89edo

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89edo

90edo →

Prime factorization	89 (prime)
Step size	13.4831¢
Fifth	52\89 (701.124¢)
Semitones (A1:m2)	8:7 (107.9¢ : 94.38¢)
Consistency limit	11
Distinct consistency limit	11

89 equal divisions of the octave (abbreviated 89edo or 89ed2), also called 89-tone equal temperament (89tet) or 89 equal temperament (89et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 89 equal parts of about 13.5 ¢ each. Each step represents a frequency ratio of 2^1/89, or the 89th root of 2.

Theory

89edo has a harmonic 3 less than a cent flat and a harmonic 5 less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out quartisma.

As an equal temperament, it tempers out 32805/32768 (schisma) in the 5-limit; 126/125, 1728/1715, and 2401/2400 in the 7-limit; and 176/175, 243/242, 441/440 and 540/539 in the 11-limit. It is an especially good tuning for the myna temperament, both in the 7-limit, tempering out 126/125 and 1728/1715, and in the 11-limit, where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament thrush, tempering out 126/125 and 176/175.

The 13-limit is a little tricky as 13 is tuned distinctly flat, tempering out 169/168, 364/363, 729/728, 832/825, and 1287/1280. 13/10 and 15/13 are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes 13/8 much sharper, conflating it with 18/11. It tempers out 144/143, 196/195, 351/350, and 352/351 instead, and supports 13-limit myna and thrush. However 58edo is a better tuning for those purposes.

The 17 and 19 are tuned fairly well, making it consistent to the no-13 21-odd-limit. The equal temperament tempers out 256/255 and 561/560 in the 17-limit; and 171/170, 361/360, 513/512, and 1216/1215 in the 19-limit.

89edo is the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic φ (i.e. (φ - 1)×1200 cents) within a fraction of a cent.

Prime harmonics

Approximation of prime harmonics in 89edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.83	+4.70	+1.96	+1.49	-4.57	+2.91	-0.88	+5.43	-4.86	+1.03
	Relative (%)	+0.0	-6.2	+34.8	+14.5	+11.1	-33.9	+21.6	-6.6	+40.3	-36.0	+7.7
Steps (reduced)		89 (0)	141 (52)	207 (29)	250 (72)	308 (41)	329 (62)	364 (8)	378 (22)	403 (47)	432 (76)	441 (85)

