111edo: Difference between revisions

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It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]].

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament, as well as [[warrior]], the {{nowrap| 46 & 65d }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===

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| 1

| 10.8

| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]

| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]

| {{UDnote|step=1}}

|-

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| 23

| 248.6

| [[15/13]]

| [[15/13]], [[22/19]]

| {{UDnote|step=23}}

|-

| 24

| 259.5

| ~~[[22/19]]~~

|

| {{UDnote|step=24}}

|-

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=== Interval mappings ===

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 655~~

~~| steps = 111.059577998833~~

~~| step size = 10.8050113427643~~

~~| tempered height = 9.038544~~

~~| pure height = 6.131468~~

~~| integral = 1.394739~~

~~| gap = 18.041165~~

~~| octave = 1199.35625904684~~

~~| consistent = 22~~

~~| distinct = 16~~

}}

== Regular temperament properties ==

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| 4/3<br>(18/17~19/18)

| [[Misty]]

|-

| 37

| 46\111<br>(1\111)

| 497.30<br>(10.81)

| 4/3<br>(169/168)

| [[Rubidium]]

|}

<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 ==

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[[Category:Listen]]

[[Category:Buzzard]]

[[Category:Warrior]]

[[Category:Semisept]]

[[Category:Orwellismic]]

[[Category:Guanyin]]

[[Category:Valinorsmic]]

@@ Line 9: / Line 9: @@
 It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]].
-It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.
+It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 & 80 }} temperament, and [[buzzard]], the {{nowrap| 53 & 58 }} temperament, as well as [[warrior]], the {{nowrap| 46 & 65d }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.
 === Prime harmonics ===
@@ Line 36: / Line 36: @@
 | 1
 | 10.8
-| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
+| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
 | {{UDnote|step=1}}
 |-
@@ Line 146: / Line 146: @@
 | 23
 | 248.6
-| [[15/13]]
+| [[15/13]], [[22/19]]
 | {{UDnote|step=23}}
 |-
 | 24
 | 259.5
-| [[22/19]]
+|
 | {{UDnote|step=24}}
 |-
@@ Line 319: / Line 319: @@
 === Interval mappings ===
 {{Q-odd-limit intervals}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 655
-| steps = 111.059577998833
-| step size = 10.8050113427643
-| tempered height = 9.038544
-| pure height = 6.131468
-| integral = 1.394739
-| gap = 18.041165
-| octave = 1199.35625904684
-| consistent = 22
-| distinct = 16
-}}
 == Regular temperament properties ==
@@ Line 535: / Line 521: @@
 | 4/3<br>(18/17~19/18)
 | [[Misty]]
+|-
+| 37
+| 46\111<br>(1\111)
+| 497.30<br>(10.81)
+| 4/3<br>(169/168)
+| [[Rubidium]]
 |}
-<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 551: / Line 543: @@
 [[Category:Listen]]
 [[Category:Buzzard]]
+[[Category:Warrior]]
 [[Category:Semisept]]
 [[Category:Orwellismic]]
 [[Category:Guanyin]]
 [[Category:Valinorsmic]]