Pentacircle clan: Difference between revisions

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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.

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

[[Subgroup]]: 2.3.7.11

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[[Comma list]]: 896/891

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

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

[[Optimal tuning]]s:

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

* [[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, ~3/2 = 703.834, ~7/4 = 969.872

* [[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 }}

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=== Overview to extensions ===

==== Subgroup extensions ====

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.

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.

==== Full 11-limit extensions ====

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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. Pentafrost adds [[245/242]]. 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.

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.

Temperaments discussed elsewhere are:

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Comma list: 352/351, 364/363

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

Subgroup-val mapping: {{mapping| 1 0 0 7 12 | 0 1 0 -4 -7 | 0 0 1 1 1 }}

Optimal tunings:

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

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

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

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

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

{{Optimal ET sequence|legend=0| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}

Badness (Sintel): 0.266

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Optimal tunings:

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

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

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

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

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

{{Optimal ET sequence|legend=0| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}

Badness (Sintel): 0.536

== Terrapyth ==

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

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

[[Subgroup]]: 2.3.5.7.11

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[[Comma list]]: 896/891, 585640/583443

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

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

[[Optimal tuning]]s:

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

* [[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, ~3/2 = 704.181, ~7/4 = 970.622

* [[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 }}

[[Badness]] (Sintel): 6.~~432~~

[[Badness]] (Sintel): 6.43

=== 13-limit ===

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Optimal tunings:

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

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

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

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

~~Optimal ET sequence:~~ {{Optimal ET sequence| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}

{{Optimal ET sequence|legend=0| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}

Badness (Sintel): 2.~~319~~

Badness (Sintel): 2.32

=== 17-limit ===

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Optimal tunings:

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

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

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

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

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

{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}

Badness (Sintel): 1.~~449~~

Badness (Sintel): 1.45

== Pentafrost ==

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

[[Optimal tuning]]s:

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

* [[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, ~3/2 = 701.903, ~7/4 = 964.614

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

: error map: {{val| 0.000 -0.052 -1.541 -4.212 +5.683 }}

* [[CEE]]: ~2 = 1200.000, ~3/2 = 702.006, ~7/4 = 964.085

* [[CEE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.006{{c}}, ~7/4 = 964.085{{c}}

: error map: {{val| 0.000 +0.051 -2.364 -4.741 +4.741 }}

[[Badness]] (Sintel): 1.~~903~~

[[Badness]] (Sintel): 1.90

=== 13-limit ===

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Mapping: {{mapping| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }}

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

Optimal tunings:

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

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

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

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

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

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

~~Optimal ET sequence:~~ {{Optimal ET sequence| 12f, 24, 29, 41 }}

Badness (Sintel): 1.~~487~~

Badness (Sintel): 1.49

=== Permafrost ===

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Mapping: {{mapping| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }}

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

Optimal tunings:

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

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

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

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

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

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

~~Optimal ET sequence:~~ {{Optimal ET sequence| 12, 17, 24, 36, 41, 77e }}

Badness (Sintel): 2.~~450~~

Badness (Sintel): 2.45

== Tolerant ==

: ''For the 7-limit version ~~of this temperament~~, see [[Miscellaneous 7-limit temperaments #Tolerant]].''

: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Tolerant]].''

[[Subgroup]]: 2.3.5.7.11

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

[[Optimal tuning]]s:

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

* [[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, ~3/2 = 704.041, ~5/4 = 387.293

* [[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 }}

{{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 288be, 375be }}

[[Badness]] (Sintel): 1.~~249~~

[[Badness]] (Sintel): 1.25

=== 13-limit ===

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Optimal tunings:

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

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

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

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

~~Optimal ET sequence:~~ {{Optimal ET sequence| 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be }}

{{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be }}

Badness (Sintel): 0.955

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Optimal tunings:

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

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

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

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

~~Optimal ET sequence:~~ {{Optimal ET sequence| 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg }}

