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The '''130 equal divisions of the octave''' ('''130edo'''), or the '''130(-tone) equal temperament''' ('''130tet''', '''130et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 130 parts of size 9.231 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
130edo is a [[zeta peak edo]], a [[zeta peak integer edo]], and a [[zeta integral edo]] but not a gap edo. 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 [[Breed family #Jove|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 [[Schismatic family #Sesquiquartififths|sesquart]] and 13-limit [[Breedsmic temperaments #Harry|harry]] temperaments.
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 complement]]s. 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 [[tempering out|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 ===
=== Prime harmonics ===
{{Primes in edo|130|columns=11}}
{{Harmonics in equal|130|columns=11}}
{{Harmonics in equal|130|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 130edo (continued)}}
 
=== Subsets and supersets ===
Since 130 factors into primes as {{nowrap| 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.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! 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, 225/224
| ''126/125'', 144/143, 169/168, 176/175, 196/195, 225/224
|-
|-
| 2
| 2
| 18.462
| 18.46
| 81/80
| 78/77, 81/80, 91/90, 99/98, 100/99, 105/104, 121/120
|-
|-
| 3
| 3
| 27.692
| 27.69
| 64/63
| 56/55, 64/63, 65/64, 66/65
|-
|-
| 4
| 4
| 36.923
| 36.92
| 49/48, 50/49
| 45/44, 49/48, 50/49, ''55/54''
|-
|-
| 5
| 5
| 46.154
| 46.15
| 36/35
| 36/35, 40/39
|-
|-
| 6
| 6
| 55.385
| 55.38
| 33/32
| 33/32
|-
|-
| 7
| 7
| 64.615
| 64.62
| 28/27, 27/26
| 27/26, 28/27
|-
|-
| 8
| 8
| 73.846
| 73.85
| 25/24
| 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
| 63/50
| 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
| 128/99
| 84/65, 128/99
|-
|-
| 49
| 49
| 452.308
| 452.31
| 13/10
| 13/10
|-
|-
| 50
| 50
| 461.538
| 461.54
| 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
| 250/189
| 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 283: Line 291:
|…
|…
|}
|}
== Notation ==
=== Ups and downs notation ===
130edo can be notated using [[Kite's ups and downs notation|ups and downs]] and quarter-tone accidentals:
{{Ups and downs sharpness|130|true}}
=== Sagittal notation ===
130edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation#Spartan single-shaft|Spartan extension]], with the apotome equal to 12 edosteps and the limma to 10 edosteps. Since the the [[243/242|rastma]] is tempered out, a SZ half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:
{| class="wikitable" data-darkreader-inline-color="" style="text-align: center;"
! colspan="2" |Steps
!0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
|-
! rowspan="3" |Symbol
!Evo+SZ
| rowspan="3" |<big>{{sagittal||//|}}</big>
| rowspan="3" |<big>{{sagittal||(}}</big>
| rowspan="3" |<big>{{sagittal|/|}}</big>
| rowspan="3" |<big>{{sagittal||)}}</big>
| rowspan="3" |<big>{{sagittal|//|}}</big>
| rowspan="3" |<big>{{sagittal|/|)}}</big>
|<big>{{Sagittal|t}}</big>
|<small>{{Sagittal|t}}<big>{{sagittal||(}}</big></small>
|<small>{{Sagittal|t}}<big>{{sagittal|/|}}</big></small>
|<small>{{Sagittal|t}}<big>{{sagittal||)}}</big></small>
|<small>{{Sagittal|t}}<big>{{sagittal|//|}}</big></small>
|<small>{{Sagittal|t}}<big>{{sagittal|/|)}}</big></small>
| rowspan="2" |<big>{{sagittal|#}}</big>
|-
!Evo
| rowspan="2" |<big>{{sagittal|/|\}}</big>
|<small>{{sagittal|#}}<big>{{sagittal|\!)}}</big></small>
|<small>{{sagittal|#}}</small><small><big>{{sagittal|\\!}}</big></small>
|<small>{{sagittal|#}}<big>{{sagittal|!)}}</big></small>
|<small>{{sagittal|#}}<big>{{sagittal|\!}}</big></small>
|<small>{{sagittal|#}}<big>{{sagittal|!(}}</big></small>
|-
!Revo
|<big>{{sagittal|(|3=\}}</big>
|<big>{{sagittal|)||(}}</big>
|<big>{{sagittal|||)}}</big>
|<big>{{sagittal|||\}}</big>
|<big>{{sagittal|/||)}}</big>
|<big>{{sagittal|/||\}}</big>
|}
Because it uses the entire Spartan extension, it allows no accidental enharmonic respellings.
== Approximation to JI ==
{{Q-odd-limit intervals|130|23}}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 297: Line 366:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 3136/3125, 19683/19600
| 2401/2400, 3136/3125, 19683/19600
| [{{val| 130 206 302 365 }}]
| {{Mapping| 130 206 302 365 }}
| -0.119
| −0.119
| 0.311
| 0.311
| 3.37
| 3.37
Line 304: Line 373:
| 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
| [{{val| 130 206 302 365 450 }}]
| {{Mapping| 130 206 302 365 450 }}
| -0.241
| −0.241
| 0.370
| 0.370
| 4.02
| 4.02
Line 311: Line 380:
| 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
| [{{val| 130 206 302 365 450 481 }}]
| {{Mapping| 130 206 302 365 450 481 }}
| -0.177
| −0.177
| 0.367
| 0.367
| 3.98
| 3.98
Line 321: Line 390:


