130edo

← 129edo

130edo

131edo →

Prime factorization	2 × 5 × 13
Step size	9.23077¢
Fifth	76\130 (701.538¢) (→38\65)
Semitones (A1:m2)	12:10 (110.8¢ : 92.31¢)
Consistency limit	15
Distinct consistency limit	15
Special properties	Zeta peak Zeta peak integer Zeta integral

130 equal divisions of the octave (abbreviated 130edo or 130ed2), also called 130-tone equal temperament (130tet) or 130 equal temperament (130et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 130 equal parts of about 9.23 ¢ each. Each step represents a frequency ratio of 2^1/130, or the 130th root of 2.

Theory

130edo is a zeta peak edo, a zeta peak integer edo, and a zeta integral edo but not a gap edo. It is distinctly consistent to the 15-odd-limit and is the first nontrivial edo to be consistent in the 14-odd-prime-sum-limit. As an equal temperament, it tempers out 2401/2400, 3136/3125, 6144/6125, and 19683/19600 in the 7-limit; 243/242, 441/440, 540/539, and 4000/3993 in the 11-limit; and 351/350, 364/363, 676/675, 729/728, 1001/1000, 1575/1573, 1716/1715, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit. 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-3 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.

Prime harmonics

Approximation of prime harmonics in 130edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.42	+1.38	+0.40	+2.53	-0.53	-3.42	-2.13	-0.58	+4.27	-0.42
	Relative (%)	+0.0	-4.5	+14.9	+4.4	+27.4	-5.7	-37.0	-23.1	-6.3	+46.2	-4.6
Steps (reduced)		130 (0)	206 (76)	302 (42)	365 (105)	450 (60)	481 (91)	531 (11)	552 (32)	588 (68)	632 (112)	644 (124)

Subsets and supersets

Since 130 factors into 2 × 5 × 13, 130edo has subset edos 2, 5, 10, 13, 26, and 65.

260edo, which divides the edostep in two, provides a strong correction for the 29th harmonic.

Intervals

Degree	Cents	Approximate Ratios
0	0.000	1/1
1	9.231	126/125, 144/143, 169/168, 176/175, 196/195, 225/224
2	18.462	78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120
3	27.692	56/55, 64/63, 65/64, 66/65
4	36.923	45/44, 49/48, 50/49, 55/54
5	46.154	36/35, 40/39
6	55.385	33/32
7	64.615	27/26, 28/27
8	73.846	25/24, 26/25
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	44/35
44	406.154	81/64
45	415.385	14/11
46	424.615	32/25
47	433.846	9/7
48	443.077	84/65, 128/99
49	452.308	13/10
50	461.538	64/49, 72/55
51	470.769	21/16
52	480.000	33/25
53	489.231	65/49
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
…	…	…

Notation

Sagittal

Steps	0	1	2	3	4	5	6	7	8	9	10	11	12
Symbol

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 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	3\130	27.69	64/63	Arch
1	7\130	64.62	26/25	Rectified hebrew
1	9\130	83.08	21/20	Sextilififths
1	19\130	175.38	72/65	Sesquiquartififths / sesquart
1	21\130	193.85	28/25	Hemiwürschmidt
1	27\130	249.23	15/13	Hemischis
1	41\130	378.46	56/45	Subpental
2	6\130	55.38	33/32	Septisuperfourth
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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

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

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

See also: Category:130edo tracks

birdshite stalactite

• wazzock (2024)

Sevish

• Narrative Wars

Gene Ward Smith

• The Paradise of Cantor play (2006)