59edo

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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. 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	20.3		^D, E♭♭♭♭	^D, vE♭
2	40.7		D♯, vE♭♭♭	^^D, E♭
3	61	29/28, 30/29	^D♯, E♭♭♭	^3D, ^E♭
4	81.4	23/22	D𝄪, vE♭♭	^4D, ^^E♭
5	101.7	17/16	^D𝄪, E♭♭	v4D♯, ^3E♭
6	122	15/14	D♯𝄪, vE♭	v3D♯, ^4E♭
7	142.4	25/23	^D♯𝄪, E♭	vvD♯, v4E
8	162.7	11/10, 34/31	D𝄪𝄪, vE	vD♯, v3E
9	183.1		E	D♯, vvE
10	203.4		^E, F♭♭♭	^D♯, vE
11	223.7	25/22, 33/29	E♯, vF♭♭	E
12	244.1	23/20	^E♯, F♭♭	^E, vF
13	264.4	7/6	E𝄪, vF♭	F
14	284.7	13/11, 20/17, 33/28	^E𝄪, F♭	^F, vG♭
15	305.1	31/26	E♯𝄪, vF	^^F, G♭
16	325.4	29/24	F	^3F, ^G♭
17	345.8		^F, G♭♭♭♭	^4F, ^^G♭
18	366.1		F♯, vG♭♭♭	v4F♯, ^3G♭
19	386.4	5/4	^F♯, G♭♭♭	v3F♯, ^4G♭
20	406.8	19/15, 24/19	F𝄪, vG♭♭	vvF♯, v4G
21	427.1	32/25	^F𝄪, G♭♭	vF♯, v3G
22	447.5	22/17	F♯𝄪, vG♭	F♯, vvG
23	467.8	17/13	^F♯𝄪, G♭	^F♯, vG
24	488.1		F𝄪𝄪, vG	G
25	508.5		G	^G, vA♭
26	528.8	19/14, 34/25	^G, A♭♭♭♭	^^G, A♭
27	549.2	11/8	G♯, vA♭♭♭	^3G, ^A♭
28	569.5	32/23	^G♯, A♭♭♭	^4G, ^^A♭
29	589.8	31/22	G𝄪, vA♭♭	v4G♯, ^3A♭
30	610.2		^G𝄪, A♭♭	v3G♯, ^4A♭
31	630.5	23/16	G♯𝄪, vA♭	vvG♯, v4A
32	650.8	16/11	^G♯𝄪, A♭	vG♯, v3A
33	671.2	25/17, 28/19	G𝄪𝄪, vA	G♯, vvA
34	691.5		A	^G♯, vA
35	711.9		^A, B♭♭♭♭	A
36	732.2	26/17, 29/19	A♯, vB♭♭♭	^A, vB♭
37	752.5	17/11	^A♯, B♭♭♭	^^A, B♭
38	772.9	25/16	A𝄪, vB♭♭	^3A, ^B♭
39	793.2	19/12, 30/19	^A𝄪, B♭♭	^4A, ^^B♭
40	813.6	8/5	A♯𝄪, vB♭	v4A♯, ^3B♭
41	833.9		^A♯𝄪, B♭	v3A♯, ^4B♭
42	854.2		A𝄪𝄪, vB	vvA♯, v4B
43	874.6		B	vA♯, v3B
44	894.9		^B, C♭♭♭	A♯, vvB
45	915.3	17/10, 22/13	B♯, vC♭♭	^A♯, vB
46	935.6	12/7	^B♯, C♭♭	B
47	955.9	33/19	B𝄪, vC♭	^B, vC
48	976.3		^B𝄪, C♭	C
49	996.6		B♯𝄪, vC	^C, vD♭
50	1016.9		C	^^C, D♭
51	1037.3	20/11, 31/17	^C, D♭♭♭♭	^3C, ^D♭
52	1057.6		C♯, vD♭♭♭	^4C, ^^D♭
53	1078	28/15	^C♯, D♭♭♭	v4C♯, ^3D♭
54	1098.3	32/17	C𝄪, vD♭♭	v3C♯, ^4D♭
55	1118.6		^C𝄪, D♭♭	vvC♯, v4D
56	1139	29/15	C♯𝄪, vD♭	vC♯, v3D
57	1159.3		^C♯𝄪, D♭	C♯, vvD
58	1179.7		C𝄪𝄪, vD	^C♯, vD
59	1200	2/1	D	D

Instruments

Lumatone

See Lumatone mapping for 59edo

Music

Francium

• "too powerful if i had social skills" from Melancholie (2023) Spotify | Bandcamp | YouTube

Ray Perlner

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