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~~]]. 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 ===

{{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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=== 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>

|{{Sagittal|t}}<big>{{sagittal||(}}</big>

|{{Sagittal|t}}<big>{{sagittal|/|}}</big>

|{{Sagittal|t}}<big>{{sagittal||)}}</big>

|{{Sagittal|t}}<big>{{sagittal|//|}}</big>

|{{Sagittal|t}}<big>{{sagittal|/|)}}</big>

| rowspan="2" |<big>{{sagittal|#}}</big>

|-

!Evo

| rowspan="2" |<big>{{sagittal|/|\}}</big>

|{{sagittal|#}}<big>{{sagittal|\!)}}</big>

|{{sagittal|#}}<big>{{sagittal|\\!}}</big>

|{{sagittal|#}}<big>{{sagittal|!)}}</big>

|{{sagittal|#}}<big>{{sagittal|\!}}</big>

|{{sagittal|#}}<big>{{sagittal|!(}}</big>

|-

! ~~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 ==

~~=== 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]]. 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]].
+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 [[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 295: / Line 297: @@
 {{Ups and downs sharpness|130|true}}
 === Sagittal notation ===
-{| class="wikitable center-all"
+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:
-! 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 ==
-=== 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 484: / 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 ==