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'''130edo''' divides the octave into 130 parts of size 9.231 cents each.
{{Infobox ET}}
{{ED intro}}


130edo is the tenth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|zeta integral edo]] but not a gap edo. It can be used to tune a variety of temperaments, including [[hemiwürschmidt]], [[sesquiquartififths]], [[harry]] and [[hemischismic]]. It also can be used to tune the rank-three temperament [[Breed_family#Jove, aka Wonder|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.
== 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 [[trivial temperament|nontrivial edo]] to be consistent in the 14-[[odd prime sum limit|odd-prime-sum-limit]]. 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 [[Schismatic family #Sesquiquartififths|sesquart]] and 13-limit [[harry]].  


7-limit commas: 2401/2400, 3136/3125, 19683/19600
=== Prime harmonics ===
{{Harmonics in equal|130|columns=9}}
{{Harmonics in equal|130|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 130edo (continued)}}


11-limit commas: 441/440, 540/539, 3136/3125, 4000/3993
=== Subsets and supersets ===
Since 130 factors into 2 × 5 × 13, 130edo has subset edos {{EDOs| 2, 5, 10, 13, 26, and 65 }}.


13-limit commas: 3136/3125, 243/242, 441/440, 351/350, 364/363
[[260edo]], which divides the edostep in two, provides a strong correction for the 29th harmonic.
 