{{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg }}

Badness (Sintel): 0.934

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: mapping generators: ~2, ~121/70, ~5

[[Optimal tuning]]s:

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

* [[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, ~121/70 = 951.871, ~5/4 = 387.243

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

: error map: {{val| 0.000 +1.787 +0.930 +0.220 +2.762 }}

{{Optimal ET sequence|legend=1| 24, 29, 34d, 53d, 58, 87, 121, 145, 179e, 208, 266e }}

[[Badness]] (Sintel): 2.~~719~~

[[Badness]] (Sintel): 2.72

=== 13-limit ===

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Optimal tunings:

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

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

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

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

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

{{Optimal ET sequence|legend=0| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}

Badness (Sintel): 0.991

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Optimal tunings:

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

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

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

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

~~Optimal ET sequence:~~ {{Optimal ET sequence| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}

{{Optimal ET sequence|legend=0| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}

Badness (Sintel): 1.~~181~~

Badness (Sintel): 1.18

== Trienparapyth ==

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.

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.

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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[[Subgroup]]: 2.3.5.7.11

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

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

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

[[Optimal tuning]]s:

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

* [[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, ~11/10 = 165.359, ~5/4 = 387.809

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

: error map: {{val| 0.000 +1.967 +1.496 +0.031 +1.851 }}

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

[[Badness]] (Sintel): 1.~~515~~

[[Badness]] (Sintel): 1.52

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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 tunings:

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

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

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

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

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

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

Badness (Sintel): 1.~~154~~

Badness (Sintel): 1.15

=== 2.3.5.7.11.13.19 subgroup ===

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Subgroup: 2.3.5.7.11.13.19

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

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

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 tunings:

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

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

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

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

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

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

Badness (Sintel): 1.~~198~~

Badness (Sintel): 1.20

=== 2.3.5.7.11.13.19.23 subgroup ===

Subgroup: 2.3.5.7.11.13.19.23

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

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

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 tunings

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

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

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

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

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

{{Optimal ET sequence|legend=0| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}

Badness (Sintel): 1.~~136~~

Badness (Sintel): 1.14

[[Category:Temperament clans]]

