Pentacircle clan: Difference between revisions

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== Trienparapyth ==

Trienparapyth can be described as the no-17's 23-limit 80 & 87 & 109 temperament. It splits the ~4/3 generator of parapythic into three ~[[11/10]]'s by tempering out [[4000/3993]] in the 11-limit and it interprets (11/10)2 accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]], 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]]. Next, for ~~8 gens~~, 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 ~~8 gens~~ is 8 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.

Trienparapyth can be described as the no-17's 23-limit 80 & 87 & 109 temperament. It splits the ~4/3 generator of parapythic into three ~[[11/10]]'s by tempering out [[4000/3993]] in the 11-limit and it interprets (11/10)2 accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]], 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]]. 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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Optimal tunings:

* CTE: 2 = 1\1, ~11/10 = 165.3975, ~5/4 = 386.7908

* CTE: ~2 = 1\1, ~11/10 = 165.3975, ~5/4 = 386.7908

* CWE: 2 = 1\1, ~11/10 = 165.3802, ~5/4 = 387.8759

* CWE: ~2 = 1\1, ~11/10 = 165.3802, ~5/4 = 387.8759

Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}

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Optimal tunings:

* CTE: 2 = 1\1, ~11/10 = 165.2990, ~5/4 = 386.3154

* CTE: ~2 = 1\1, ~11/10 = 165.2990, ~5/4 = 386.3154

* CWE: 2 = 1\1, ~11/10 = 165.2976, ~5/4 = 387.7451

* CWE: ~2 = 1\1, ~11/10 = 165.2976, ~5/4 = 387.7451

Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}

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Optimal tunings

* CTE: 2 = 1\1, ~11/10 = 165.2579, ~5/4 = 386.1446

* CTE: ~2 = 1\1, ~11/10 = 165.2579, ~5/4 = 386.1446

* CWE: 2 = 1\1, ~11/10 = 165.2679, ~5/4 = 387.7240

* CWE: ~2 = 1\1, ~11/10 = 165.2679, ~5/4 = 387.7240

Optimal ET sequence: {{Optimal ET sequence| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}

Badness: 1.04 × 10-3

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 == Trienparapyth ==
-Trienparapyth can be described as the no-17's 23-limit 80 & 87 & 109 temperament. It splits the ~4/3 generator of parapythic into three ~[[11/10]]'s by tempering out [[4000/3993]] in the 11-limit and it interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]], 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]]. Next, for 8 gens, 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 8 gens is 8 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.
+Trienparapyth can be described as the no-17's 23-limit 80 & 87 & 109 temperament. It splits the ~4/3 generator of parapythic into three ~[[11/10]]'s by tempering out [[4000/3993]] in the 11-limit and it interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]], 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]]. 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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 Optimal tunings:
-* CTE: 2 = 1\1, ~11/10 = 165.3975, ~5/4 = 386.7908
+* CTE: ~2 = 1\1, ~11/10 = 165.3975, ~5/4 = 386.7908
-* CWE: 2 = 1\1, ~11/10 = 165.3802, ~5/4 = 387.8759
+* CWE: ~2 = 1\1, ~11/10 = 165.3802, ~5/4 = 387.8759
 Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}
@@ Line 271: / Line 271: @@
 Optimal tunings:
-* CTE: 2 = 1\1, ~11/10 = 165.2990, ~5/4 = 386.3154
+* CTE: ~2 = 1\1, ~11/10 = 165.2990, ~5/4 = 386.3154
-* CWE: 2 = 1\1, ~11/10 = 165.2976, ~5/4 = 387.7451
+* CWE: ~2 = 1\1, ~11/10 = 165.2976, ~5/4 = 387.7451
 Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}
@@ Line 288: / Line 288: @@
 Optimal tunings
-* CTE: 2 = 1\1, ~11/10 = 165.2579, ~5/4 = 386.1446
+* CTE: ~2 = 1\1, ~11/10 = 165.2579, ~5/4 = 386.1446
-* CWE: 2 = 1\1, ~11/10 = 165.2679, ~5/4 = 387.7240
+* CWE: ~2 = 1\1, ~11/10 = 165.2679, ~5/4 = 387.7240
 Optimal ET sequence: {{Optimal ET sequence| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
 Badness: 1.04 × 10<sup>-3</sup>