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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|130}}
{{ED intro}}


== Theory ==
== Theory ==
Line 6: Line 6:


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|130|columns=12}}
{{Harmonics in equal|130|columns=9}}
{{Harmonics in equal|130|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 130edo (continued)}}
{{Harmonics in equal|130|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 130edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 130 factors into {{Factorisation|130}}, 130edo has subset edos {{EDOs| 2, 5, 10, 13, 26, and 65 }}.
Since 130 factors into 2 × 5 × 13, 130edo has subset edos {{EDOs| 2, 5, 10, 13, 26, and 65 }}.


[[260edo]], which divides the edostep in two, provides a strong correction for the 29th harmonic.
[[260edo]], which divides the edostep in two, provides a strong correction for the 29th harmonic.
Line 19: Line 19:
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios
! Approximate ratios
|-
|-
| 0
| 0
| 0.000
| 0.00
| 1/1
| 1/1
|-
|-
| 1
| 1
| 9.231
| 9.23
| ''126/125'', 144/143, 169/168, 176/175, 196/195, 225/224
| ''126/125'', 144/143, 169/168, 176/175, 196/195, 225/224
|-
|-
| 2
| 2
| 18.462
| 18.46
| 78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120
| 78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120
|-
|-
| 3
| 3
| 27.692
| 27.69
| 56/55, 64/63, 65/64, 66/65
| 56/55, 64/63, 65/64, 66/65
|-
|-
| 4
| 4
| 36.923
| 36.92
| 45/44, 49/48, 50/49, ''55/54''
| 45/44, 49/48, 50/49, ''55/54''
|-
|-
| 5
| 5
| 46.154
| 46.15
| 36/35, 40/39
| 36/35, 40/39
|-
|-
| 6
| 6
| 55.385
| 55.38
| 33/32
| 33/32
|-
|-
| 7
| 7
| 64.615
| 64.62
| 27/26, 28/27
| 27/26, 28/27
|-
|-
| 8
| 8
| 73.846
| 73.85
| 25/24, 26/25
| 25/24, 26/25
|-
|-
| 9
| 9
| 83.077
| 83.08
| 21/20, 22/21
| 21/20, 22/21
|-
|-
| 10
| 10
| 92.308
| 92.31
| 135/128
| 135/128
|-
|-
| 11
| 11
| 101.538
| 101.54
| 35/33
| 35/33
|-
|-
| 12
| 12
| 110.769
| 110.77
| 16/15
| 16/15
|-
|-
| 13
| 13
| 120.000
| 120.00
| 15/14
| 15/14
|-
|-
| 14
| 14
| 129.231
| 129.23
| 14/13
| 14/13
|-
|-
| 15
| 15
| 138.462
| 138.46
| 13/12
| 13/12
|-
|-
| 16
| 16
| 147.692
| 147.69
| 12/11
| 12/11
|-
|-
| 17
| 17
| 156.923
| 156.92
| 35/32
| 35/32
|-
|-
| 18
| 18
| 166.154
| 166.15
| 11/10
| 11/10
|-
|-
| 19
| 19
| 175.385
| 175.38
| 72/65
| 72/65
|-
|-
| 20
| 20
| 184.615
| 184.62
| 10/9
| 10/9
|-
|-
| 21
| 21
| 193.846
| 193.85
| 28/25
| 28/25
|-
|-
| 22
| 22
| 203.077
| 203.08
| 9/8
| 9/8
|-
|-
| 23
| 23
| 212.308
| 212.31
| 44/39
| 44/39
|-
|-
| 24
| 24
| 221.538
| 221.54
| 25/22
| 25/22
|-
|-
| 25
| 25
| 230.769
| 230.77
| 8/7
| 8/7
|-
|-
| 26
| 26
| 240.000
| 240.00
| 55/48
| 55/48
|-
|-
| 27
| 27
| 249.231
| 249.23
| 15/13
| 15/13
