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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|130}}
{{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 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]].  
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]]. 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]]<nowiki/>s.
 
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]].  


=== Prime harmonics ===
=== Prime harmonics ===
Line 19: Line 21:
! 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 291: Line 293:


== Notation ==
== 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 ===
=== Sagittal notation ===
{| class="wikitable center-all"
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:
! Steps
{| class="wikitable" data-darkreader-inline-color="" style="text-align: center;"
| 0
! colspan="2" |Steps
| 1
!0
| 2
! 1
| 3
! 2
| 4
! 3
| 5
! 4
| 6
! 5
| 7
! 6
| 8
! 7
| 9
! 8
| 10
! 9
| 11
! 10
| 12
! 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>
|-
|-
! Symbol
!Revo
| [[File:Sagittal natural.png]]
|<big>{{sagittal|(|3=\}}</big>
| [[File:Sagittal nai.png]]
|<big>{{sagittal|)||(}}</big>
| [[File:Sagittal pai.png]]
|<big>{{sagittal|||)}}</big>
| [[File:Sagittal tai.png]]
|<big>{{sagittal|||\}}</big>
| [[File:Sagittal phai.png]]
|<big>{{sagittal|/||)}}</big>
| [[File:Sagittal patai.png]]
|<big>{{sagittal|/||\}}</big>
| [[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]]
|}
|}
Because it uses the entire Spartan extension, it allows no accidental enharmonic respellings.


== 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 378:
| 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 385:
| 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 392:
| 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 426:
| 83.08
| 83.08
| 21/20
| 21/20
| [[Sextilififths]]
| [[Sextilifourths]]
|-
|-
| 1
| 1
Line 478: Line 486:
| 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 498:
| 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 498: Line 506:
| [[Bosonic]]
| [[Bosonic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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 ==
Line 564: Line 572:
| [[Octave]] (2/1, 0{{c}})
| [[Octave]] (2/1, 0{{c}})
|}
|}
== Instruments ==
[[Lumatone mapping for 130edo]]


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/130edo"