Pentacircle clan: Difference between revisions

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== 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

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== 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)2 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)7 onwards. We may simplify (11/10)7 as [[16/9|(4/3)2]]([[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)7~45/44 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)9/(11/10)/2 = (4/3)3/(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)2 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)2 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)7 onwards. We may simplify (11/10)7 as [[16/9|(4/3)2]]([[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)7~45/44 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)9/(11/10)/2 = (4/3)3/(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)2 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)8 so that ~[[20/7]] is reached at (11/10)11, and this is how the last generator can be either 5 or 7.

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[[Category:Pentacircle clan| ]]

[[Category:Rank 3]]

~~[[Category:Tolerant]]~~

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 == 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
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 == 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.
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 [[Category:Pentacircle clan| ]] <!-- main article -->
 [[Category:Rank 3]]
-[[Category:Tolerant]]