Subsets and supersets

89edo is the 24th prime edo, following 83edo and before 97edo.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	13.5		^D, ^^E♭♭
2	27		^^D, ^3E♭♭
3	40.4	41/40, 42/41	^3D, v4E♭
4	53.9	31/30, 32/31, 33/32, 34/33	^4D, v3E♭
5	67.4	27/26	v3D♯, vvE♭
6	80.9	22/21	vvD♯, vE♭
7	94.4	19/18, 37/35	vD♯, E♭
8	107.9	33/31	D♯, ^E♭
9	121.3	15/14, 29/27	^D♯, ^^E♭
10	134.8	40/37	^^D♯, ^3E♭
11	148.3	12/11, 37/34	^3D♯, v4E
12	161.8	34/31	^4D♯, v3E
13	175.3	21/19, 31/28, 41/37	v3D𝄪, vvE
14	188.8	29/26	vvD𝄪, vE
15	202.2	9/8	E
16	215.7	17/15	^E, ^^F♭
17	229.2	8/7	^^E, ^3F♭
18	242.7	23/20, 38/33	^3E, v4F
19	256.2	22/19, 36/31	^4E, v3F
20	269.7	7/6	v3E♯, vvF
21	283.1	20/17, 33/28	vvE♯, vF
22	296.6	19/16, 32/27	F
23	310.1		^F, ^^G♭♭
24	323.6	41/34	^^F, ^3G♭♭
25	337.1	17/14	^3F, v4G♭
26	350.6	38/31	^4F, v3G♭
27	364	21/17, 37/30	v3F♯, vvG♭
28	377.5	41/33	vvF♯, vG♭
29	391		vF♯, G♭
30	404.5	24/19	F♯, ^G♭
31	418	14/11	^F♯, ^^G♭
32	431.5	41/32	^^F♯, ^3G♭
33	444.9	22/17, 31/24	^3F♯, v4G
34	458.4	30/23	^4F♯, v3G
35	471.9	21/16	v3F𝄪, vvG
36	485.4	41/31	vvF𝄪, vG
37	498.9	4/3	G
38	512.4	39/29	^G, ^^A♭♭
39	525.8	23/17, 42/31	^^G, ^3A♭♭
40	539.3	15/11, 41/30	^3G, v4A♭
41	552.8	11/8	^4G, v3A♭
42	566.3		v3G♯, vvA♭
43	579.8	7/5	vvG♯, vA♭
44	593.3	31/22, 38/27	vG♯, A♭
45	606.7	27/19	G♯, ^A♭
46	620.2	10/7	^G♯, ^^A♭
47	633.7		^^G♯, ^3A♭
48	647.2	16/11	^3G♯, v4A
49	660.7	22/15, 41/28	^4G♯, v3A
50	674.2	31/21, 34/23	v3G𝄪, vvA
51	687.6		vvG𝄪, vA
52	701.1	3/2	A
53	714.6		^A, ^^B♭♭
54	728.1	32/21, 35/23	^^A, ^3B♭♭
55	741.6	23/15	^3A, v4B♭
56	755.1	17/11	^4A, v3B♭
57	768.5		v3A♯, vvB♭
58	782	11/7	vvA♯, vB♭
59	795.5	19/12	vA♯, B♭
60	809		A♯, ^B♭
61	822.5	37/23	^A♯, ^^B♭
62	836	34/21	^^A♯, ^3B♭
63	849.4	31/19	^3A♯, v4B
64	862.9	28/17	^4A♯, v3B
65	876.4		v3A𝄪, vvB
66	889.9		vvA𝄪, vB
67	903.4	27/16, 32/19	B
68	916.9	17/10	^B, ^^C♭
69	930.3	12/7	^^B, ^3C♭
70	943.8	19/11, 31/18	^3B, v4C
71	957.3	33/19, 40/23	^4B, v3C
72	970.8	7/4	v3B♯, vvC
73	984.3	30/17	vvB♯, vC
74	997.8	16/9	C
75	1011.2		^C, ^^D♭♭
76	1024.7	38/21	^^C, ^3D♭♭
77	1038.2	31/17	^3C, v4D♭
78	1051.7	11/6	^4C, v3D♭
79	1065.2	37/20	v3C♯, vvD♭
80	1078.7	28/15, 41/22	vvC♯, vD♭
81	1092.1		vC♯, D♭
82	1105.6	36/19	C♯, ^D♭
83	1119.1	21/11	^C♯, ^^D♭
84	1132.6		^^C♯, ^3D♭
85	1146.1	31/16, 33/17	^3C♯, v4D
86	1159.6	41/21	^4C♯, v3D
87	1173		v3C𝄪, vvD
88	1186.5		vvC𝄪, vD
89	1200	2/1	D

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-141 89⟩	[⟨89 141]]	+0.262	0.262	1.95
2.3.5	32805/32768, 10077696/9765625	[⟨89 141 207]]	−0.500	1.098	8.15
2.3.5.7	126/125, 1728/1715, 32805/32768	[⟨89 141 207 250]]	−0.550	0.955	7.08
2.3.5.7.11	126/125, 176/175, 243/242, 16384/16335	[⟨89 141 207 250 308]]	−0.526	0.855	6.35

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	13\89	175.28	72/65	Sesquiquartififths / sesquart
1	21\89	283.15	13/11	Neominor
1	23\89	310.11	6/5	Myna
1	29\89	391.01	5/4	Amigo
1	37\89	498.87	4/3	Grackle

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

• Myna7
• Myna11
• Myna15

Music

Francium

• Singing Golden Myna (2022) – myna[11] in 89edo