59edo

The 59 equal division divides the octave into 59 equal steps of 20.339 cents each.

← 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

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.

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

Degrees	Cents	Approximate Ratios
		2.9.5.21.11.17 Subgroup	Full 11-limit in Patent Val
0	0.000	1/1	1/1
1	20.339	81/80	50/49, 99/98
2	40.678	128/125	49/48
3	61.017	648/625	25/24, 81/80, 36/35, 33/32
4	81.356	21/20, 22/21
5	101.695	17/16, 18/17	16/15
6	122.034		15/14
7	142.373
8	162.712	11/10	10/9, 11/10, 12/11
9	183.051	10/9
10	203.390	9/8
11	223.729		9/8, 8/7
12	244.068
13	264.407		7/6
14	284.746	20/17
15	305.085
16	325.424		6/5, 11/9
17	345.763	11/9
18	366.102	21/17
19	386.441	5/4	5/4
20	406.780	81/64
21	427.119	32/25	32/25, 14/11
22	447.458	22/17	9/7
23	467.797	21/16
24	488.136		4/3, 21/16
25	508.475
26	528.814
27	549.153	11/8	27/20, 11/8, 15/11
28	569.492	25/18
29	589.831	45/32	7/5
30	610.169	64/45	10/7
31	630.508	36/25
32	650.847	16/11	40/27, 16/11, 22/15
33	671.186
34	691.525
35	711.864		3/2, 32/21
36	732.203	32/21
37	752.542	17/11	14/9
38	772.881	25/16	25/16, 11/7
39	793.220	128/81
40	813.559	8/5	8/5
41	833.898	34/21
42	854.237	18/11
43	874.576		5/3, 18/11
44	894.915
45	915.254	17/10
46	935.593		12/7
47	955.932
48	976.271		16/9, 7/4
49	996.610	16/9
50	1016.949	9/5
51	1037.288	20/11	9/5, 11/6, 20/11
52	1057.627
53	1077.966		28/15
54	1098.305	17/9, 32/17	15/8
55	1118.644	40/21, 21/11
56	1138.983	625/324	48/25, 160/81, 35/18, 64/33
57	1159.322	125/64	96/49
58	1179.661	160/81	49/25, 196/99
59	1200.000	2/1	2/1

Music

• Chinchillian Fugue (First mode of Porcupine 7 scale, 59EDO) by Ray Perlner