@@ Line 7: / Line 7: @@
 {{Main| Parapyth }}
-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.
+Parapyth, by the original definition, is the [[2.3.7.11.13 subgroup|2.3.7.11.13-subgroup]] temperament tempering out [[352/351]] and [[364/363]]. We begin by looking at the [[2.3.7.11 subgroup|2.3.7.11]] [[restriction]] thereof.
 [[Subgroup]]: 2.3.7.11
@@ Line 13: / Line 13: @@
 [[Comma list]]: 896/891
-{{mapping|legend=1| 1 0 0 7 | 0 1 0 -4 | 0 0 1 1 }}
+{{Mapping|legend=2| 1 0 0 7 | 0 1 0 -4 | 0 0 1 1 }}
+: mapping generators: ~2, ~3, ~7
-: sval mapping generators: ~2, ~3, ~7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.576, ~7/4 = 967.554
+* [[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, ~3/2 = 703.834, ~7/4 = 969.872
+* [[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 }}
@@ Line 29: / Line 28: @@
 === Overview to extensions ===
 ==== Subgroup extensions ====
-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.
+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.
 ==== Full 11-limit extensions ====
@@ Line 36: / Line 35: @@
 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. Pentafrost adds [[245/242]]. 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.
+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.
 Temperaments discussed elsewhere are:
@@ Line 57: / Line 56: @@
 Comma list: 352/351, 364/363
-Sval mapping: {{mapping| 1 0 0 7 12 | 0 1 0 -4 -7 | 0 0 1 1 1 }}
+Subgroup-val mapping: {{mapping| 1 0 0 7 12 | 0 1 0 -4 -7 | 0 0 1 1 1 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 703.786, ~7/4 = 967.665
+* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.786{{c}}, ~7/4 = 967.665{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 703.856, ~7/4 = 969.907
+* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.856{{c}}, ~7/4 = 969.907{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}
+{{Optimal ET sequence|legend=0| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}
 Badness (Sintel): 0.266
@@ Line 75: / Line 74: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 703.978, ~7/4 = 968.399
+* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.978{{c}}, ~7/4 = 968.399{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 704.032, ~7/4 = 970.605
+* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.032{{c}}, ~7/4 = 970.605{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}
+{{Optimal ET sequence|legend=0| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}
 Badness (Sintel): 0.536
 == Terrapyth ==
-Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.
+Terrapyth tempers out the leapday comma, and can be described as {{nowrap| 29 & 46 & 121 }}.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 89: / Line 88: @@
 [[Comma list]]: 896/891, 585640/583443
-[[Mapping]]: {{mapping| 1 0 -31 0 7 | 0 1 21 0 -4 | 0 0 0 1 1 }}
+{{Mapping|legend=1| 1 0 -31 0 7 | 0 1 21 0 -4 | 0 0 0 1 1 }}
 : mapping generators: ~2, ~3, ~7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 704.102, ~7/4 = 968.390
+* [[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, ~3/2 = 704.181, ~7/4 = 970.622
+* [[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 }}
-[[Badness]] (Sintel): 6.432
+[[Badness]] (Sintel): 6.43
 === 13-limit ===
@@ Line 111: / Line 109: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 704.099, ~7/4 = 968.601
+* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.099{{c}}, ~7/4 = 968.601{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 704.169, ~7/4 = 970.843
+* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.169{{c}}, ~7/4 = 970.843{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
+{{Optimal ET sequence|legend=0| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
-Badness (Sintel): 2.319
+Badness (Sintel): 2.32
 === 17-limit ===
@@ Line 126: / Line 124: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 704.096, ~7/4 = 968.504
+* CTE: ~2 = 1200.000{{c}}, ~3/2 = 704.096{{c}}, ~7/4 = 968.504{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 704.163, ~7/4 = 970.662
+* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.163{{c}}, ~7/4 = 970.662{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
+{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
-Badness (Sintel): 1.449
+Badness (Sintel): 1.45
 == Pentafrost ==
@@ Line 141: / Line 139: @@
 {{Mapping|legend=1| 1 0 15 0 7 | 0 1 -8 0 -4 | 0 0 0 1 1 }}
 : mapping generators: ~2, ~3, ~7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 701.883, ~7/4 = 964.864
