# 59edo

 ← 58edo 59edo 60edo →
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

Lumatone

Francium
Ray Perlner