130edo

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← 129edo130edo131edo →
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 0 1 2 3 4 5 6 7 8 9 10 11 12
Symbol Sagittal natural.png Sagittal nai.png Sagittal pai.png Sagittal tai.png Sagittal phai.png Sagittal patai.png Sagittal pakai.png Sagittal jakai.png Sagittal sharp phao.png Sagittal sharp tao.png Sagittal sharp pao.png Sagittal sharp nao.png Sagittal sharp.png

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
Sevish
Gene Ward Smith