Pentacircle clan: Difference between revisions

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.

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.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: [[896/891], [[3388/3375]]

{{Mapping|legend=1| 1 2 0 2 1 | 0 -3 0 -11 1 | 0 0 1 1 1 }}

 

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1\1, ~11/10 = 165.4134, ~5/4 = 386.8872

* [[CWE]]: ~2 = 1\1, ~11/10 = 165.3593, ~5/4 = 387.8093

[[Badness]]: 1.26 × 10^-3

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 1001/1000

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

Optimal tunings:

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

* 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 }}

Badness: 1.23 × 10^-3

=== 2.3.5.7.11.13.19 subgroup ===

Subgroup: 2.3.5.7.11.13.19

Comma list: 352/351, 364/363, 1001/1000, 1521/1520

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

Optimal tunings:

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

* 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 }}

Badness: 1.22 × 10^-3

=== 2.3.5.7.11.13.19.23 subgroup ===

Subgroup: 2.3.5.7.11.13.19.23

Comma list: 352/351, 364/363, 1001/1000, 1521/1520, 2300/2299

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

Optimal tunings

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

* 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