Olympic clan: Difference between revisions

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[[Comma list]]: 131072/130977

: mapping generators: ~2, ~3, ~7

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=== Overview to extensions ===

The second comma in the comma list determines how we extend olympian to include ~~the harmonic~~ 5. Akea adds [[385/384]], and finds ~~the harmonic~~ 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Orthoschismic adds [[32805/32768]], and finds ~~the harmonic~~ 5 on the chain of fifths. Cassaschismic adds [[19712/19683]] ~~with~~ an independent generator ~~for harmonic 5~~. Pessoal adds [[9801/9800]], splitting the octave into two. Lif adds [[2401/2400]], splitting the perfect fifth into two. Baffin adds [[5632/5625]], splitting the perfect twelfth into two. Lux adds [[3025/3024]], splitting the ~21/16 into two. Hera adds [[6144/6125]] or [[8019/8000]], splitting the ~21/16 into three. ~~Finally, sophia~~ adds 42875/42768, splitting the ~8/7 into three. ~~These all~~ have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]].

The second comma in the comma list determines how we extend olympian to include prime 5. Akea adds [[385/384]], and finds prime 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Orthoschismic adds [[32805/32768]], and finds prime 5 on the chain of fifths. Cassaschismic adds [[19712/19683]], giving prime 5 an independent generator.

Pessoal adds [[9801/9800]], splitting the octave into two. Lif adds [[2401/2400]], splitting the perfect fifth into two. Baffin adds [[5632/5625]], splitting the perfect twelfth into two. Lux adds [[3025/3024]], splitting the ~21/16 into two. Hera adds [[6144/6125]] or [[8019/8000]], splitting the ~21/16 into three. Sophia adds [[42875/42768]], splitting the ~8/7 into three. Finally, raglympic adds [[4375/4374]], splitting the ~27/7 into four.

All of them have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]].

Temperaments discussed elsewhere are:

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* ''[[Baffin]]'' (+5632/5625) → [[Vishdelismic clan #Baffin|Vishdelismic clan]]

* ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]

* ''[[Raglympic]]'' (+4375/4374) → [[Ragismic family #Raglympic|Ragismic family]]

* ''[[Orthoschismic]]'' (+540/539) → [[Schismic rank-3 family #Orthoschismic|Schismic rank-3 family]]

* [[Cassaschismic]] (+19712/19683) → [[Septischismic family #Undecimal septischismic (cassaschismic)|Septischismic family]]

Considered below are ~~[[#Orthoschismic|orthoschismic]],~~ [[#Pessoal|pessoal]], [[#Lux|lux]], and [[#Sophia|sophia]].

Considered below are [[#Pessoal|pessoal]], [[#Lux|lux]], and [[#Sophia|sophia]].

== Pessoal ==

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Comma list: 1716/1715, 2080/2079, 4096/4095

~~Mapping:~~ {{~~mapping~~| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}

{{Mapping|legend=0| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}

Optimal tunings:

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Comma list: 2080/2079, 3025/3024, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}

{{Mapping|legend=0| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}

Optimal tunings:

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Comma list: 2080/2079, 4096/4095, 13720/13689

~~Mapping:~~ {{~~mapping~~| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}

{{Mapping|legend=0| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}

Optimal tunings:

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Comma list: 595/594, 833/832, 1156/1155, 4096/4095

~~Mapping:~~ {{~~mapping~~| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}

{{Mapping|legend=0| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}

Optimal tunings:

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[[Category:Olympic clan| ]]

[[Category:Olympic| ]]

[[Category:Temperament clans]]

[[Category:Catalogs of rank-3 temperaments]]

