130edo

130edo divides the octave into 130 parts of size 9.231 cents each.

Theory

130edo is a zeta peak edo, a zeta peak integer edo, and a zeta integral edo but not a gap edo. It can be used to tune a variety of temperaments, including hemiwürschmidt, sesquiquartififths, harry and hemischis. It also can be used to tune the rank-three temperament jove, tempering out 243/242 and 441/440, plus 364/363 for the 13-limit and 595/594 for the 17-limit. It gives the optimal patent val for 11-limit hemiwürschmidt and sesquart and 13-limit harry temperaments.

Intervals

Degree	Cents	Approximate Ratios
0	0.000	1/1
1	9.231	126/125, 225/224
2	18.462	81/80
3	27.692	64/63
4	36.923	49/48, 50/49
5	46.154	36/35
6	55.385	33/32
7	64.615	28/27, 27/26
8	73.846	25/24
9	83.077	21/20, 22/21
10	92.308	135/128
11	101.538	35/33
12	110.769	16/15
13	120.000	15/14
14	129.231	14/13
15	138.462	13/12
16	147.692	12/11
17	156.923	35/32
18	166.154	11/10
19	175.385	72/65
20	184.615	10/9
21	193.846	28/25
22	203.077	9/8
23	212.308	44/39
24	221.538	25/22
25	230.769	8/7
26	240.000	55/48
27	249.231	15/13
28	258.462	64/55
29	267.692	7/6
30	276.923	75/64
31	286.154	13/11
32	295.385	32/27
33	304.615	25/21
34	313.846	6/5
35	323.077	65/54
36	332.308	40/33
37	341.538	39/32
38	350.769	11/9, 27/22
39	360.000	16/13
40	369.231	26/21
41	378.462	56/45
42	387.692	5/4
43	396.923	63/50
44	406.154	81/64
45	415.385	14/11
46	424.615	32/25
47	433.846	9/7
48	443.077	128/99
49	452.308	13/10
50	461.538	72/55
51	470.769	21/16
52	480.000	33/25
53	489.231	250/189
54	498.462	4/3
55	507.692	75/56
56	516.923	27/20
57	526.154	65/48
58	535.385	15/11
59	544.615	48/35
60	553.846	11/8
61	563.077	18/13
62	572.308	25/18
63	581.538	7/5
64	590.769	45/32
65	600.000	99/70, 140/99
…	…	…

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5.7	2401/2400, 3136/3125, 19683/19600	[⟨130 206 302 365]]	-0.119	0.311	3.37
2.3.5.7.11	243/242, 441/440, 3136/3125, 4000/3993	[⟨130 206 302 365 450]]	-0.241	0.370	4.02
2.3.5.7.11.13	243/242, 351/350, 364/363, 441/440, 3136/3125	[⟨130 206 302 365 450 481]]	-0.177	0.367	3.98

Rank-2 temperaments

Note: temperaments supported by 65et are not included.

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	3\130	27.69	64/63	Arch
1	9\130	83.08	21/20	Sextilififths
1	19\130	175.38	72/65	Sesquiquartififths / sesquart
1	21\130	193.85	28/25	Didacus / hemiwürschmidt
1	27\130	249.23	15/13	Hemischis
1	41\130	378.46	56/45	Subpental
2	6\130	55.38	33/32	Biscapade
2	9\130	83.08	21/20	Harry
2	17\130	156.92	35/32	Bison
2	19\130	175.38	448/405	Bisesqui
2	54\130 (11\130)	498.46 (101.54)	4/3 (35/33)	Bischismic
5	27\130 (1\130)	249.23 (9.23)	81/70 (176/175)	Hemipental
10	27\130 (1\130)	249.23 (9.23)	15/13 (176/175)	Decoid
10	54\130 (2\130)	498.46 (18.46)	4/3 (81/80)	Decal
26	54\130 (1\130)	498.46 (9.23)	4/3 (225/224)	Bosonic

Scales

14-tone temperament of "Narrative Wars"
as an example of using 130-EDO:

Step	Cents	Distance to the nearest JI interval (selected ratios)
13 (13/130)	120.000	15/14 (+0.557 ¢)
7 (20/130)	184.615	10/9 (+2.211 ¢)
9 (29/130)	267.692	7/6 (+0,821 ¢)
9 (38/130)	350.769	11/9 (+3.361 ¢)
9 (47/130)	433.846	9/7 (-1.238 ¢)
7 (54/130)	498.462	4/3 (+0.417 ¢)
13 (67/130)	618.462	10/7 (+0.974 ¢)
9 (76/130)	701.538	3/2 (-0.417 ¢)
7 (83/130)	766.154	14/9 (+1.238 ¢)
13 (96/130)	886.154	5/3 (+1.795 ¢)
5 (101/130)	932.308	12/7 (-0.821 ¢)
13 (114/130)	1052.308	11/6 (+2.945 ¢)
7 (121/130)	1116.923	21/11 (-2.540 ¢)
9 (130/130)	1200.000	Octave (2/1, ±0 ¢)

Music

• The Paradise of Cantor play by Gene Ward Smith
• "Narrative Wars" by Sevish (uses a 14-tone (13 7 9 9 9 7 13 9 7 13 5 13 7 9) subset of 130-EDO, from the 2016 compilation album "Next Xen")