130edo

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← 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

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 distinctly consistent to the 15-odd-limit. It is also almost consistent in the no-29 31-odd-limit, missing 19/11 (50.5%), 25/19 (52.9%), 17/11 (64,4%), 25/17 (66.8%), and octave complements. It is a zeta peak edo, a zeta peak integer edo, and a zeta integral edo but not a zeta gap edo.

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)
Approximation of prime harmonics in 130edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -2.11 -4.45 -3.83 -0.89 +3.42 +2.37 +0.04 +3.77 -4.31 +2.98 -4.54
Relative (%) -22.9 -48.2 -41.4 -9.7 +37.0 +25.6 +0.4 +40.8 -46.7 +32.3 -49.1
Steps
(reduced) 		677
(27) 		696
(46) 		705
(55) 		722
(72) 		745
(95) 		765
(115) 		771
(121) 		789
(9) 		799
(19) 		805
(25) 		819
(39)

Subsets and supersets

Since 130 factors into primes as 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 0.00 1/1
1 9.23 126/125, 144/143, 169/168, 176/175, 196/195, 225/224
2 18.46 78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120
3 27.69 56/55, 64/63, 65/64, 66/65
4 36.92 45/44, 49/48, 50/49, 55/54
5 46.15 36/35, 40/39
6 55.38 33/32
7 64.62 27/26, 28/27
8 73.85 25/24, 26/25
9 83.08 21/20, 22/21
10 92.31 135/128
11 101.54 35/33
12 110.77 16/15
13 120.00 15/14
14 129.23 14/13
15 138.46 13/12
16 147.69 12/11
17 156.92 35/32
18 166.15 11/10
19 175.38 72/65
20 184.62 10/9
21 193.85 28/25
22 203.08 9/8
23 212.31 44/39
24 221.54 25/22
25 230.77 8/7
26 240.00 55/48
27 249.23 15/13
28 258.46 64/55
29 267.69 7/6
30 276.92 75/64
31 286.15 13/11
32 295.38 32/27
33 304.62 25/21
34 313.85 6/5
35 323.08 65/54
36 332.31 40/33
37 341.54 39/32
38 350.77 11/9, 27/22
39 360.00 16/13
40 369.23 26/21
41 378.46 56/45
42 387.69 5/4
43 396.92 44/35
44 406.15 81/64
45 415.38 14/11
46 424.62 32/25
47 433.85 9/7
48 443.08 84/65, 128/99
49 452.31 13/10
50 461.54 64/49, 72/55
51 470.77 21/16
52 480.00 33/25
53 489.23 65/49
54 498.46 4/3
55 507.69 75/56
56 516.92 27/20
57 526.15 65/48
58 535.38 15/11
59 544.62 48/35
60 553.85 11/8
61 563.08 18/13
62 572.31 25/18
63 581.54 7/5
64 590.77 45/32
65 600.00 99/70, 140/99
… … …

Notation

Ups and downs notation

130edo can be notated using ups and downs and quarter-tone accidentals:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Sharp symbol   
 
  
   
    
   
  
   
    
     
    
   
  
   
    
     
    
   
  
   
    
     
    
   
  
   
Flat symbol
 
  
   
    
   
  
   
    
     
    
   
  
   
    
     
    
   
  
   
    
     
    
   
  
   

Sagittal notation

130edo can be notated in Sagittal using the Spartan extension, with the apotome equal to 12 edosteps and the limma to 10 edosteps. Since the the rastma is tempered out, a SZ half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:

Steps 0 1 2 3 4 5 6 7 8 9 10 11 12
Symbol Evo+SZ
Evo
Revo

Because it uses the entire Spartan extension, it allows no accidental enharmonic respellings.

Approximation to JI

The following tables show how 23-odd-limit intervals are represented in 130edo. Prime harmonics are in bold; inconsistent intervals are in italics.