@@ Line 9: / Line 9: @@
 [[Comma list]]: 131072/130977
-{{Mapping|legend=2| 1 0 0 17 | 0 1 0 -5 | 0 0 1 -2 }}
+{{Mapping|legend=3| 1 0 0 17 | 0 1 0 -5 | 0 0 1 -2 }}
 : mapping generators: ~2, ~3, ~7
@@ Line 23: / Line 23: @@
 === Overview to extensions ===
-The second comma in the comma list determines how we extend olympian to include the harmonic 5. Akea adds [[385/384]], and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Orthoschismic adds [[32805/32768]], and finds the harmonic 5 on the chain of fifths. Cassaschismic adds [[19712/19683]] with an independent generator for harmonic 5. Pessoal adds [[9801/9800]], splitting the octave into two. Lif adds [[2401/2400]], splitting the perfect fifth into two. Baffin adds [[5632/5625]], splitting the perfect twelfth into two. Lux adds [[3025/3024]], splitting the ~21/16 into two. Hera adds [[6144/6125]] or [[8019/8000]], splitting the ~21/16 into three. Finally, sophia adds 42875/42768, splitting the ~8/7 into three. These all have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]].
+The second comma in the comma list determines how we extend olympian to include prime 5. Akea adds [[385/384]], and finds prime 5 by equating the [[81/80|syntonic comma (81/80)]] with the septimal comma. Orthoschismic adds [[32805/32768]], and finds prime 5 on the chain of fifths. Cassaschismic adds [[19712/19683]], giving prime 5 an independent generator.
+Pessoal adds [[9801/9800]], splitting the octave into two. Lif adds [[2401/2400]], splitting the perfect fifth into two. Baffin adds [[5632/5625]], splitting the perfect twelfth into two. Lux adds [[3025/3024]], splitting the ~21/16 into two. Hera adds [[6144/6125]] or [[8019/8000]], splitting the ~21/16 into three. Sophia adds [[42875/42768]], splitting the ~8/7 into three. Finally, raglympic adds [[4375/4374]], splitting the ~27/7 into four.
+All of them have neat extensions to the 13-limit via tempering out both [[2080/2079]] and [[4096/4095]].
 Temperaments discussed elsewhere are:
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 * ''[[Baffin]]'' (+5632/5625) → [[Vishdelismic clan #Baffin|Vishdelismic clan]]
 * ''[[Hera]]'' (+6144/6125) → [[Porwell family #Hera|Porwell family]]
+* ''[[Raglympic]]'' (+4375/4374) → [[Ragismic family #Raglympic|Ragismic family]]
 * ''[[Orthoschismic]]'' (+540/539) → [[Schismic rank-3 family #Orthoschismic|Schismic rank-3 family]]
 * [[Cassaschismic]] (+19712/19683) → [[Septischismic family #Undecimal septischismic (cassaschismic)|Septischismic family]]
-Considered below are [[#Orthoschismic|orthoschismic]], [[#Pessoal|pessoal]], [[#Lux|lux]], and [[#Sophia|sophia]].
+Considered below are [[#Pessoal|pessoal]], [[#Lux|lux]], and [[#Sophia|sophia]].
 == Pessoal ==
@@ Line 62: / Line 67: @@
 Comma list: 1716/1715, 2080/2079, 4096/4095
-Mapping: {{mapping| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}
+{{Mapping|legend=0| 2 0 1 10 14 13 | 0 1 0 -1 -3 -1 | 0 0 3 -1 2 -2 }}
 Optimal tunings:
@@ Line 99: / Line 104: @@
 Comma list: 2080/2079, 3025/3024, 4096/4095
-Mapping: {{mapping| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}
+{{Mapping|legend=0| 1 0 -5 4 9 13 | 0 1 4 -1 -3 -5 | 0 0 5 2 -4 -7 }}
 Optimal tunings:
@@ Line 134: / Line 139: @@
 Comma list: 2080/2079, 4096/4095, 13720/13689
-Mapping: {{mapping| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}
+{{Mapping|legend=0| 1 0 2 3 11 7 | 0 1 0 0 -5 -2 | 0 0 5 -3 6 -2 }}
 Optimal tunings:
@@ Line 149: / Line 154: @@
 Comma list: 595/594, 833/832, 1156/1155, 4096/4095
-Mapping: {{mapping| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}
+{{Mapping|legend=0| 1 0 2 3 11 7 7 | 0 1 0 0 -5 -2 -2 | 0 0 5 -3 6 -2 4 }}
 Optimal tunings:
@@ Line 160: / Line 165: @@
 [[Category:Olympic clan| ]] <!-- main article -->
+[[Category:Olympic| ]] <!-- key article -->
 [[Category:Temperament clans]]
 [[Category:Catalogs of rank-3 temperaments]]

Olympic clan: Difference between revisions

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