17-limit commas: 221/220, 364/363, 442/441, 595/594, 1275/1274, 4913/4875


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degree
! Degree
! Cents
! Cents
! Associated Temperament
! Approximate ratios
|-
|-
| 0
| 0
| 0.000
| 0.00
|  
| 1/1
|-
|-
| 1
| 1
| 9.231
| 9.23
|  
| ''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
|-
|-
| 3
| 3
| 27.692
| 27.69
|  
| 56/55, 64/63, 65/64, 66/65
|-
|-
| 4
| 4
| 36.923
| 36.92
|  
| 45/44, 49/48, 50/49, ''55/54''
|-
|-
| 5
| 5
| 46.154
| 46.15
|  
| 36/35, 40/39
|-
|-
| 6
| 6
| 55.385
| 55.38
|  
| 33/32
|-
|-
| 7
| 7
| 64.615
| 64.62
|  
| 27/26, 28/27
|-
|-
| 8
| 8
| 73.846
| 73.85
|  
| 25/24, 26/25
|-
|-
| 9
| 9
| 83.077
| 83.08
| [[Harry]]
| 21/20, 22/21
|-
|-
| 10
| 10
| 92.308
| 92.31
|  
| 135/128
|-
|-
| 11
| 11
| 101.538
| 101.54
|  
| 35/33
|-
|-
| 12
| 12
| 110.769
| 110.77
|  
| 16/15
|-
|-
| 13
| 13
| 120.000
| 120.00
|  
| 15/14
|-
|-
| 14
| 14
| 129.231
| 129.23
|  
| 14/13
|-
|-
| 15
| 15
| 138.462
| 138.46
|  
| 13/12
|-
|-
| 16
| 16
| 147.692
| 147.69
|  
| 12/11
|-
|-
| 17
| 17
| 156.923
| 156.92
|  
| 35/32
|-
|-
| 18
| 18
| 166.154
| 166.15
|  
| 11/10
|-
|-
| 19
| 19
| 175.385
| 175.38
| [[Schismatic_family #Sesquiquartififths|Sesquart]]
| 72/65
|-
|-
| 20
| 20
| 184.615
| 184.62
|  
| 10/9
|-
|-
| 21
| 21
| 193.846
| 193.85
| [[Hemiwürschmidt]]
| 28/25
|-
|-
| 22
| 22
| 203.077
| 203.08
|  
| 9/8
|-
|-
| 23
| 23
| 212.308
| 212.31
|  
| 44/39
|-
|-
| 24
| 24
| 221.538
| 221.54
|  
| 25/22
|-
|-
| 25
| 25
| 230.769
| 230.77
|  
| 8/7
|-
|-
| 26
| 26
| 240.000
| 240.00
|  
| 55/48
|-
|-
| 27
| 27
| 249.231
| 249.23
| [[Hemischismic]]
| 15/13
|-
|-
| 28
| 28
| 258.462
| 258.46
|  
| 64/55
|-
|-
| 29
| 29
| 267.692
| 267.69
|  
| 7/6
|-
|-
| 30
| 30
| 276.923
| 276.92
|  
| 75/64
|-
|-
| 31
| 31
| 286.154
| 286.15
|  
| 13/11
|-
|-
| 32
| 32
| 295.385
| 295.38
|  
| 32/27
|-
|-
| 33
| 33
| 304.615
| 304.62
|  
| 25/21
|-
|-
| 34
| 34
| 313.846
| 313.85
|  
| 6/5
|-
|-
| 35
| 35
| 323.077
| 323.08
|  
| 65/54
|-
|-
| 36
| 36
| 332.308
| 332.31
|  
| 40/33
|-
|-
| 37
| 37
| 341.538
| 341.54
|  
| 39/32
|-
|-
| 38
| 38
| 350.769
| 350.77
|  
| 11/9, 27/22
|-
|-
| 39
| 39
| 360.000
| 360.00
|  
| 16/13
|-
|-
| 40
| 40
| 369.231
| 369.23
|  
| 26/21
|-
|-
| 41
| 41
| 378.462
| 378.46
|  
| 56/45
|-
|-
| 42
| 42
| 387.692
| 387.69
|  
| 5/4
|-
|-
| 43
| 43
| 396.923
| 396.92
|  
| 44/35
|-
|-
| 44
| 44
| 406.154
| 406.15
|  
| 81/64
|-
|-
| 45
| 45
| 415.385
| 415.38
|  
| 14/11
|-
|-
| 46
| 46
| 424.615
| 424.62
|  
| 32/25
|-
|-
| 47
| 47
| 433.846
| 433.85
|  
| 9/7
|-
|-
| 48
| 48
| 443.077
| 443.08
|  
| 84/65, 128/99
|-
|-
| 49
| 49
| 452.308
| 452.31
|  
| 13/10
|-
|-
| 50
| 50
| 461.538
| 461.54
|  
| 64/49, ''72/55''
|-
|-
| 51
| 51
| 470.769
| 470.77
|  
| 21/16
|-
|-
| 52
| 52
| 480.000
| 480.00
|  
| 33/25
|-
|-
| 53
| 53
| 489.231
| 489.23
|  
| 65/49
|-
|-
| 54
| 54
| 498.462
| 498.46
|  
| 4/3
|-
|-
| 55
| 55
| 507.692
| 507.69
|  
| 75/56
|-
|-
| 56
| 56
| 516.923
| 516.92
|  
| 27/20
|-
|-
| 57
| 57
| 526.154
| 526.15
|  
| 65/48
|-
|-
| 58
| 58
| 535.385
| 535.38
|  
| 15/11
|-
|-
| 59
| 59
| 544.615
| 544.62
|  
| 48/35
|-
|-
| 60
| 60
| 553.846
| 553.85
|  
| 11/8
|-
|-
| 61
| 61
| 563.077
| 563.08
|  
| 18/13
|-
|-
| 62
| 62
| 572.308
| 572.31
|  
| 25/18
|-
|-
| 63
| 63
| 581.538
| 581.54
|  
| 7/5
|-
|-
| 64
| 64
| 590.769
| 590.77
|  
| 45/32
|-
|-
| 65
| 65
| 600.000
| 600.00
|  
| 99/70, 140/99
|-
|-
|…
|…
Line 287: Line 289:
|…
|…
|}
|}
== Notation ==
=== Sagittal notation ===
{| class="wikitable center-all"
! Steps
| 0
| 1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
| 11
| 12
|-
! Symbol
| [[File:Sagittal natural.png]]
| [[File:Sagittal nai.png]]
| [[File:Sagittal pai.png]]
| [[File:Sagittal tai.png]]
| [[File:Sagittal phai.png]]
| [[File:Sagittal patai.png]]
| [[File:Sagittal pakai.png]]
| [[File:Sagittal jakai.png]]
| [[File:Sagittal sharp phao.png]]
| [[File:Sagittal sharp tao.png]]
| [[File:Sagittal sharp pao.png]]
| [[File:Sagittal sharp nao.png]]
| [[File:Sagittal sharp.png]]
|}
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 796
| steps = 130.003910460506
| step size = 9.23049157328654
| tempered height = 10.355108
| pure height = 10.339572
| integral = 1.634018
| gap = 19.594551
| octave = 1199.96390452725
| consistent = 16
| distinct = 16
}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2401/2400, 3136/3125, 19683/19600
| {{Mapping| 130 206 302 365 }}
| −0.119
| 0.311
| 3.37
|-
| 2.3.5.7.11
| 243/242, 441/440, 3136/3125, 4000/3993
| {{Mapping| 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
| {{Mapping| 130 206 302 365 450 481 }}
| −0.177
| 0.367
| 3.98
|}
=== Rank-2 temperaments ===
Note: temperaments supported by [[65edo|65et]] are not included.
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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<br>(11\130)
| 498.46<br>(101.54)
| 4/3<br>(35/33)
| [[Bischismic]]
|-
| 5
| 27\130<br>(1\130)
| 249.23<br>(9.23)
| 81/70<br>(176/175)
| [[Hemiquintile]]
|-
| 10
| 27\130<br>(1\130)
| 249.23<br>(9.23)
| 15/13<br>(176/175)
| [[Decoid]]
|-
| 10
| 54\130<br>(2\130)
| 498.46<br>(18.46)
| 4/3<br>(81/80)
| [[Decile]]
|-
| 26
| 54\130<br>(1\130)
| 498.46<br>(9.23)
| 4/3<br>(225/224)
| [[Bosonic]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Scales ==
{| class="wikitable"
|+ style="font-size: 105%;" | 14-tone temperament of "Narrative Wars"<br />as an example of using 130edo:
|-
! Step
! Cents
! Distance to the nearest JI interval<br />(selected ratios)
|-
| 13 (13/130)
| 120.000
| [[15/14]] (+0.557{{c}})
|-
| 7 (20/130)
| 184.615
| [[10/9]] (+2.211{{c}})
|-
| 9 (29/130)
| 267.692
| [[7/6]] (+0,821{{c}})
|-
| 9 (38/130)
| 350.769
| [[11/9]] (+3.361{{c}})
|-
| 9 (47/130)
| 433.846
| [[9/7]] (−1.238{{c}})
|-
| 7 (54/130)
| 498.462
| [[4/3]] (+0.417{{c}})
|-
| 13 (67/130)
| 618.462
| [[10/7]] (+0.974{{c}})
|-
| 9 (76/130)
| 701.538
| [[3/2]] (−0.417{{c}})
|-
| 7 (83/130)
| 766.154
| [[14/9]] (+1.238{{c}})
|-
| 13 (96/130)
| 886.154
| [[5/3]] (+1.795{{c}})
|-
| 5 (101/130)
| 932.308
| [[12/7]] (−0.821{{c}})
|-
| 13 (114/130)
| 1052.308
| [[11/6]] (+2.945{{c}})
|-
| 7 (121/130)
| 1116.923
| [[21/11]] (−2.540{{c}})
|-
| 9 (130/130)
| 1200.000
| [[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}}
 
; [[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]]
Retrieved from "https://en.xen.wiki/w/130edo"