Pentacircle clan: Difference between revisions

* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.576{{c}}, ~7/4 = 967.554{{c}}

: [[error map]]: {{val| 0.000 +1.621 -1.272 +1.937 }}

* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.834{{c}}, ~7/4 = 969.872{{c}}

: error map: {{val| 0.000 +1.879 +1.046 +3.216 }}

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.786{{c}}, ~7/4 = 967.665{{c}}

* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.856{{c}}, ~7/4 = 969.907{{c}}

Sval mapping: {{mapping| 1 0 0 7 12 -13 | 0 1 0 -4 -7 9 | 0 0 1 1 1 1 }}

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.978{{c}}, ~7/4 = 968.399{{c}}

* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.032{{c}}, ~7/4 = 970.605{{c}}

* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.102{{c}}, ~7/4 = 968.390{{c}}

: [[error map]]: {{val| 0.000 +2.147 -0.163 -0.436 +0.662 }}

* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.181{{c}}, ~7/4 = 970.622{{c}}

: error map: {{val| 0.000 +2.226 +1.496 +1.796 +2.578 }}

{{Optimal ET sequence|legend=1| 17c, 29, 46, 92de, 121, 167, 288be }}

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.099{{c}}, ~7/4 = 968.601{{c}}

* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.169{{c}}, ~7/4 = 970.843{{c}}

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.096{{c}}, ~7/4 = 968.504{{c}}

* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.163{{c}}, ~7/4 = 970.662{{c}}

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.

[[Comma list]]: [[245/242]], [[896/891]]

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.883{{c}}, ~7/4 = 964.864{{c}}

: [[error map]]: {{val| 0.000 -0.072 -1.375 -3.962 +6.016 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.903{{c}}, ~7/4 = 964.614{{c}}

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

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.106{{c}}, ~7/4 = 962.655{{c}}

* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.145{{c}}, ~7/4 = 962.175{{c}}

* CEE: ~2 = 1200.000{{c}}, ~3/2 = 702.360{{c}}, ~7/4 = 962.210{{c}}

Comma list: [[144/143]], [[245/242]], [[896/891]]

* CTE: 2 = 1200.000{{c}}, ~3/2 = 701.783{{c}}, ~7/4 = 966.113{{c}}

* CWE: 2 = 1200.000{{c}}, ~3/2 = 701.753{{c}}, ~7/4 = 966.445{{c}}

* CEE: 2 = 1200.000{{c}}, ~3/2 = 701.770{{c}}, ~7/4 = 965.771{{c}}

* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.642{{c}}, ~5/4 = 386.223{{c}}

: [[error map]]: {{val| 0.000 +1.687 -0.091 -1.207 +1.732 }}

* [[POTE]]: ~2 = 1200.000{{c}}, ~3/2 = 704.041{{c}}, ~5/4 = 387.293{{c}}

: error map: {{val| 0.000 +2.086 +0.979 +1.042 +2.385 }}

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.705{{c}}, ~5/4 = 386.616{{c}}

* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.961{{c}}, ~5/4 = 386.983{{c}}

* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.891{{c}}, ~5/4 = 387.424{{c}}

* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.083{{c}}, ~5/4 = 387.327{{c}}

* [[CTE]]: ~2 = 1200.000{{c}}, ~121/70 = 951.837{{c}}, ~5/4 = 386.405{{c}}

: [[error map]]: {{val| 0.000 +1.718 +0.091 -1.198 +1.617 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~121/70 = 951.871{{c}}, ~5/4 = 387.243{{c}}

* CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.852{{c}}, ~5/4 = 386.089{{c}}

* CWE: ~2 = 1200.000{{c}}, ~26/15 = 951.881{{c}}, ~5/4 = 387.104{{c}}

* CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.802{{c}}, ~5/4 = 386.991{{c}}

* CWE: ~2 = 1200.000{{c}}, ~26/15 = 951.879{{c}}, ~5/4 = 387.723{{c}}

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.

* [[CTE]]: ~2 = 1200.000{{c}}, ~11/10 = 165.413{{c}}, ~5/4 = 386.887{{c}}

: [[error map]]: {{val| 0.000 +1.805 +0.574 -1.486 +0.983 }}

* [[CWE]]: ~2 = 1200.000{{c}}, ~11/10 = 165.359{{c}}, ~5/4 = 387.809{{c}}

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

* CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.398{{c}}, ~5/4 = 386.791{{c}}

* CWE: ~2 = 1200.000{{c}}, ~11/10 = 165.380{{c}}, ~5/4 = 387.876{{c}}

{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}

* CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.299{{c}}, ~5/4 = 386.315{{c}}

* CWE: ~2 = 1200.000{{c}}, ~11/10 = 165.298{{c}}, ~5/4 = 387.745{{c}}

{{Optimal ET sequence|legend=0| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}

Optimal tunings  

* CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.258{{c}}, ~5/4 = 386.145{{c}}

* CWE: ~2 = 1200.000{{c}}, ~11/10 = 165.268{{c}}, ~5/4 = 387.724{{c}}