Pentacircle clan: Difference between revisions

Parapyth, by the original definition, is the 2.3.7.11.13 [[subgroup temperament]] tempering out [[352/351]] and [[364/363]]. We begin by looking at the 2.3.7.11 [[restriction]] thereof.  

{{mapping|legend=1| 1 0 0 7 | 0 1 0 -4 | 0 0 1 1 }}

: sval mapping generators: ~2, ~3, ~7

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8345, ~7/4 = 969.8722

[[Badness]]: 0.0205 × 10^-3

By tempering out 896/891, we have mapped 14/11 to the major third, suggesting a slightly sharp fifth. This makes the minor third very close to the flat-of-Pythagorean [[13/11]], and extending the temperament to include harmonic 13 this way implies we temper out [[352/351]]. In fact, 896/891 = (352/351)([[364/363]]), so it is a very natural interpretation, giving rise to the 2.3.7.11.13 subgroup temperament shown below.  

Pele adds [[441/440]] and finds the harmonic 5 by equating the [[81/80|syntonic comma (81/80)]] with the [[64/63|septimal comma (64/63)]]. Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds [[245/243]] or [[385/384]], a traditional RTT favorite. Apollo adds [[100/99]] or [[225/224]], and is even simpler than sensamagic. Uni adds [[540/539]]. Melpomene adds [[56/55]] or [[81/80]]. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth.  

Julius aka varda adds [[176/175]], splitting the octave into two. Parahemif adds [[243/242]], splitting the perfect fifth into two. Kujuku adds 14700/14641, splitting the perfect twelfth into two. Tolerant adds 2200/2187, splitting the ~33/32 into two. Finally, canta adds 472392/471625, splitting the ~14/9 into three.  

* ''[[Melpomene]]'' → [[Didymus rank three family #Melpomene|Didymus rank-3 family]]

* ''[[Pele]]'' → [[Hemifamity family #Pele|Hemifamity family]]

* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]

* ''[[Parahemif]]'' → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8563, ~7/4 = 969.9074

{{Optimal ET sequence|legend=1| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}

Badness: 0.101 × 10^-3

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0315, ~7/4 = 970.6051

{{Optimal ET sequence|legend=1| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}

Badness: 0.325 × 10^-3

Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.  

[[Mapping]]: {{mapping| 1 0 -31 0 7 | 0 1 21 0 -4 | 0 0 0 1 1 }}

: mapping generators: ~2, ~3, ~7

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217

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

[[Badness]]: 5.35 × 10^-3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432

{{Optimal ET sequence|legend=1| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}

Badness: 2.48 × 10^-3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620

{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}

Badness: 1.52 × 10^-3

== Tolerant ==

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 179200/177147

{{Mapping|legend=1| 1 0 0 -10 | 0 1 0 11 | 0 0 1 -2 }}

: mapping generators: ~2, ~3, ~5

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.9571, ~5/4 = 386.8863

[[Badness]]: 0.165 × 10^-3

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.0412, ~5/4 = 387.2927

{{Optimal ET sequence|legend=1| 41, 80, 87, 121, 167, 208, 334be, 375be, 542bce }}

[[Badness]]: 1.039 × 10^-3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9605, ~5/4 = 386.9831

{{Optimal ET sequence|legend=1| 41, 46, 80, 87, 121, 167, 208, 375be, 583bef }}

Badness: 1.021 × 10^-3

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0831, ~5/4 = 387.3269

{{Optimal ET sequence|legend=1| 41, 46, 75e, 80, 87, 121, 167, 288beg }}

Badness: 0.982 × 10^-3

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.4956, ~5/4 = 386.7868

[[Badness]]: 2.26 × 10^-3

Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8367, ~5/4 = 386.4048

{{Optimal ET sequence|legend=1| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}

Badness: 1.06 × 10^-3

Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8015, ~5/4 = 386.9912

 

{{Optimal ET sequence|legend=1| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}

Badness: 1.24 × 10^-3

[[Category:Temperament clans]]

Trienparapyth can be described as the no-17's 23-limit 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.

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

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

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

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

[[Badness]]: 1.26 × 10^-3

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

* 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

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

Comma list: [[286/285]], [[352/351]], [[364/363]], [[400/399]]

* 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

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

Comma list: [[208/207]], [[286/285]], [[352/351]], [[364/363]], [[400/399]]

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

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. In [[aberrismic theory]], it supports [[compdye]], the '''5L7m12s''' scale structure.

 

Subgroup: 2.3.5.7.11

 

Comma list: [[32805/32768]], [[896/891]]