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Pentacircle clan: Difference between revisions

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[[Comma list]]: 896/891
[[Comma list]]: 896/891


{{Mapping|legend=2| 1 0 0 7 | 0 1 0 -4 | 0 0 1 1 }}
{{Mapping|legend=3| 1 0 0 7 | 0 1 0 -4 | 0 0 1 1 }}
: mapping generators: ~2, ~3, ~7
: mapping generators: ~2, ~3, ~7


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* ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]
* ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]
* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
* [[Pele]] → [[Hemifamity family #Pele|Hemifamity family]]
* [[Pele]] → [[Aberschismic family #Pele|Aberschismic family]]
* ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]]
* ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]]
* ''[[Varda]]'' → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Varda]]'' → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif|Rastmic rank-3 clan]]
* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif|Rastmic rank-3 clan]]
* ''[[Canta]]'' → [[Canou family #Canta|Canou family]]
* ''[[Canta]]'' → [[Cathartic family #Canta|Cathartic family]]


Considered below are tolerant, kujuku, and terrapyth.  
Considered below are tolerant, kujuku, and terrapyth.  
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Comma list: 352/351, 364/363
Comma list: 352/351, 364/363


Subgroup-val mapping: {{mapping| 1 0 0 7 12 | 0 1 0 -4 -7 | 0 0 1 1 1 }}
{{Mapping|legend=2| 1 0 0 7 12 | 0 1 0 -4 -7 | 0 0 1 1 1 }}


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


Subgroup-val mapping: {{mapping| 1 0 0 7 12 -13 | 0 1 0 -4 -7 9 | 0 0 1 1 1 1 }}
{{Mapping|legend=2| 1 0 0 7 12 -13 | 0 1 0 -4 -7 9 | 0 0 1 1 1 1 }}


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


Mapping: {{mapping| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}
{{Mapping|legend=0| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}


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


Mapping: {{mapping| 1 0 -31 0 7 12 -13 | 0 1 0 21 0 4 -7 9 | 0 0 0 1 1 1 1 1 }}
{{Mapping|legend=0| 1 0 -31 0 7 12 -13 | 0 1 0 21 0 4 -7 9 | 0 0 0 1 1 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
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== Pentafrost ==
== Pentafrost ==
Pentafrost tempers out the [[245/242|frostma]] in addition to 896/891 which also means that the [[schisma]] is tempered out, mapping prime 5 to 8 [[4/3|perfect fourths]] and -1 octaves.
Pentafrost tempers out the [[245/242|frostma]] in addition to 896/891 which also means that the [[schisma]] is tempered out, mapping prime 5 to eight [[4/3|perfect fourths]] minus an octave.
 
It was named by [[Tristan Bay]] in 2024 as a portmanteau of ''pentacircle'' and ''frost''.  


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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Comma list: 105/104, 245/242, 352/351
Comma list: 105/104, 245/242, 352/351


Mapping: {{mapping| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }}
{{Mapping|legend=0| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }}


Optimal tunings:
Optimal tunings:
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Comma list: 144/143, 245/242, 896/891
Comma list: 144/143, 245/242, 896/891


Mapping: {{mapping| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }}
{{Mapping|legend=0| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }}


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


Mapping: {{mapping| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}
{{Mapping|legend=0| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}


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


Mapping: {{mapping| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}
{{Mapping|legend=0| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}


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


Mapping: {{mapping| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}
{{Mapping|legend=0| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 0.991
Badness (Sintel): 0.991
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 ===
=== 17-limit ===
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Comma list: 256/255, 352/351, 364/363, 676/675
Comma list: 256/255, 352/351, 364/363, 676/675


Mapping: {{mapping| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}
{{Mapping|legend=0| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}


