59edo

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← 58edo59edo60edo →
Prime factorization 59 (prime)
Step size 20.339¢ 
Fifth 35\59 (711.864¢)
Semitones (A1:m2) 9:2 (183.1¢ : 40.68¢)
Dual sharp fifth 35\59 (711.864¢)
Dual flat fifth 34\59 (691.525¢)
Dual major 2nd 10\59 (203.39¢)
Consistency limit 7
Distinct consistency limit 7

59 equal divisions of the octave (abbreviated 59edo or 59ed2), also called 59-tone equal temperament (59tet) or 59 equal temperament (59et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each. Each step represents a frequency ratio of 21/59, or the 59th root of 2.

Theory

59edo's best fifth is stretched about 9.91 cents from the just interval, and yet its major third is nearly pure (stretched only 0.127 cents), as the denominator of a convergent to log25. It is a good porcupine tuning, giving in fact the optimal patent val for 11-limit porcupine. This patent val tempers out 250/243 in the 5-limit, 64/63 and 16875/16807 in the 7-limit, and 55/54, 100/99 and 176/175 in the 11-limit. As every other step of 118edo, 59edo is an excellent tuning for the 2.9.5.21.11 11-limit 2*59 subgroup, on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the 50&59 temperament with a subminor third generator provides an interesting temperament.

Using the flat fifth instead of the sharp one allows for the 12&35 temperament, which is a kind of bizarre cousin to garibaldi temperament with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for flattone temperament in the 59bc val, a variant of meantone with flat fifths.

59edo is the 17th prime edo.


Approximation of odd harmonics in 59edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27
Error Absolute (¢) +9.91 +0.13 +7.45 -0.52 -2.17 -6.63 +10.04 -3.26 +7.57 -2.98 +2.23 +0.25 +9.39
Relative (%) +48.7 +0.6 +36.6 -2.6 -10.6 -32.6 +49.3 -16.0 +37.2 -14.7 +11.0 +1.2 +46.2
Steps
(reduced)
94
(35)
137
(19)
166
(48)
187
(10)
204
(27)
218
(41)
231
(54)
241
(5)
251
(15)
259
(23)
267
(31)
274
(38)
281
(45)

Intervals

Steps Cents Approximate Ratios Ups and Downs Notation
(Dual Flat Fifth 34\59)
Ups and Downs Notation
(Dual Sharp Fifth 35\59)
0 0 1/1 D D
1 20.339 65/64, 78/77 ^D, E♭♭♭♭ ^D, vE♭
2 40.678 40/39, 49/48, 77/75 D♯, vE♭♭♭ ^^D, E♭
3 61.017 26/25, 33/32, 80/77 ^D♯, E♭♭♭ ^3D, v8E
4 81.356 D𝄪, vE♭♭ ^4D, v7E
5 101.695 35/33, 55/52 ^D𝄪, E♭♭ ^5D, v6E
6 122.034 15/14 D♯𝄪, vE♭ ^6D, v5E
7 142.373 49/45 ^D♯𝄪, E♭ ^7D, v4E
8 162.712 11/10, 35/32 D𝄪𝄪, vE ^8D, v3E
9 183.051 39/35 E D♯, vvE
10 203.39 28/25, 44/39 ^E, F♭♭♭ ^D♯, vE
11 223.729 8/7, 25/22 E♯, vF♭♭ E
12 244.068 ^E♯, F♭♭ ^E, vF
13 264.407 7/6, 64/55 E𝄪, vF♭ F
14 284.746 13/11, 33/28 ^E𝄪, F♭ ^F, vG♭
15 305.085 E♯𝄪, vF ^^F, G♭
16 325.424 40/33, 77/64 F ^3F, v8G
17 345.763 39/32, 60/49 ^F, G♭♭♭♭ ^4F, v7G
18 366.102 16/13 F♯, vG♭♭♭ ^5F, v6G
19 386.441 5/4 ^F♯, G♭♭♭ ^6F, v5G
20 406.78 F𝄪, vG♭♭ ^7F, v4G
21 427.119 32/25, 50/39, 77/60 ^F𝄪, G♭♭ ^8F, v3G
22 447.458 13/10 F♯𝄪, vG♭ F♯, vvG
23 467.797 ^F♯𝄪, G♭ ^F♯, vG
24 488.136 33/25 F𝄪𝄪, vG G
25 508.475 66/49, 75/56 G ^G, vA♭
26 528.814 49/36 ^G, A♭♭♭♭ ^^G, A♭
27 549.153 11/8, 48/35 G♯, vA♭♭♭ ^3G, v8A
28 569.492 39/28 ^G♯, A♭♭♭ ^4G, v7A
29 589.831 7/5, 55/39 G𝄪, vA♭♭ ^5G, v6A
30 610.169 10/7, 78/55 ^G𝄪, A♭♭ ^6G, v5A
31 630.508 56/39 G♯𝄪, vA♭ ^7G, v4A
32 650.847 16/11, 35/24 ^G♯𝄪, A♭ ^8G, v3A
33 671.186 65/44, 72/49 G𝄪𝄪, vA G♯, vvA
34 691.525 49/33 A ^G♯, vA
35 711.864 50/33 ^A, B♭♭♭♭ A
36 732.203 75/49 A♯, vB♭♭♭ ^A, vB♭
37 752.542 20/13, 77/50 ^A♯, B♭♭♭ ^^A, B♭
38 772.881 25/16, 39/25 A𝄪, vB♭♭ ^3A, v8B
39 793.22 ^A𝄪, B♭♭ ^4A, v7B
40 813.559 8/5, 77/48 A♯𝄪, vB♭ ^5A, v6B
41 833.898 13/8 ^A♯𝄪, B♭ ^6A, v5B
42 854.237 49/30, 64/39 A𝄪𝄪, vB ^7A, v4B
43 874.576 33/20 B ^8A, v3B
44 894.915 ^B, C♭♭♭ A♯, vvB
45 915.254 22/13, 56/33 B♯, vC♭♭ ^A♯, vB
46 935.593 12/7, 55/32 ^B♯, C♭♭ B
47 955.932 B𝄪, vC♭ ^B, vC
48 976.271 7/4, 44/25 ^B𝄪, C♭ C
49 996.61 25/14, 39/22 B♯𝄪, vC ^C, vD♭
50 1016.949 70/39 C ^^C, D♭
51 1037.288 20/11, 64/35 ^C, D♭♭♭♭ ^3C, v8D
52 1057.627 C♯, vD♭♭♭ ^4C, v7D
53 1077.966 28/15 ^C♯, D♭♭♭ ^5C, v6D
54 1098.305 66/35 C𝄪, vD♭♭ ^6C, v5D
55 1118.644 ^C𝄪, D♭♭ ^7C, v4D
56 1138.983 25/13, 64/33, 77/40 C♯𝄪, vD♭ ^8C, v3D
57 1159.322 39/20 ^C♯𝄪, D♭ C♯, vvD
58 1179.661 77/39 C𝄪𝄪, vD ^C♯, vD
59 1200 2/1 D D

Instruments

Lumatone

See Lumatone mapping for 59edo

Music

Francium
Ray Perlner