89edo

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← 88edo89edo90edo →
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 21/89, 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.

89et tempers out the commas 126/125, 1728/1715, 32805/32768, 2401/2400, 176/175, 243/242, 441/440 and 540/539. 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.

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 -6 +35 +15 +11 -34 +22 -7 +40 -36 +8
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, and 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.

Interval table

Steps Cents Ups and downs notation Approximate ratios
0 0 D 1/1
1 13.4831 ↑D, ↓6E♭
2 26.9663 ↑↑D, ↓5E♭ 56/55, 64/63, 65/64, 66/65
3 40.4494 ↑3D, ↓4E♭ 45/44, 49/48, 77/75
4 53.9326 ↑4D, ↓3E♭ 33/32, 65/63
5 67.4157 ↑5D, ↓↓E♭ 27/26, 28/27, 80/77
6 80.8989 ↑6D, ↓E♭ 21/20, 22/21
7 94.382 ↑7D, E♭
8 107.865 D♯, ↓7E 16/15
9 121.348 ↑D♯, ↓6E 15/14, 77/72
10 134.831 ↑↑D♯, ↓5E 13/12
11 148.315 ↑3D♯, ↓4E 12/11, 49/45
12 161.798 ↑4D♯, ↓3E 11/10
13 175.281 ↑5D♯, ↓↓E 72/65
14 188.764 ↑6D♯, ↓E 49/44
15 202.247 E 9/8, 55/49
16 215.73 ↑E, ↓6F
17 229.213 ↑↑E, ↓5F 8/7
18 242.697 ↑3E, ↓4F
19 256.18 ↑4E, ↓3F 65/56
20 269.663 ↑5E, ↓↓F 7/6
21 283.146 ↑6E, ↓F 33/28
22 296.629 F 32/27, 77/65
23 310.112 ↑F, ↓6G♭
24 323.596 ↑↑F, ↓5G♭ 65/54, 77/64
25 337.079 ↑3F, ↓4G♭ 40/33, 63/52
26 350.562 ↑4F, ↓3G♭ 11/9, 27/22, 49/40, 60/49
27 364.045 ↑5F, ↓↓G♭ 16/13
28 377.528 ↑6F, ↓G♭ 56/45, 81/65
29 391.011 ↑7F, G♭ 5/4, 44/35
30 404.494 F♯, ↓7G 81/64
31 417.978 ↑F♯, ↓6G 14/11, 33/26, 80/63
32 431.461 ↑↑F♯, ↓5G 9/7, 77/60
33 444.944 ↑3F♯, ↓4G
34 458.427 ↑4F♯, ↓3G 64/49
35 471.91 ↑5F♯, ↓↓G 21/16, 55/42
36 485.393 ↑6F♯, ↓G 65/49
37 498.876 G 4/3
38 512.36 ↑G, ↓6A♭ 66/49
39 525.843 ↑↑G, ↓5A♭ 65/48
40 539.326 ↑3G, ↓4A♭ 15/11
41 552.809 ↑4G, ↓3A♭ 11/8
42 566.292 ↑5G, ↓↓A♭ 18/13
43 579.775 ↑6G, ↓A♭ 7/5
44 593.258 ↑7G, A♭ 45/32
45 606.742 G♯, ↓7A 64/45
46 620.225 ↑G♯, ↓6A 10/7, 63/44
47 633.708 ↑↑G♯, ↓5A 13/9, 81/56
48 647.191 ↑3G♯, ↓4A 16/11
49 660.674 ↑4G♯, ↓3A 22/15
50 674.157 ↑5G♯, ↓↓A 65/44
51 687.64 ↑6G♯, ↓A 49/33
52 701.124 A 3/2
53 714.607 ↑A, ↓6B♭
54 728.09 ↑↑A, ↓5B♭ 32/21
55 741.573 ↑3A, ↓4B♭ 49/32, 75/49
56 755.056 ↑4A, ↓3B♭ 65/42
57 768.539 ↑5A, ↓↓B♭ 14/9, 81/52
58 782.022 ↑6A, ↓B♭ 11/7, 52/33, 63/40
59 795.506 ↑7A, B♭
60 808.989 A♯, ↓7B 8/5, 35/22
61 822.472 ↑A♯, ↓6B 45/28, 77/48
62 835.955 ↑↑A♯, ↓5B 13/8
63 849.438 ↑3A♯, ↓4B 18/11, 44/27, 49/30, 80/49
64 862.921 ↑4A♯, ↓3B 33/20
65 876.404 ↑5A♯, ↓↓B
66 889.888 ↑6A♯, ↓B
67 903.371 B 27/16
68 916.854 ↑B, ↓6C 56/33
69 930.337 ↑↑B, ↓5C 12/7, 77/45
70 943.82 ↑3B, ↓4C
71 957.303 ↑4B, ↓3C
72 970.787 ↑5B, ↓↓C 7/4
73 984.27 ↑6B, ↓C
74 997.753 C 16/9
75 1011.24 ↑C, ↓6D♭
76 1024.72 ↑↑C, ↓5D♭ 65/36
77 1038.2 ↑3C, ↓4D♭ 20/11
78 1051.69 ↑4C, ↓3D♭ 11/6, 81/44
79 1065.17 ↑5C, ↓↓D♭ 24/13
80 1078.65 ↑6C, ↓D♭ 28/15
81 1092.13 ↑7C, D♭ 15/8
82 1105.62 C♯, ↓7D
83 1119.1 ↑C♯, ↓6D 21/11, 40/21
84 1132.58 ↑↑C♯, ↓5D 27/14, 52/27, 77/40
85 1146.07 ↑3C♯, ↓4D 64/33
86 1159.55 ↑4C♯, ↓3D
87 1173.03 ↑5C♯, ↓↓D 55/28, 63/32, 65/33
88 1186.52 ↑6C♯, ↓D
89 1200 D 2/1

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 		Temperament
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

Scales

Music

Francium
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