|-
|-
| 28
| 28
| 258.462
| 258.46
| 64/55
| 64/55
|-
|-
| 29
| 29
| 267.692
| 267.69
| 7/6
| 7/6
|-
|-
| 30
| 30
| 276.923
| 276.92
| 75/64
| 75/64
|-
|-
| 31
| 31
| 286.154
| 286.15
| 13/11
| 13/11
|-
|-
| 32
| 32
| 295.385
| 295.38
| 32/27
| 32/27
|-
|-
| 33
| 33
| 304.615
| 304.62
| 25/21
| 25/21
|-
|-
| 34
| 34
| 313.846
| 313.85
| 6/5
| 6/5
|-
|-
| 35
| 35
| 323.077
| 323.08
| 65/54
| 65/54
|-
|-
| 36
| 36
| 332.308
| 332.31
| 40/33
| 40/33
|-
|-
| 37
| 37
| 341.538
| 341.54
| 39/32
| 39/32
|-
|-
| 38
| 38
| 350.769
| 350.77
| 11/9, 27/22
| 11/9, 27/22
|-
|-
| 39
| 39
| 360.000
| 360.00
| 16/13
| 16/13
|-
|-
| 40
| 40
| 369.231
| 369.23
| 26/21
| 26/21
|-
|-
| 41
| 41
| 378.462
| 378.46
| 56/45
| 56/45
|-
|-
| 42
| 42
| 387.692
| 387.69
| 5/4
| 5/4
|-
|-
| 43
| 43
| 396.923
| 396.92
| 44/35
| 44/35
|-
|-
| 44
| 44
| 406.154
| 406.15
| 81/64
| 81/64
|-
|-
| 45
| 45
| 415.385
| 415.38
| 14/11
| 14/11
|-
|-
| 46
| 46
| 424.615
| 424.62
| 32/25
| 32/25
|-
|-
| 47
| 47
| 433.846
| 433.85
| 9/7
| 9/7
|-
|-
| 48
| 48
| 443.077
| 443.08
| 84/65, 128/99
| 84/65, 128/99
|-
|-
| 49
| 49
| 452.308
| 452.31
| 13/10
| 13/10
|-
|-
| 50
| 50
| 461.538
| 461.54
| 64/49, ''72/55''
| 64/49, ''72/55''
|-
|-
| 51
| 51
| 470.769
| 470.77
| 21/16
| 21/16
|-
|-
| 52
| 52
| 480.000
| 480.00
| 33/25
| 33/25
|-
|-
| 53
| 53
| 489.231
| 489.23
| 65/49
| 65/49
|-
|-
| 54
| 54
| 498.462
| 498.46
| 4/3
| 4/3
|-
|-
| 55
| 55
| 507.692
| 507.69
| 75/56
| 75/56
|-
|-
| 56
| 56
| 516.923
| 516.92
| 27/20
| 27/20
|-
|-
| 57
| 57
| 526.154
| 526.15
| 65/48
| 65/48
|-
|-
| 58
| 58
| 535.385
| 535.38
| 15/11
| 15/11
|-
|-
| 59
| 59
| 544.615
| 544.62
| 48/35
| 48/35
|-
|-
| 60
| 60
| 553.846
| 553.85
| 11/8
| 11/8
|-
|-
| 61
| 61
| 563.077
| 563.08
| 18/13
| 18/13
|-
|-
| 62
| 62
| 572.308
| 572.31
| 25/18
| 25/18
|-
|-
| 63
| 63
| 581.538
| 581.54
| 7/5
| 7/5
|-
|-
| 64
| 64
| 590.769
| 590.77
| 45/32
| 45/32
|-
|-
| 65
| 65
| 600.000
| 600.00
| 99/70, 140/99
| 99/70, 140/99
|-
|-
Line 326: Line 326:
== Approximation to JI ==
== Approximation to JI ==
=== Zeta peak index ===
=== Zeta peak index ===
{| class="wikitable center-all"
{{ZPI
|-
| zpi = 796
! colspan="3" | Tuning
| steps = 130.003910460506
! colspan="3" | Strength
| step size = 9.23049157328654
! colspan="2" | Closest edo
| tempered height = 10.355108
! colspan="2" | Integer limit
| pure height = 10.339572
|-
| integral = 1.634018
! ZPI
| gap = 19.594551
! Steps per octave
| octave = 1199.96390452725
! Step size (cents)
| consistent = 16
! Height
| distinct = 16
! Integral
}}
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[796zpi]]
| 130.003910460506
| 9.23049157328654
| 10.355108
| 1.634018
| 19.594551
| 130edo
| 1199.96390452725
| 16
| 16
|}


