Pentacircle clan: Difference between revisions

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[[Category:Rank 3]]

[[Category:Tolerant]]

== Trienparapyth ==

Trienparapyth (which can be thought of as the no-17's 23-limit[80&87&109] temperament) splits the ~4/3 generator of parapythic into three ~[[11/10]]'s (tempering out [[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|S20/S22]]), or optionally less accurately as ~[[17/14]], though because this mapping only really makes much sense in [[80edo]] it isn't included here; however, its connection to parapyth structure comes from later in the gen. 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 the pinkanberry, [[1521/1520|S39]]. 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 [[4000/3993|S10/S11]] and [[896/891]] defines trienparapyth in the 11-limit (also tempering [[3388/3375]]), the 13-limit adds [[352/351]], the no-17's 19-limit [[1521/1520|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 the same as parapythic in 2.3.7.11.13 (as in that subgroup it tempers the same commas), but the independent generator of 7 is 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 (''in a sense'') the independent generator represents both 5 and 7 simultaneously, though the mapping uses 5 (which makes more sense for appraisal, as in the 13-limit only the 5 uses trienparapyth rather than parapythic, so that this temperament makes more sense in higher limits).

Subgroup: [[11-limit|2.3.5.7.11]]

Comma list: [[4000/3993]], [[896/891]]

Mapping: {{mapping| 1 2 0 2 1 | 0 -3 0 -11 1 | 0 0 1 1 1 }}

: Mapping generators: ~2, ~11/10, ~5

Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.413 (~3/2 = 703.761), ~5/4 = 386.887 (~7/4 = 967.340)

{{Optimal ET sequence|legend=1| 7d, 14e, 15d, 22, 51, 58, 80, 87, 145, 167, 312ce, 479bce }}

Badness: 0.00032498807511038835

Badness (Dirichlet): 1.515

=== 13-limit ===

Subgroup: [[13-limit|2.3.5.7.11.13]]

Comma list: [[4000/3993]], [[896/891]], [[352/351]], [[1521/1520]]

Mapping: {{mapping| 1 2 0 2 1 0 | 0 -3 0 -11 1 10 | 0 0 1 1 1 1 }}

: Mapping generators: ~2, ~11/10, ~5

Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.398 (~3/2 = 703.806), ~5/4 = 386.791 (7/4 = 967.418)

{{Optimal ET sequence|legend=1| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}

Badness: 0.00035577335441867717

Badness (Dirichlet): 1.154

=== no-17's 19-limit ===

Note [[109edo]] is a good patent val tuning not listed in the optimal ET sequence here.

Subgroup: 2.3.5.7.11.13.19 (no-17's [[19-limit]])

Comma list: [[4000/3993]], [[896/891]], [[352/351]], [[1521/1520]]

Mapping: {{mapping| 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

Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.299 (~3/2 = 704.103), ~5/4 = 386.315 (~7/4 = 968.027)

{{Optimal ET sequence|legend=1| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}

Badness: 0.00040867444151208805

Badness (Dirichlet): 1.198

=== no-17's 23-limit ===

Subgroup: 2.3.5.7.11.13.19.23 (no-17's [[23-limit]])

Comma list: [[4000/3993]], [[896/891]], [[352/351]], [[1521/1520]], [[2300/2299]]

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 generators: ~2, ~11/10, ~5

Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.258 (~3/2 = 704.226), ~5/4 = 386.145 (~7/4 = 968.308)

{{Optimal ET sequence|legend=1| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}

Badness: 0.0003920458395201445

Badness (Dirichlet): 1.136

@@ Line 220: / Line 220: @@
 [[Category:Rank 3]]
 [[Category:Tolerant]]
+== Trienparapyth ==
+Trienparapyth (which can be thought of as the no-17's 23-limit[80&87&109] temperament) splits the ~4/3 generator of parapythic into three ~[[11/10]]'s (tempering out [[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|S20/S22]]), or optionally less accurately as ~[[17/14]], though because this mapping only really makes much sense in [[80edo]] it isn't included here; however, its connection to parapyth structure comes from later in the gen. 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 the pinkanberry, [[1521/1520|S39]]. 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 [[4000/3993|S10/S11]] and [[896/891]] defines trienparapyth in the 11-limit (also tempering [[3388/3375]]), the 13-limit adds [[352/351]], the no-17's 19-limit [[1521/1520|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 the same as parapythic in 2.3.7.11.13 (as in that subgroup it tempers the same commas), but the independent generator of 7 is 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 (''in a sense'') the independent generator represents both 5 and 7 simultaneously, though the mapping uses 5 (which makes more sense for appraisal, as in the 13-limit only the 5 uses trienparapyth rather than parapythic, so that this temperament makes more sense in higher limits).
+Subgroup: [[11-limit|2.3.5.7.11]]
+Comma list: [[4000/3993]], [[896/891]]
+Mapping: {{mapping| 1 2 0 2 1 | 0 -3 0 -11 1 | 0 0 1 1 1 }}
+: Mapping generators: ~2, ~11/10, ~5
+Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.413 (~3/2 = 703.761), ~5/4 = 386.887 (~7/4 = 967.340)
+{{Optimal ET sequence|legend=1| 7d, 14e, 15d, 22, 51, 58, 80, 87, 145, 167, 312ce, 479bce }}
+Badness: 0.00032498807511038835
+Badness (Dirichlet): 1.515
+=== 13-limit ===
+Subgroup: [[13-limit|2.3.5.7.11.13]]
+Comma list: [[4000/3993]], [[896/891]], [[352/351]], [[1521/1520]]
+Mapping: {{mapping| 1 2 0 2 1 0 | 0 -3 0 -11 1 10 | 0 0 1 1 1 1 }}
+: Mapping generators: ~2, ~11/10, ~5
+Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.398 (~3/2 = 703.806), ~5/4 = 386.791 (7/4 = 967.418)
+{{Optimal ET sequence|legend=1| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}
+Badness: 0.00035577335441867717
+Badness (Dirichlet): 1.154
+=== no-17's 19-limit ===
+Note [[109edo]] is a good patent val tuning not listed in the optimal ET sequence here.
+Subgroup: 2.3.5.7.11.13.19 (no-17's [[19-limit]])
+Comma list: [[4000/3993]], [[896/891]], [[352/351]], [[1521/1520]]
+Mapping: {{mapping| 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
+Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.299 (~3/2 = 704.103), ~5/4 = 386.315 (~7/4 = 968.027)
+{{Optimal ET sequence|legend=1| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}
+Badness: 0.00040867444151208805
+Badness (Dirichlet): 1.198
+=== no-17's 23-limit ===
+Subgroup: 2.3.5.7.11.13.19.23 (no-17's [[23-limit]])
+Comma list: [[4000/3993]], [[896/891]], [[352/351]], [[1521/1520]], [[2300/2299]]
+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 generators: ~2, ~11/10, ~5
+Optimal tuning (CTE): 2 = 1\1, ~11/10 = 165.258 (~3/2 = 704.226), ~5/4 = 386.145 (~7/4 = 968.308)
+{{Optimal ET sequence|legend=1| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
+Badness: 0.0003920458395201445
+Badness (Dirichlet): 1.136