23-odd-limit intervals in 130edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
21/16, 32/21 0.012 0.1
23/13, 26/23 0.054 0.6
13/12, 24/13 0.111 1.2
23/12, 24/23 0.166 1.8
23/18, 36/23 0.251 2.7
13/9, 18/13 0.305 3.3
7/4, 8/7 0.405 4.4
3/2, 4/3 0.417 4.5
21/13, 26/21 0.516 5.6
13/8, 16/13 0.528 5.7
15/14, 28/15 0.557 6.0
23/21, 42/23 0.570 6.2
23/16, 32/23 0.582 6.3
7/6, 12/7 0.821 8.9
9/8, 16/9 0.833 9.0
13/7, 14/13 0.933 10.1
15/8, 16/15 0.962 10.4
7/5, 10/7 0.974 10.5
23/14, 28/23 0.987 10.7
11/10, 20/11 1.150 12.5
9/7, 14/9 1.238 13.4
19/17, 34/19 1.289 14.0
19/18, 36/19 1.295 14.0
5/4, 8/5 1.379 14.9
21/20, 40/21 1.390 15.1
15/13, 26/15 1.490 16.1
23/15, 30/23 1.544 16.7
23/19, 38/23 1.546 16.8
15/11, 22/15 1.566 17.0
19/13, 26/19 1.601 17.3
19/12, 24/19 1.712 18.5
5/3, 6/5 1.795 19.4
13/10, 20/13 1.906 20.7
23/20, 40/23 1.961 21.2
21/19, 38/21 2.117 22.9
11/7, 14/11 2.123 23.0
19/16, 32/19 2.128 23.1
9/5, 10/9 2.212 24.0
11/8, 16/11 2.528 27.4
19/14, 28/19 2.533 27.4
21/11, 22/21 2.540 27.5
17/9, 18/17 2.584 28.0
23/17, 34/23 2.835 30.7
17/13, 26/17 2.889 31.3
11/6, 12/11 2.945 31.9
17/12, 24/17 3.000 32.5
13/11, 22/13 3.056 33.1
19/15, 30/19 3.090 33.5
23/22, 44/23 3.110 33.7
17/11, 22/17 3.286 35.6
11/9, 18/11 3.361 36.4
21/17, 34/21 3.405 36.9
17/16, 32/17 3.417 37.0
19/10, 20/19 3.507 38.0
17/14, 28/17 3.822 41.4
17/15, 30/17 4.379 47.4
17/10, 20/17 4.435 48.0
19/11, 22/19 4.574 49.6
23-odd-limit intervals in 130edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
21/16, 32/21 0.012 0.1
23/13, 26/23 0.054 0.6
13/12, 24/13 0.111 1.2
23/12, 24/23 0.166 1.8
23/18, 36/23 0.251 2.7
13/9, 18/13 0.305 3.3
7/4, 8/7 0.405 4.4
3/2, 4/3 0.417 4.5
21/13, 26/21 0.516 5.6
13/8, 16/13 0.528 5.7
15/14, 28/15 0.557 6.0
23/21, 42/23 0.570 6.2
23/16, 32/23 0.582 6.3
7/6, 12/7 0.821 8.9
9/8, 16/9 0.833 9.0
13/7, 14/13 0.933 10.1
15/8, 16/15 0.962 10.4
7/5, 10/7 0.974 10.5
23/14, 28/23 0.987 10.7
11/10, 20/11 1.150 12.5
9/7, 14/9 1.238 13.4
19/17, 34/19 1.289 14.0
19/18, 36/19 1.295 14.0
5/4, 8/5 1.379 14.9
21/20, 40/21 1.390 15.1
15/13, 26/15 1.490 16.1
23/15, 30/23 1.544 16.7
23/19, 38/23 1.546 16.8
15/11, 22/15 1.566 17.0
19/13, 26/19 1.601 17.3
19/12, 24/19 1.712 18.5
5/3, 6/5 1.795 19.4
13/10, 20/13 1.906 20.7
23/20, 40/23 1.961 21.2
21/19, 38/21 2.117 22.9
11/7, 14/11 2.123 23.0
19/16, 32/19 2.128 23.1
9/5, 10/9 2.212 24.0
11/8, 16/11 2.528 27.4
19/14, 28/19 2.533 27.4
21/11, 22/21 2.540 27.5
17/9, 18/17 2.584 28.0
23/17, 34/23 2.835 30.7
17/13, 26/17 2.889 31.3
11/6, 12/11 2.945 31.9
17/12, 24/17 3.000 32.5
13/11, 22/13 3.056 33.1
19/15, 30/19 3.090 33.5
23/22, 44/23 3.110 33.7
11/9, 18/11 3.361 36.4
21/17, 34/21 3.405 36.9
17/16, 32/17 3.417 37.0
19/10, 20/19 3.507 38.0
17/14, 28/17 3.822 41.4
17/15, 30/17 4.379 47.4
19/11, 22/19 4.657 50.4
17/10, 20/17 4.796 52.0
17/11, 22/17 5.945 64.4

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* 		Temperament
1 3\130 27.69 64/63 Arch
1 7\130 64.62 26/25 Rectified hebrew
1 9\130 83.08 21/20 Sextilifourths
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) 		Hemiquintile
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) 		Decile
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 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 ¢)

Instruments

Lumatone mapping for 130edo

Music

See also: Category:130edo tracks
birdshite stalactite
Sevish
Gene Ward Smith
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