	-33.0	-4.7	+4.7	-43.0	+36.1	+19.4	+30.3	+18.2	+47.6	-7.3
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	Approximate ratios	Ups and downs notation (Dual flat fifth 51\88)	Ups and downs notation (Dual sharp fifth 52\88)
0	0	1/1	D	D
1	13.6		^D, ^^E♭♭♭	^D, v3E♭
2	27.3		^^D, vvE♭♭	^^D, vvE♭
3	40.9	41/40, 42/41	vvD♯, vE♭♭	^3D, vE♭
4	54.5	32/31	vD♯, E♭♭	^4D, E♭
5	68.2	25/24	D♯, ^E♭♭	^5D, ^E♭
6	81.8	21/20, 22/21	^D♯, ^^E♭♭	^6D, ^^E♭
7	95.5	37/35	^^D♯, vvE♭	v5D♯, ^3E♭
8	109.1		vvD𝄪, vE♭	v4D♯, ^4E♭
9	122.7		vD𝄪, E♭	v3D♯, ^5E♭
10	136.4	40/37	D𝄪, ^E♭	vvD♯, v6E
11	150	12/11	^D𝄪, ^^E♭	vD♯, v5E
12	163.6	11/10	^^D𝄪, vvE	D♯, v4E
13	177.3	31/28, 41/37	vvD♯𝄪, vE	^D♯, v3E
14	190.9	19/17, 29/26	E	^^D♯, vvE
15	204.5		^E, ^^F♭♭	^3D♯, vE
16	218.2	42/37	^^E, vvF♭	E
17	231.8	8/7	vvE♯, vF♭	^E, v3F
18	245.5		vE♯, F♭	^^E, vvF
19	259.1		E♯, ^F♭	^3E, vF
20	272.7	34/29, 41/35	^E♯, ^^F♭	F
21	286.4		^^E♯, vvF	^F, v3G♭
22	300	19/16, 25/21	vvE𝄪, vF	^^F, vvG♭
23	313.6	6/5	F	^3F, vG♭
24	327.3		^F, ^^G♭♭♭	^4F, G♭
25	340.9	28/23, 39/32	^^F, vvG♭♭	^5F, ^G♭
26	354.5	38/31	vvF♯, vG♭♭	^6F, ^^G♭
27	368.2		vF♯, G♭♭	v5F♯, ^3G♭
28	381.8		F♯, ^G♭♭	v4F♯, ^4G♭
29	395.5	39/31	^F♯, ^^G♭♭	v3F♯, ^5G♭
30	409.1		^^F♯, vvG♭	vvF♯, v6G
31	422.7		vvF𝄪, vG♭	vF♯, v5G
32	436.4		vF𝄪, G♭	F♯, v4G
33	450		F𝄪, ^G♭	^F♯, v3G
34	463.6	17/13	^F𝄪, ^^G♭	^^F♯, vvG
35	477.3		^^F𝄪, vvG	^3F♯, vG
36	490.9		vvF♯𝄪, vG	G
37	504.5		G	^G, v3A♭
38	518.2	31/23	^G, ^^A♭♭♭	^^G, vvA♭
39	531.8		^^G, vvA♭♭	^3G, vA♭
40	545.5	26/19	vvG♯, vA♭♭	^4G, A♭
41	559.1		vG♯, A♭♭	^5G, ^A♭
42	572.7	32/23, 39/28	G♯, ^A♭♭	^6G, ^^A♭
43	586.4		^G♯, ^^A♭♭	v5G♯, ^3A♭
44	600		^^G♯, vvA♭	v4G♯, ^4A♭
45	613.6		vvG𝄪, vA♭	v3G♯, ^5A♭
46	627.3	23/16	vG𝄪, A♭	vvG♯, v6A
47	640.9		G𝄪, ^A♭	vG♯, v5A
48	654.5	19/13, 35/24	^G𝄪, ^^A♭	G♯, v4A
49	668.2		^^G𝄪, vvA	^G♯, v3A
50	681.8		vvG♯𝄪, vA	^^G♯, vvA
51	695.5		A	^3G♯, vA
52	709.1		^A, ^^B♭♭♭	A
53	722.7		^^A, vvB♭♭	^A, v3B♭
54	736.4	26/17	vvA♯, vB♭♭	^^A, vvB♭
55	750	37/24	vA♯, B♭♭	^3A, vB♭
56	763.6		A♯, ^B♭♭	^4A, B♭
57	777.3		^A♯, ^^B♭♭	^5A, ^B♭
58	790.9		^^A♯, vvB♭	^6A, ^^B♭
59	804.5	35/22	vvA𝄪, vB♭	v5A♯, ^3B♭
60	818.2		vA𝄪, B♭	v4A♯, ^4B♭
61	831.8		A𝄪, ^B♭	v3A♯, ^5B♭
62	845.5	31/19	^A𝄪, ^^B♭	vvA♯, v6B
63	859.1	23/14, 41/25	^^A𝄪, vvB	vA♯, v5B
64	872.7		vvA♯𝄪, vB	A♯, v4B
65	886.4	5/3	B	^A♯, v3B
66	900	32/19, 37/22, 42/25	^B, ^^C♭♭	^^A♯, vvB
67	913.6	39/23	^^B, vvC♭	^3A♯, vB
68	927.3	29/17, 41/24	vvB♯, vC♭	B
69	940.9		vB♯, C♭	^B, v3C
70	954.5		B♯, ^C♭	^^B, vvC
71	968.2	7/4	^B♯, ^^C♭	^3B, vC
72	981.8	37/21	^^B♯, vvC	C
73	995.5		vvB𝄪, vC	^C, v3D♭
74	1009.1	34/19	C	^^C, vvD♭
75	1022.7		^C, ^^D♭♭♭	^3C, vD♭
76	1036.4	20/11	^^C, vvD♭♭	^4C, D♭
77	1050	11/6	vvC♯, vD♭♭	^5C, ^D♭
78	1063.6	37/20	vC♯, D♭♭	^6C, ^^D♭
79	1077.3	41/22	C♯, ^D♭♭	v5C♯, ^3D♭
80	1090.9		^C♯, ^^D♭♭	v4C♯, ^4D♭
81	1104.5		^^C♯, vvD♭	v3C♯, ^5D♭
82	1118.2	21/11, 40/21	vvC𝄪, vD♭	vvC♯, v6D
83	1131.8		vC𝄪, D♭	vC♯, v5D
84	1145.5	31/16	C𝄪, ^D♭	C♯, v4D
85	1159.1	41/21	^C𝄪, ^^D♭	^C♯, v3D
86	1172.7		^^C𝄪, vvD	^^C♯, vvD
87	1186.4		vvC♯𝄪, vD	^3C♯, vD
88	1200	2/1	D	D