{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Temperaments
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
Line 333: Line 403:
| 64/63
| 64/63
| [[Arch]]
| [[Arch]]
|-
| 1
| 7\130
| 64.62
| 26/25
| [[Rectified hebrew]]
|-
|-
| 1
| 1
Line 338: Line 414:
| 83.08
| 83.08
| 21/20
| 21/20
| [[Sextilififths]]
| [[Sextilifourths]]
|-
|-
| 1
| 1
Line 350: Line 426:
| 193.85
| 193.85
| 28/25
| 28/25
| [[Didacus]] / [[hemiwürschmidt]]
| [[Hemiwürschmidt]]
|-
|-
| 1
| 1
Line 368: Line 444:
| 55.38
| 55.38
| 33/32
| 33/32
| [[Biscapade]]
| [[Septisuperfourth]]
|-
|-
| 2
| 2
Line 398: Line 474:
| 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 410: Line 486:
| 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 418: Line 494:
| [[Bosonic]]
| [[Bosonic]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
{| class="wikitable"
{| class="wikitable"
|+14-tone temperament of "Narrative Wars"<br>as an example of using 130-EDO:
|+ style="font-size: 105%;" | 14-tone temperament of "Narrative Wars"<br />as an example of using 130edo:
|-
! Step
! Step
! Cents
! Cents
! Distance to the nearest JI interval<br>(selected ratios)
! Distance to the nearest JI interval<br />(selected ratios)
|-
|-
| 13 (13/130)
| 13 (13/130)
| 120.000
| 120.000
| [[15/14]] (+0.557 ¢)
| [[15/14]] (+0.557{{c}})
|-
|-
| 7 (20/130)
| 7 (20/130)
| 184.615
| 184.615
| [[10/9]] (+2.211 ¢)
| [[10/9]] (+2.211{{c}})
|-
|-
| 9 (29/130)
| 9 (29/130)
| 267.692
| 267.692
| [[7/6]] (+0,821 ¢)
| [[7/6]] (+0,821{{c}})
|-
|-
| 9 (38/130)
| 9 (38/130)
| 350.769
| 350.769
| [[11/9]] (+3.361 ¢)
| [[11/9]] (+3.361{{c}})
|-
|-
| 9 (47/130)
| 9 (47/130)
| 433.846
| 433.846
| [[9/7]] (-1.238 ¢)
| [[9/7]] (−1.238{{c}})
|-
|-
| 7 (54/130)
| 7 (54/130)
| 498.462
| 498.462
| [[4/3]] (+0.417 ¢)
| [[4/3]] (+0.417{{c}})
|-
|-
| 13 (67/130)
| 13 (67/130)
| 618.462
| 618.462
| [[10/7]] (+0.974 ¢)
| [[10/7]] (+0.974{{c}})
|-
|-
| 9 (76/130)
| 9 (76/130)
| 701.538
| 701.538
| [[3/2]] (-0.417 ¢)
| [[3/2]] (−0.417{{c}})
|-
|-
| 7 (83/130)
| 7 (83/130)
| 766.154
| 766.154
| [[14/9]] (+1.238 ¢)
| [[14/9]] (+1.238{{c}})
|-
|-
| 13 (96/130)
| 13 (96/130)
| 886.154
| 886.154
| [[5/3]] (+1.795 ¢)
| [[5/3]] (+1.795{{c}})
|-
|-
| 5 (101/130)
| 5 (101/130)
| 932.308
| 932.308
| [[12/7]] (-0.821 ¢)
| [[12/7]] (−0.821{{c}})
|-
|-
| 13 (114/130)
| 13 (114/130)
| 1052.308
| 1052.308
| [[11/6]] (+2.945 ¢)
| [[11/6]] (+2.945{{c}})
|-
|-
| 7 (121/130)
| 7 (121/130)
| 1116.923
| 1116.923
| [[21/11]] (-2.540 ¢)
| [[21/11]] (−2.540{{c}})
|-
|-
| 9 (130/130)
| 9 (130/130)
| 1200.000
| 1200.000
| [[Octave]] (2/1, ±0 ¢)
| [[Octave]] (2/1, 0{{c}})
|}
|}
== Instruments ==
[[Lumatone mapping for 130edo]]


== Music ==
== Music ==
* [http://www.archive.org/details/TheParadiseOfCantor The Paradise of Cantor] [http://www.archive.org/download/TheParadiseOfCantor/cantor.mp3 play] by [[Gene Ward Smith]]
{{Catrel|130edo tracks}}
* [https://youtu.be/30UQVYWnsDU "Narrative Wars"] by Sevish (uses a 14-tone (13 7 9 9 9 7 13 9 7 13 5 13 7 9) subset of 130-EDO, from the 2016 compilation album "Next Xen")    
 
; [[birdshite stalactite]]
* [https://www.youtube.com/watch?v=q41n5XI6YA4 ''wazzock''] (2024)
 
; [[Sevish]]
* [https://www.youtube.com/watch?v=30UQVYWnsDU Narrative Wars]
 
; [[Gene Ward Smith]]
* [https://www.archive.org/details/TheParadiseOfCantor ''The Paradise of Cantor''] [https://www.archive.org/download/TheParadiseOfCantor/cantor.mp3 play] (2006)


[[Category:Equal divisions of the octave]]
[[Category:Harry]]
[[Category:Harry]]
[[Category:Hemischismic]]
[[Category:Hemischis]]
[[Category:Hemiwürschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Sesquiquartififths]]
[[Category:Sesquiquartififths]]
[[Category:Zeta]]

Latest revision as of 08:47, 6 July 2026

← 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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