130edo: Difference between revisions

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== 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]]. 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.

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 [[~~Schismatic family #Sesquiquartififths|~~sesquart]] and 13-limit [[harry]].

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 ===

{{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 ===

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.

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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;"

!~~Degree~~

! colspan="2" |Steps

|0

!0

~~| +~~1

! 1

~~| +~~2

! 2

~~| +~~3

! 3

~~| +~~4

! 4

~~| +~~5

! 5

~~| +~~6

! 6

~~| +~~7

! 7

~~| +~~8

! 8

~~| +~~9

! 9

~~| +~~10

! 10

~~| +~~11

! 11

~~| +~~12

! 12

|-

! rowspan="3" |Symbol

!Evo+SZ

| rowspan="3" |<big>{{sagittal||//|}}</big>

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|}

*

Because it uses the entire Spartan extension, it allows no accidental enharmonic respellings.

~~See [[Sagittal notation#Revo|apotome complements]] for equivalent~~ accidental ~~pairs~~.

== 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 ==

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| [[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 ==

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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.
+edo 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 [[Schismatic family #Sesquiquartififths|sesquart]] and 13-limit [[harry]].
+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 ===
-{{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 ===
-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.
@@ Line 299: / Line 299: @@
 edo 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;"
-!Degree
+! colspan="2" |Steps
-|0
+!0
-| +1
+! 1
-| +2
+! 2
-| +3
+! 3
-| +4
+! 4
-| +5
+! 5
-| +6
+! 6
-| +7
+! 7
-| +8
+! 8
-| +9
+! 9
-| +10
+! 10
-| +11
+! 11
-| +12
+! 12
 |-
+! rowspan="3" |Symbol
 !Evo+SZ
 | rowspan="3" |<big>{{sagittal||//|}}</big>
@@ Line 346: / Line 347: @@
 |}
-*
+Because it uses the entire Spartan extension, it allows no accidental enharmonic respellings.
-See [[Sagittal notation#Revo|apotome complements]] for equivalent accidental pairs.
 == Approximation to JI ==
-=== Zeta peak index ===
+{{Q-odd-limit intervals|130|23}}
-{{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 ==
@@ Line 507: / Line 494: @@
 | [[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 ==