# 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

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 21/130, 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. 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.

### 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, 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

## Notation

### Sagittal

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

## 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 ¢)