Pentacircle clan: Difference between revisions
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The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. | 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]]. | ||
For the rank-4 pentacircle temperament, see [[Rank-4 temperament #Pentacircle (896/891)]]. | |||
== Parapyth == | == Parapyth == | ||
{{Main| Parapyth }} | {{Main| Parapyth }} | ||
Parapyth is the | 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 | ||
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[[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 === | ||
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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 | ||
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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> | ||
[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||