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 (59edo), or 59-tone equal temperament (59tet), 59 equal temperament (59et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 59 equal parts of about 20.3 ¢ each.

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 (%) +49 +1 +37 -3 -11 -33 +49 -16 +37 -15 +11 +1 +46
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 Ups and downs notation
(dual flat fifth 34\59)
Ups and downs notation
(dual sharp fifth 35\59)
Approximate ratios
0 0 D D 1/1, 55/54, 64/63
1 20.339 ^D, Ebbbb ^D, vEb 50/49, 65/63, 65/64, 78/77
2 40.678 D#, vEbbb ^^D, Eb 28/27, 40/39, 49/48, 56/55, 66/65, 77/75
3 61.0169 ^D#, Ebbb ^3D, v8E 22/21, 25/24, 26/25, 33/32, 36/35, 45/44, 80/77, 81/80
4 81.3559 Dx, vEbb ^4D, v7E 52/49
5 101.695 ^Dx, Ebb ^5D, v6E 16/15, 21/20, 27/26, 35/33, 55/52, 77/72
6 122.034 D#x, vEb ^6D, v5E 13/12, 15/14, 81/77
7 142.373 ^D#x, Eb ^7D, v4E 14/13, 49/45
8 162.712 Dxx, vE ^8D, v3E 10/9, 11/10, 12/11, 27/25, 35/32
9 183.051 E D#, vvE 39/35, 54/49, 55/49
10 203.39 ^E, Fbbb ^D#, vE 28/25, 44/39, 49/44, 72/65
11 223.729 E#, vFbb E 8/7, 9/8, 25/22, 52/45, 55/48
12 244.068 ^E#, Fbb ^E, vF 65/56
13 264.407 Ex, vFb F 7/6, 15/13, 32/27, 63/55, 64/55
14 284.746 ^Ex, Fb ^F, vGb 13/11, 25/21, 33/28, 65/54, 75/64, 81/70
15 305.085 E#x, vF ^^F, Gb 77/65
16 325.424 F ^3F, v8G 6/5, 11/9, 40/33, 77/64
17 345.763 ^F, Gbbbb ^4F, v7G 26/21, 39/32, 60/49
18 366.102 F#, vGbbb ^5F, v6G 16/13, 49/40, 56/45, 63/52
19 386.441 ^F#, Gbbb ^6F, v5G 5/4, 27/22, 44/35, 80/63
20 406.78 Fx, vGbb ^7F, v4G 49/39
21 427.119 ^Fx, Gbb ^8F, v3G 14/11, 32/25, 33/26, 35/27, 50/39, 63/50, 77/60
22 447.458 F#x, vGb F#, vvG 9/7, 13/10, 55/42, 64/49, 81/64
23 467.797 ^F#x, Gb ^F#, vG 65/49
24 488.136 Fxx, vG G 4/3, 21/16, 33/25, 72/55
25 508.475 G ^G, vAb 65/48, 66/49, 75/56
26 528.814 ^G, Abbbb ^^G, Ab 35/26, 49/36
27 549.153 G#, vAbbb ^3G, v8A 11/8, 15/11, 25/18, 27/20, 48/35
28 569.492 ^G#, Abbb ^4G, v7A 39/28
29 589.831 Gx, vAbb ^5G, v6A 7/5, 18/13, 55/39, 64/45, 77/54
30 610.169 ^Gx, Abb ^6G, v5A 10/7, 13/9, 45/32, 78/55
31 630.508 G#x, vAb ^7G, v4A 56/39
32 650.847 ^G#x, Ab ^8G, v3A 16/11, 22/15, 35/24, 36/25, 40/27, 63/44, 75/52
33 671.186 Gxx, vA G#, vvA 52/35, 65/44, 72/49, 81/56
34 691.525 A ^G#, vA 49/33, 77/52
35 711.864 ^A, Bbbbb A 3/2, 32/21, 50/33, 55/36
36 732.203 A#, vBbbb ^A, vBb 65/42, 75/49
37 752.542 ^A#, Bbbb ^^A, Bb 14/9, 20/13, 49/32, 77/50
38 772.881 Ax, vBbb ^3A, v8B 11/7, 25/16, 39/25, 52/33, 54/35
39 793.22 ^Ax, Bbb ^4A, v7B 78/49
40 813.559 A#x, vBb ^5A, v6B 8/5, 35/22, 44/27, 63/40, 77/48
41 833.898 ^A#x, Bb ^6A, v5B 13/8, 45/28, 80/49
42 854.237 Axx, vB ^7A, v4B 21/13, 49/30, 64/39
43 874.576 B ^8A, v3B 5/3, 18/11, 33/20, 81/50
44 894.915 ^B, Cbbb A#, vvB 81/49
45 915.254 B#, vCbb ^A#, vB 22/13, 42/25, 56/33, 77/45
46 935.593 ^B#, Cbb B 12/7, 26/15, 27/16, 55/32, 75/44
47 955.932 Bx, vCb ^B, vC
48 976.271 ^Bx, Cb C 7/4, 16/9, 44/25, 45/26
49 996.61 B#x, vC ^C, vDb 25/14, 39/22, 65/36
50 1016.95 C ^^C, Db 49/27, 70/39
51 1037.29 ^C, Dbbbb ^3C, v8D 9/5, 11/6, 20/11, 50/27, 64/35
52 1057.63 C#, vDbbb ^4C, v7D 13/7
53 1077.97 ^C#, Dbbb ^5C, v6D 24/13, 28/15
54 1098.31 Cx, vDbb ^6C, v5D 15/8, 40/21, 52/27, 66/35
55 1118.64 ^Cx, Dbb ^7C, v4D 49/26
56 1138.98 C#x, vDb ^8C, v3D 21/11, 25/13, 35/18, 48/25, 64/33, 77/40
57 1159.32 ^C#x, Db C#, vvD 27/14, 39/20, 55/28, 65/33
58 1179.66 Cxx, vD ^C#, vD 49/25, 77/39
59 1200 D D 2/1, 55/27, 63/32

Francium
Ray Perlner