Optimal tunings:  
Optimal tunings:  
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== Trienparapyth ==
== Trienparapyth ==
Trienparapyth can be described as the no-17's 23-limit {{nowrap| 80 & 87 & 109 }} temperament. It splits the ~4/3 generator of [[#Parapythic|parapythic]] into three ~[[11/10]]'s by tempering out [[4000/3993|4000/3993 = S10/S11]] in the 11-limit and it interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299|2300/2299 = S20/S22]], or optionally less accurately as ~[[17/14]], though because this mapping only really makes much sense in [[80edo]] it is not included here; however, its connection to parapyth structure comes from later in the generator chain; specifically, from (11/10)<sup>7</sup> onwards. We may simplify (11/10)<sup>7</sup> as [[16/9|(4/3)<sup>2</sup>]]([[11/10]]) = [[88/45]], the octave-complement of [[45/44]]. Notice that parapythic wants a slightly flat ~4/3 corresponding to an 11/10 being tuned anywhere from around just (in an extremely sharp-for-parapyth tuning) to a little less than 1-cent sharp, a very narrow tuning range; therefore 88/45 is flattened so that 2/(11/10)<sup>7</sup>~45/44 is sharpened so that we can equate it with [[40/39]], tempering out (40/39)/(45/44) = [[352/351]], and because we know we want prime 19 later on, we equate this with [[39/38]] by tempering out the pinkanberry, [[1521/1520|1521/1520 = S39]]. Next, for eight generator steps, observe that (11/10)<sup>9</sup>/(11/10)/2 = (4/3)<sup>3</sup>/(11/10)/2 = ([[32/27]])/(11/10) = 320/297 is sharp of [[15/14]] by [[896/891]], which is reasonable to equate it with because in an optimal tuning 11/10 will be very slightly sharp so that the interval of eight generator steps is eight times as sharp. Thus, tempering out [[896/891]] and [[4000/3993]] defines trienparapyth in the 11-limit, which also tempers out [[3388/3375]], the 13-limit adds [[352/351]], the no-17's 19-limit equates 40/39 with 39/38 and the no-17's 23-limit equates 23/19 with (11/10)<sup>2</sup> as already mentioned.
Named by [[Godtone]] in 2024, trienparapyth can be described as the {{nowrap| 58 & 80 & 87 }} temperament, with an extension to the no-17's 23-limit. It splits the ~4/3 generator of parapythic into three [[~]][[11/10]]'s by tempering out [[4000/3993]] ([[S-expression|S10/S11]]) in the 11-limit. It further interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]] ([[S-expression|S20/S22]]), or optionally less accurately as ~[[17/14]], though because this mapping only really makes much sense in [[80edo]] it is not included here; however, its connection to parapyth structure comes from later in the generator chain; specifically, from (11/10)<sup>7</sup> onwards. We may simplify (11/10)<sup>7</sup> as [[16/9|(4/3)<sup>2</sup>]]([[11/10]]) = [[88/45]], the octave-complement of [[45/44]]. Notice that parapythic wants a slightly flat ~4/3 corresponding to an 11/10 being tuned anywhere from around just (in an extremely sharp-for-parapyth tuning) to a little less than 1-cent sharp, a very narrow tuning range; therefore 88/45 is flattened so that 2/(11/10)<sup>7</sup>~45/44 is sharpened so that we can equate it with [[40/39]], tempering out (40/39)/(45/44) = [[352/351]], and because we know we want prime 19 later on, we equate this with [[39/38]] by tempering out the pinkanberry, [[1521/1520]] ({{S|39}}). Next, for eight generator steps, observe that (11/10)<sup>9</sup>/(11/10)/2 = (4/3)<sup>3</sup>/(11/10)/2 = ([[32/27]])/(11/10) = 320/297 is sharp of [[15/14]] by [[896/891]], which is reasonable to equate it with because in an optimal tuning 11/10 will be very slightly sharp so that the interval of eight generator steps is eight times as sharp. Thus, tempering out [[896/891]] and [[4000/3993]] defines trienparapyth in the 11-limit, which also tempers out [[3388/3375]], the 13-limit adds [[352/351]], the no-17's 19-limit equates 40/39 with 39/38 and the no-17's 23-limit equates 23/19 with (11/10)<sup>2</sup> as already mentioned.


Structurally, trienparapyth is three copies of parapyth with the independent generator of 7 connected to an equivalent independent generator for 5 through the ~[[15/7]] reached at (11/10)<sup>8</sup> so that ~[[20/7]] is reached at (11/10)<sup>11</sup>, and this is how the last generator can be either 5 or 7.
Structurally, trienparapyth is three copies of parapyth with the independent generator of 7 connected to an equivalent independent generator for 5 through the ~[[15/7]] reached at (11/10)<sup>8</sup> so that ~[[20/7]] is reached at (11/10)<sup>11</sup>, and this is how the last generator can be either 5 or 7.
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Comma list: 352/351, 364/363, 1001/1000
Comma list: 352/351, 364/363, 1001/1000


Mapping: {{mapping| 1 2 0 2 1 0 | 0 -3 0 -11 1 10 | 0 0 1 1 1 1 }}
{{Mapping|legend=0| 1 2 0 2 1 0 | 0 -3 0 -11 1 10 | 0 0 1 1 1 1 }}
: mapping generators: ~2, ~11/10, ~5
: mapping generators: ~2, ~11/10, ~5


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Comma list: 286/285, 352/351, 364/363, 400/399
Comma list: 286/285, 352/351, 364/363, 400/399


Mapping: {{mapping| 1 2 0 2 1 0 0 | 0 -3 0 -11 1 10 14 | 0 0 1 1 1 1 1 }}
{{Mapping|legend=0| 1 2 0 2 1 0 0 | 0 -3 0 -11 1 10 14 | 0 0 1 1 1 1 1 }}
: mapping generators: ~2, ~11/10, ~5
: mapping generators: ~2, ~11/10, ~5


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Comma list: 208/207, 286/285, 352/351, 364/363, 400/399
Comma list: 208/207, 286/285, 352/351, 364/363, 400/399


Mapping: {{mapping| 1 2 0 2 1 0 0 0 | 0 -3 0 -11 1 10 14 16 | 0 0 1 1 1 1 1 1 }}
{{Mapping|legend=0| 1 2 0 2 1 0 0 0 | 0 -3 0 -11 1 10 14 16 | 0 0 1 1 1 1 1 1 }}
: mapping generators: ~2, ~11/10, ~5
: mapping generators: ~2, ~11/10, ~5


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Badness (Sintel): 1.14
Badness (Sintel): 1.14


[[Category:Pentacircle clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pentacircle clan| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
[[Category:Tolerant]]
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