Pentacircle clan: Difference between revisions

Line 12:

[[Comma list]]: 896/891

[[Sval]] [[mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 0 1 1 }}]

[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]

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

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=== Overview to extensions ===

==== Subgroup extensions ====

By tempering out 896/891, we have mapped 14/11 to the major third, suggesting a slightly sharp fifth. This makes the minor third very close to the flat-of-Pythagorean [[13/11]], and extending the temperament to include harmonic 13 this way implies we temper out [[352/351]]. In fact, 896/891 = (352/351)(364/363), so it is a very natural interpretation, giving rise to the 2.3.7.11.13 subgroup temperament shown below.

By tempering out 896/891, we have mapped 14/11 to the major third, suggesting a slightly sharp fifth. This makes the minor third very close to the flat-of-Pythagorean [[13/11]], and extending the temperament to include harmonic 13 this way implies we temper out [[352/351]]. In fact, 896/891 = (352/351)([[364/363]]), so it is a very natural interpretation, giving rise to the 2.3.7.11.13 subgroup temperament shown below.

==== Full 11-limit extensions ====

The second comma in the comma list determines how we extend parapyth to include the harmonic 5.

Pele adds [[441/440]] and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the [[64/63|septimal comma]]. Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds [[245/243]] or [[385/384]], a traditional RTT favorite. Apollo adds [[100/99]] or [[225/224]], and is even simpler than sensamagic. Uni adds [[540/539]]. Melpomene adds [[56/55]] or [[81/80]]. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth.

Pele adds [[441/440]] and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the [[64/63|septimal comma (64/63)]]. Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds [[245/243]] or [[385/384]], a traditional RTT favorite. Apollo adds [[100/99]] or [[225/224]], and is even simpler than sensamagic. Uni adds [[540/539]]. Melpomene adds [[56/55]] or [[81/80]]. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth.

Julius aka varda adds [[176/175]], splitting the octave into two. Parahemif adds [[243/242]], splitting the perfect fifth into two. Kujuku adds 14700/14641, splitting the perfect twelfth into two. Tolerant adds 2200/2187, splitting the ~33/32 into two. Finally, canta adds 472392/471625, splitting the ~14/9 into three.

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* ''[[Melpomene]]'' → [[Didymus rank three family #Melpomene|Didymus rank-3 family]]

* ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]

* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]

* ''[[Parahemif]]'' → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]

* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]

* ''[[Pele]]'' → [[Hemifamity family #Pele|Hemifamity family]]

* ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]]

* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]

* ''[[Parahemif]]'' → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]

* ''[[Canta]]'' → [[Canou family #Canta|Canou family]]

Line 50:

Comma list: 352/351, 364/363

Sval mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 -4 -7 }}, {{val| 0 0 0 1 1 1 }}]

Sval mapping: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8563, ~7/4 = 969.9074

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Comma list: 352/351, 364/363, 442/441

Sval mapping: [{{val| 1 0 0 0 7 12 -13 }}, {{val| 0 1 0 0 -4 -7 9 }}, {{val| 0 0 0 1 1 1 1 }}]

Sval mapping: [{{val| 1 0 0 7 12 -13 }}, {{val| 0 1 0 -4 -7 9 }}, {{val| 0 0 1 1 1 1 }}]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0315, ~7/4 = 970.6051

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: mapping generators: ~2, ~121/70, ~5

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.~~8367~~, ~5/4 = 386.~~4048~~

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.4956, ~5/4 = 386.7868

{{Val list|legend=1| 24, 29, 34d, 53d, 58, 87, 121, 145, 179e, 208, 266e }}

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Mapping: [{{val| 1 0 0 -13 -6 -1 }}, {{val| 0 2 0 17 9 3 }}, {{val| 0 0 1 1 1 1 }}]

Optimal tuning (CTE): ~2 = 1\1, ~~~121~~/70 = 951.8367, ~5/4 = 386.4048

Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8367, ~5/4 = 386.4048

Optimal GPV sequence: {{Val list| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}

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Mapping: [{{val| 1 0 0 -13 -6 -1 8 }}, {{val| 0 2 0 17 9 3 -2 }}, {{val| 0 0 1 1 1 1 -1 }}]

Optimal tuning (CTE): ~2 = 1\1, ~~~121~~/70 = 951.8015, ~5/4 = 386.9912

Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8015, ~5/4 = 386.9912

Optimal GPV sequence: {{Val list| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}