== Regular temperament properties ==
== Regular temperament properties ==
Line 370: Line 353:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 3136/3125, 19683/19600
| 2401/2400, 3136/3125, 19683/19600
| {{mapping| 130 206 302 365 }}
| {{Mapping| 130 206 302 365 }}
| −0.119
| −0.119
| 0.311
| 0.311
Line 377: Line 360:
| 2.3.5.7.11
| 2.3.5.7.11
| 243/242, 441/440, 3136/3125, 4000/3993
| 243/242, 441/440, 3136/3125, 4000/3993
| {{mapping| 130 206 302 365 450 }}
| {{Mapping| 130 206 302 365 450 }}
| −0.241
| −0.241
| 0.370
| 0.370
Line 384: Line 367:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 243/242, 351/350, 364/363, 441/440, 3136/3125
| 243/242, 351/350, 364/363, 441/440, 3136/3125
| {{mapping| 130 206 302 365 450 481 }}
| {{Mapping| 130 206 302 365 450 481 }}
| −0.177
| −0.177
| 0.367
| 0.367
Line 418: Line 401:
| 83.08
| 83.08
| 21/20
| 21/20
| [[Sextilififths]]
| [[Sextilifourths]]
|-
|-
| 1
| 1
Line 478: Line 461:
| 249.23<br>(9.23)
| 249.23<br>(9.23)
| 81/70<br>(176/175)
| 81/70<br>(176/175)
| [[Hemipental]]
| [[Hemiquintile]]
|-
|-
| 10
| 10
Line 490: Line 473:
| 498.46<br>(18.46)
| 498.46<br>(18.46)
| 4/3<br>(81/80)
| 4/3<br>(81/80)
| [[Decal]]
| [[Decile]]
|-
|-
| 26
| 26
Line 564: Line 547:
| [[Octave]] (2/1, 0{{c}})
| [[Octave]] (2/1, 0{{c}})
|}
|}
== Instruments ==
[[Lumatone mapping for 130edo]]


== Music ==
== Music ==

Latest revision as of 10:27, 11 May 2025

← 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 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
Error Absolute (¢) +0.00 -0.42 +1.38 +0.40 +2.53 -0.53 -3.42 -2.13 -0.58
Relative (%) +0.0 -4.5 +14.9 +4.4 +27.4 -5.7 -37.0 -23.1 -6.3
Steps
(reduced) 		130
(0) 		206
(76) 		302
(42) 		365
(105) 		450
(60) 		481
(91) 		531
(11) 		552
(32) 		588
(68)
Approximation of prime harmonics in 130edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) +4.27 -0.42 -2.11 -4.45 -3.83 -0.89 +3.42 +2.37 +0.04
Relative (%) +46.2 -4.6 -22.9 -48.2 -41.4 -9.7 +37.0 +25.6 +0.4
Steps
(reduced) 		632
(112) 		644
(124) 		677
(27) 		696
(46) 		705
(55) 		722
(72) 		745
(95) 		765
(115) 		771
(121)

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 0.00 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

Sagittal notation

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

Approximation to JI

Zeta peak index

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
796zpi 130.00391 9.230492 10.355108 10.339572 1.634018 19.594551 1199.963905 −0.036095 16 16

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
Retrieved from "https://en.xen.wiki/index.php?title=130edo&oldid=195935"