Pentacircle clan: Difference between revisions

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The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. But we can start with the rank-4 temperament.  
The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. This have the effect of identifying [[14/11]] at the [[81/64|Pythagorean major third]].  


Temperaments discussed elsewhere are:
For the rank-4 pentacircle temperament, see [[Rank-4 temperament #Pentacircle (896/891)]].  
* ''[[Melpomene]]'', {56/55, 81/80} → [[Didymus rank three family #Melpomene|Didymus rank-3 family]]
* ''[[Apollo]]'', {100/99, 225/224} → [[Marvel family #Apollo|Marvel family]]
* ''[[Julius]]'' or ''[[varda]]'', {176/175, 896/891} → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
* ''[[Parahemif]]'', {243/242, 896/891} → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]
* [[Sensamagic]], {245/243, 385/384} → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
* ''[[Pele]]'', {441/440, 896/891} → [[Hemifamity family #Pele|Hemifamity family]]
* ''[[Uni]]'', {540/539, 896/891} → [[Hemimage family #Uni|Hemimage family]]
 
Considered below, in addition to the no-5 subgroup temperament ''parapyth'', are ''tolerant'' and ''terrapyth''.
 
== Pentacircle ==
{{See also| Rank-4 temperament #Pentacircle (896/891) }}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 896/891
 
[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 1 }}]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
 
{{Val list|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
 
[[Badness]]: 0.0658 × 10<sup>-6</sup>


== Parapyth ==
== Parapyth ==
{{Main| Parapyth }}
{{Main| Parapyth }}


Parapyth is the no-5 [[subgroup temperament]] tempering out [[352/351]] and [[364/363]] in the 2.3.7.11.13 subgroup.  
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
[[Subgroup]]: 2.3.7.11
Line 45: Line 21:


[[Badness]]: 0.0205 × 10<sup>-3</sup>
[[Badness]]: 0.0205 × 10<sup>-3</sup>
=== 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 [[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.
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:
* ''[[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]]
* ''[[Canta]]'' → [[Canou family #Canta|Canou family]]
Considered below are tolerant, kujuku, and terrapyth.


=== 2.3.7.11.13 subgroup ===
=== 2.3.7.11.13 subgroup ===
Line 59: Line 58:
Badness: 0.101 × 10<sup>-3</sup>
Badness: 0.101 × 10<sup>-3</sup>


=== Etypyth ===
==== Etypyth ====
Subgroup: 2.3.7.11.13.17
Subgroup: 2.3.7.11.13.17


Line 71: Line 70:


Badness: 0.325 × 10<sup>-3</sup>
Badness: 0.325 × 10<sup>-3</sup>
== 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]]: [{{val| 1 0 -31 0 7 }}, {{val| 0 1 21 0 -4 }}, {{val| 0 0 0 1 1 }}]
: mapping generators: ~2, ~3, ~7
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217
{{Val list|legend=1| 17c, 29, 46, 92de, 121, 167, 288be }}
[[Badness]]: 5.35 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 9295/9261
Mapping: [{{val| 1 0 -31 0 7 12 }}, {{val| 0 1 0 21 0 4 -7 }}, {{val| 0 0 0 1 1 1 }}]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432
Optimal GPV sequence: {{Val list| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
Badness: 2.48 × 10<sup>-3</sup>
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 364/363, 442/441, 715/714
Mapping: [{{val| 1 0 -31 0 7 12 -13 }}, {{val| 0 1 0 21 0 4 -7 9 }}, {{val| 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: {{Val list| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
Badness: 1.52 × 10<sup>-3</sup>


== Tolerant ==
== Tolerant ==
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Badness: 1.24 × 10<sup>-3</sup>
Badness: 1.24 × 10<sup>-3</sup>
== 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]]: [{{val| 1 0 -31 0 7 }}, {{val| 0 1 21 0 -4 }}, {{val| 0 0 0 1 1 }}]
: mapping generators: ~2, ~3, ~7
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217
{{Val list|legend=1| 17c, 29, 46, 92de, 121, 167, 288be }}
[[Badness]]: 5.35 × 10<sup>-3</sup>
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 364/363, 9295/9261
Mapping: [{{val| 1 0 -31 0 7 12 }}, {{val| 0 1 0 11 7 4 -7 }}, {{val| 0 0 1 -2 -2 -2 1 }}]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432
Optimal GPV sequence: {{Val list| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
Badness: 2.48 × 10<sup>-3</sup>
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 364/363, 442/441, 715/714
Mapping: [{{val| 1 0 -31 0 7 12 -13 }}, {{val| 0 1 0 11 7 4 -7 9 }}, {{val| 0 0 1 -2 -2 -2 1 1 }}]
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620
Optimal GPV sequence: {{Val list| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
Badness: 1.52 × 10<sup>-3</sup>


[[Category:Temperament clans]]
[[Category:Temperament clans]]

Revision as of 12:34, 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 0 7], ⟨0 1 0 0 -4], ⟨0 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. 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 0 7 12], ⟨0 1 0 0 -4 -7], ⟨0 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 0 7 12 -13], ⟨0 1 0 0 -4 -7 9], ⟨0 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 = 951.8367, ~5/4 = 386.4048

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, ~121/70 = 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, ~121/70 = 951.8015, ~5/4 = 386.9912

Optimal GPV sequence: Template:Val list

Badness: 1.24 × 10-3

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