@@ Line 12: / Line 12: @@
 [[Comma list]]: 896/891
-[[Sval]] [[mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 0 1 1 }}]
+[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
 : sval mapping generators: ~2, ~3, ~7
@@ Line 24: / Line 24: @@
 === Overview to extensions ===
 ==== Subgroup extensions ====
-By tempering out 896/891, we have mapped 14/11 to the major third, suggesting a slightly sharp fifth. This makes the minor third very close to the flat-of-Pythagorean [[13/11]], and extending the temperament to include harmonic 13 this way implies we temper out [[352/351]]. In fact, 896/891 = (352/351)(364/363), so it is a very natural interpretation, giving rise to the 2.3.7.11.13 subgroup temperament shown below.
+By tempering out 896/891, we have mapped 14/11 to the major third, suggesting a slightly sharp fifth. This makes the minor third very close to the flat-of-Pythagorean [[13/11]], and extending the temperament to include harmonic 13 this way implies we temper out [[352/351]]. In fact, 896/891 = (352/351)([[364/363]]), so it is a very natural interpretation, giving rise to the 2.3.7.11.13 subgroup temperament shown below.
 ==== Full 11-limit extensions ====
 The second comma in the comma list determines how we extend parapyth to include the harmonic 5.
-Pele adds [[441/440]] and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the [[64/63|septimal comma]]. Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds [[245/243]] or [[385/384]], a traditional RTT favorite. Apollo adds [[100/99]] or [[225/224]], and is even simpler than sensamagic. Uni adds [[540/539]]. Melpomene adds [[56/55]] or [[81/80]]. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth.
+Pele adds [[441/440]] and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the [[64/63|septimal comma (64/63)]]. Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds [[245/243]] or [[385/384]], a traditional RTT favorite. Apollo adds [[100/99]] or [[225/224]], and is even simpler than sensamagic. Uni adds [[540/539]]. Melpomene adds [[56/55]] or [[81/80]]. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth.
 Julius aka varda adds [[176/175]], splitting the octave into two. Parahemif adds [[243/242]], splitting the perfect fifth into two. Kujuku adds 14700/14641, splitting the perfect twelfth into two. Tolerant adds 2200/2187, splitting the ~33/32 into two. Finally, canta adds 472392/471625, splitting the ~14/9 into three.
@@ Line 36: / Line 36: @@
 * ''[[Melpomene]]'' → [[Didymus rank three family #Melpomene|Didymus rank-3 family]]
 * ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]
-* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
-* ''[[Parahemif]]'' → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]
 * [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
 * ''[[Pele]]'' → [[Hemifamity family #Pele|Hemifamity family]]
 * ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]]
+* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
+* ''[[Parahemif]]'' → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]
 * ''[[Canta]]'' → [[Canou family #Canta|Canou family]]
@@ Line 50: / Line 50: @@
 Comma list: 352/351, 364/363
-Sval mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 -4 -7 }}, {{val| 0 0 0 1 1 1 }}]
+Sval mapping: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
 Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8563, ~7/4 = 969.9074
@@ Line 63: / Line 63: @@
 Comma list: 352/351, 364/363, 442/441
-Sval mapping: [{{val| 1 0 0 0 7 12 -13 }}, {{val| 0 1 0 0 -4 -7 9 }}, {{val| 0 0 0 1 1 1 1 }}]
+Sval mapping: [{{val| 1 0 0 7 12 -13 }}, {{val| 0 1 0 -4 -7 9 }}, {{val| 0 0 1 1 1 1 }}]
 Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0315, ~7/4 = 970.6051
@@ Line 180: / Line 180: @@
 : mapping generators: ~2, ~121/70, ~5
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.8367, ~5/4 = 386.4048
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.4956, ~5/4 = 386.7868
 {{Val list|legend=1| 24, 29, 34d, 53d, 58, 87, 121, 145, 179e, 208, 266e }}
@@ Line 193: / Line 193: @@
 Mapping: [{{val| 1 0 0 -13 -6 -1 }}, {{val| 0 2 0 17 9 3 }}, {{val| 0 0 1 1 1 1 }}]
-Optimal tuning (CTE): ~2 = 1\1, ~121/70 = 951.8367, ~5/4 = 386.4048
+Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8367, ~5/4 = 386.4048
 Optimal GPV sequence: {{Val list| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}
@@ Line 208: / Line 208: @@
 Mapping: [{{val| 1 0 0 -13 -6 -1 8 }}, {{val| 0 2 0 17 9 3 -2 }}, {{val| 0 0 1 1 1 1 -1 }}]
-Optimal tuning (CTE): ~2 = 1\1, ~121/70 = 951.8015, ~5/4 = 386.9912
+Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8015, ~5/4 = 386.9912
 Optimal GPV sequence: {{Val list| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}