59edo

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59edo

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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 2^1/59, or the 59th root of 2.

Theory

59edo's best fifth is stretched about 9.91 cents from the just interval, and yet its 5/4 is nearly pure (stretched only 0.127 ¢), 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.

Using the flat fifth instead of the sharp one allows for the 12 & 35 temperament, which is a kind of bizarre cousin to garibaldi 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 flattertone temperament in the 59bcd val, a variant of meantone with very flat fifths.

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.

Odd harmonics

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)

Subsets and supersets

59edo is the 17th prime edo, following 53edo and before 61edo. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

Intervals

Steps	Cents	Approximate ratios (2.9.5.21.11.39.17-subgroup)	Ratios of 3, 7, 13 (tending sharp)	Ratios of 3, 7, 13 (tending flat)
0	0.0	1/1
1	20.3	81/80
2	40.7	40/39, 45/44
3	61.0	27/26, 28/27
4	81.4	21/20, 22/21
5	101.7	17/16, 18/17, 35/33
6	122.0	15/14, 14/13
7	142.4	13/12
8	162.7	11/10
9	183.1	10/9
10	203.4	9/8, 44/39
11	223.7	25/22	8/7
12	244.1	15/13, 39/34		8/7
13	264.4	7/6, 64/55
14	284.7	20/17, 33/28
15	305.1	25/21
16	325.4
17	345.8	11/9, 39/32, 128/105	16/13
18	366.1	21/17		16/13
19	386.4	5/4
20	406.8	81/64
21	427.1	32/25, 50/39
22	447.5	22/17, 35/27, 128/99
23	467.8	21/16, 64/49
24	488.1	45/34, 85/64	4/3
25	508.5	35/26		4/3
26	528.8	34/25
27	549.2	11/8, 48/35
28	569.5	25/18
29	589.8	45/32, 128/91
30	610.2	64/45, 91/64
31	630.5	36/25
32	650.8	16/11, 35/24
33	671.2	25/17
34	691.5	52/35		3/2
35	711.9	68/45, 128/85	3/2
36	732.2	32/21, 49/32
37	752.5	17/11, 54/35, 99/64
38	772.9	25/16, 39/25
39	793.2	128/81
40	813.6	8/5
41	833.9	34/21		13/8
42	854.2	18/11, 64/39, 105/64	13/8
43	874.6
44	894.9	42/25
45	915.3	17/10, 56/33
46	935.6	12/7, 55/32
47	955.9	26/15, 68/39		7/4
48	976.3	44/25	7/4
49	996.6	16/9, 39/22
50	1016.9	9/5
51	1037.3	20/11
52	1057.6	24/13
53	1078.0	13/7, 28/15
54	1098.3	17/9, 32/17, 66/35
55	1118.6	21/11, 40/21
56	1139.0	27/14, 52/27
57	1159.3	39/20, 88/45
58	1179.7	160/81
59	1200.0	2/1

Todo: complete table

Notation

Sagittal notation

Best fifth notation

This notation uses the same sagittal sequence as 66-EDO.

Evo flavor

Revo flavor

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's primary comma (the comma it exactly represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it approximately represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

Second-best fifth notation

This notation uses the same sagittal sequence as EDOs 45 and 52.

Evo flavor

Revo flavor

Evo-SZ flavor

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.

Octave stretch or compression

59edo’s approximations of 3/1, 7/1 and 11/1 are improved by 93edt, a stretched-octave version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

211ed12 is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers compressed octaves, then 153ed6 is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

Scales

Porcupine scales

• Porcupine[7]: 8 8 8 11 8 8 8
• Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
• Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
• Antechinus (nonoctave period)

Instruments

Lumatone

See Lumatone mapping for 59edo.

Music

Bryan Deister

• Microtonal improvisation in 59edo (2025)
• icosa - Oliver Buckland (microtonal cover in 59edo) (2025)
• Le Ciel - Malice Mizer (microtonal cover in 59edo) (2026)

Francium

• "too powerful if i had social skills" from Melancholie (2023) – Spotify | Bandcamp | YouTube
• "Stay Away From The Fog" from Void (2025) – Spotify | Bandcamp | YouTube

Budjarn Lambeth

• The Odd Effects of Breathing the Fairy Dust (2026)

Ray Perlner

• Chinchillian Fugue – first mode of the Porcupine[7] scale in 59edo