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== 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 [[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 ===
{{Harmonics in equal|130|columns=9}}
{{Harmonics in equal|130|columns=11}}
{{Harmonics in equal|130|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 130edo (continued)}}
{{Harmonics in equal|130|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 130edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 130 factors into 2 × 5 × 13, 130edo has subset edos {{EDOs| 2, 5, 10, 13, 26, and 65 }}.
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.
[[260edo]], which divides the edostep in two, provides a strong correction for the 29th harmonic.
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 ===
{{Q-odd-limit intervals|130|23}}
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! 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 366:
| 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 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
| {{mapping| 130 206 302 365 450 }}
| {{Mapping| 130 206 302 365 450 }}
| −0.241
| −0.241
| 0.370
| 0.370
Line 384: 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
| {{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 414:
| 83.08
| 83.08
| 21/20
| 21/20
| [[Sextilififths]]
| [[Sextilifourths]]
|-
|-
| 1
| 1
Line 478: 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 490: 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 498: Line 494:
| [[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 560:
| [[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"