Pentacircle clan: Difference between revisions

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[[Comma list]]: 896/891
[[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
: sval mapping generators: ~2, ~3, ~7
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=== Overview to extensions ===
=== Overview to extensions ===
==== Subgroup 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 ====
==== Full 11-limit extensions ====
The second comma in the comma list determines how we extend parapyth to include the harmonic 5.  
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.  
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]]
* ''[[Melpomene]]'' → [[Didymus rank three family #Melpomene|Didymus rank-3 family]]
* ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel 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]]
* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
* ''[[Pele]]'' → [[Hemifamity family #Pele|Hemifamity family]]
* ''[[Pele]]'' → [[Hemifamity family #Pele|Hemifamity family]]
* ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage 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]]
* ''[[Canta]]'' → [[Canou family #Canta|Canou family]]


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Comma list: 352/351, 364/363
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
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
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
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0315, ~7/4 = 970.6051
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: mapping generators: ~2, ~121/70, ~5
: 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 }}
{{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 }}]
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 }}
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 }}]
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 }}
Optimal GPV sequence: {{Val list| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}

Revision as of 13:26, 3 May 2023

The pentacircle clan of rank-3 temperaments tempers out the pentacircle comma, 896/891. This have the effect of identifying 14/11 at the Pythagorean major third.

For the rank-4 pentacircle temperament, see Rank-4 temperament #Pentacircle (896/891).

Parapyth

Main article: Parapyth

Parapyth, by the original definition, is the 2.3.7.11.13 subgroup temperament tempering out 352/351 and 364/363. We begin by looking at the 2.3.7.11 restriction thereof.

Subgroup: 2.3.7.11

Comma list: 896/891

Sval mapping: [⟨1 0 0 7], ⟨0 1 0 -4], ⟨0 0 1 1]]

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8345, ~7/4 = 969.8722

Template:Val list

Badness: 0.0205 × 10-3

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.

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 syntonic comma (81/80) with the 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.

Temperaments discussed elsewhere are:

Considered below are tolerant, kujuku, and terrapyth.

2.3.7.11.13 subgroup

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

Sval mapping: [⟨1 0 0 7 12], ⟨0 1 0 -4 -7], ⟨0 0 1 1 1]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.101 × 10-3

Etypyth

Subgroup: 2.3.7.11.13.17

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

Sval mapping: [⟨1 0 0 7 12 -13], ⟨0 1 0 -4 -7 9], ⟨0 0 1 1 1 1]]

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

Optimal GPV sequence: Template:Val list

Badness: 0.325 × 10-3

Terrapyth

Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 585640/583443

Mapping: [⟨1 0 -31 0 7], ⟨0 1 21 0 -4], ⟨0 0 0 1 1]]

mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217

Template:Val list

Badness: 5.35 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 9295/9261

Mapping: [⟨1 0 -31 0 7 12], ⟨0 1 0 21 0 4 -7], ⟨0 0 0 1 1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432

Optimal GPV sequence: Template:Val list

Badness: 2.48 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 364/363, 442/441, 715/714

Mapping: [⟨1 0 -31 0 7 12 -13], ⟨0 1 0 21 0 4 -7 9], ⟨0 0 0 1 1 1 1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620

Optimal GPV sequence: Template:Val list

Badness: 1.52 × 10-3

Tolerant

7-limit

Subgroup: 2.3.5.7

Comma list: 179200/177147

Mapping: [⟨1 0 0 -10], ⟨0 1 0 11], ⟨0 0 1 -2]]

mapping generators: ~2, ~3, ~5

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9571, ~5/4 = 386.8863

Template:Val list

Badness: 0.165 × 10-3

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187

Mapping: [⟨1 0 0 -10 -3], ⟨0 1 0 11 7], ⟨0 0 1 -2 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0412, ~5/4 = 387.2927

Template:Val list

Badness: 1.039 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363

Mapping: [⟨1 0 0 -10 -3 2], ⟨0 1 0 11 7 4], ⟨0 0 1 -2 -2 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9605, ~5/4 = 386.9831

Optimal GPV sequence: Template:Val list

Badness: 1.021 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363

Mapping: [⟨1 0 0 -10 -3 2 8], ⟨0 1 0 11 7 4 -1], ⟨0 0 1 -2 -2 -2 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0831, ~5/4 = 387.3269

Optimal GPV sequence: Template:Val list

Badness: 0.982 × 10-3

Kujuku

Kujuku splits the perfect twelfth into two. Scott Dakota has aliased this temperament SQPP (for semiquartal parapyth).

Subgroup: 2.3.5.7.11

Comma list: 896/891, 14700/14641

Mapping: [⟨1 0 0 -13 -6], ⟨0 2 0 17 9], ⟨0 0 1 1 1]]

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

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

Template:Val list

Badness: 2.26 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 676/675

Mapping: [⟨1 0 0 -13 -6 -1], ⟨0 2 0 17 9 3], ⟨0 0 1 1 1 1]]

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

Optimal GPV sequence: Template:Val list

Badness: 1.06 × 10-3

Complexity spectrum: 15/13, 4/3, 13/10, 9/8, 13/11, 15/11, 12/11, 11/9, 11/8, 14/11, 16/13, 16/15, 11/10, 13/12, 9/7, 5/4, 18/13, 7/6, 6/5, 8/7, 10/9, 14/13, 15/14, 7/5

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 676/675

Mapping: [⟨1 0 0 -13 -6 -1 8], ⟨0 2 0 17 9 3 -2], ⟨0 0 1 1 1 1 -1]]

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

Optimal GPV sequence: Template:Val list

Badness: 1.24 × 10-3

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