+* [[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, ~3/2 = 701.903, ~7/4 = 964.614
+* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.903{{c}}, ~7/4 = 964.614{{c}}
 : error map: {{val| 0.000 -0.052 -1.541 -4.212 +5.683 }}
-* [[CEE]]: ~2 = 1200.000, ~3/2 = 702.006, ~7/4 = 964.085
+* [[CEE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.006{{c}}, ~7/4 = 964.085{{c}}
 : error map: {{val| 0.000 +0.051 -2.364 -4.741 +4.741 }}
 {{Optimal ET sequence|legend=1| 12, 24, 29, 36, 41, 106d }}
-[[Badness]] (Sintel): 1.903
+[[Badness]] (Sintel): 1.90
 === 13-limit ===
@@ Line 162: / Line 159: @@
 Mapping: {{mapping| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }}
 : mapping generators: ~2, ~3, ~7
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 702.106, ~7/4 = 962.655
+* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.106{{c}}, ~7/4 = 962.655{{c}}
-* CWE: ~2 = 1200.000, ~3/2 = 702.145, ~7/4 = 962.175
+* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.145{{c}}, ~7/4 = 962.175{{c}}
-* CEE: ~2 = 1200.000, ~3/2 = 702.360, ~7/4 = 962.210
+* CEE: ~2 = 1200.000{{c}}, ~3/2 = 702.360{{c}}, ~7/4 = 962.210{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 12f, 24, 29, 41 }}
+{{Optimal ET sequence|legend=0| 12f, 24, 29, 41 }}
-Badness (Sintel): 1.487
+Badness (Sintel): 1.49
 === Permafrost ===
@@ Line 180: / Line 176: @@
 Mapping: {{mapping| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }}
 : mapping generators: ~2, ~3, ~7
 Optimal tunings:
-* CTE: 2 = 1200.000, ~3/2 = 701.783, ~7/4 = 966.113
+* CTE: 2 = 1200.000{{c}}, ~3/2 = 701.783{{c}}, ~7/4 = 966.113{{c}}
-* CWE: 2 = 1200.000, ~3/2 = 701.753, ~7/4 = 966.445
+* CWE: 2 = 1200.000{{c}}, ~3/2 = 701.753{{c}}, ~7/4 = 966.445{{c}}
-* CEE: 2 = 1200.000, ~3/2 = 701.770, ~7/4 = 965.771
+* CEE: 2 = 1200.000{{c}}, ~3/2 = 701.770{{c}}, ~7/4 = 965.771{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 12, 17, 24, 36, 41, 77e }}
+{{Optimal ET sequence|legend=0| 12, 17, 24, 36, 41, 77e }}
-Badness (Sintel): 2.450
+Badness (Sintel): 2.45
 == Tolerant ==
-: ''For the 7-limit version of this temperament, see [[Miscellaneous 7-limit temperaments #Tolerant]].''
+: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Tolerant]].''
 [[Subgroup]]: 2.3.5.7.11
@@ Line 200: / Line 195: @@
 {{Mapping|legend=1| 1 0 0 -10 -3 | 0 1 0 11 7 | 0 0 1 -2 -2 }}
 : mapping generators: ~2, ~3, ~5
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.642, ~5/4 = 386.223
+* [[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, ~3/2 = 704.041, ~5/4 = 387.293
+* [[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 }}
 {{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 288be, 375be }}
-[[Badness]] (Sintel): 1.249
+[[Badness]] (Sintel): 1.25
 === 13-limit ===
@@ Line 221: / Line 215: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 703.705, ~5/4 = 386.616
+* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.705{{c}}, ~5/4 = 386.616{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 703.961, ~5/4 = 386.983
+* POTE: ~2 = 1200.000{{c}}, ~3/2 = 703.961{{c}}, ~5/4 = 386.983{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be }}
+{{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be }}
 Badness (Sintel): 0.955
@@ Line 236: / Line 230: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~3/2 = 703.891, ~5/4 = 387.424
+* CTE: ~2 = 1200.000{{c}}, ~3/2 = 703.891{{c}}, ~5/4 = 387.424{{c}}
-* POTE: ~2 = 1200.000, ~3/2 = 704.083, ~5/4 = 387.327
+* POTE: ~2 = 1200.000{{c}}, ~3/2 = 704.083{{c}}, ~5/4 = 387.327{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg }}
+{{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg }}
 Badness (Sintel): 0.934
@@ Line 251: / Line 245: @@
 {{Mapping|legend=1| 1 0 0 -13 -6 | 0 2 0 17 9 | 0 0 1 1 1 }}
 : mapping generators: ~2, ~121/70, ~5
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~121/70 = 951.837, ~5/4 = 386.405
+* [[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, ~121/70 = 951.871, ~5/4 = 387.243
+* [[CWE]]: ~2 = 1200.000{{c}}, ~121/70 = 951.871{{c}}, ~5/4 = 387.243{{c}}
 : error map: {{val| 0.000 +1.787 +0.930 +0.220 +2.762 }}
 {{Optimal ET sequence|legend=1| 24, 29, 34d, 53d, 58, 87, 121, 145, 179e, 208, 266e }}
-[[Badness]] (Sintel): 2.719
+[[Badness]] (Sintel): 2.72
 === 13-limit ===
@@ Line 272: / Line 265: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~26/15 = 951.852, ~5/4 = 386.089
+* CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.852{{c}}, ~5/4 = 386.089{{c}}
-* CWE: ~2 = 1200.000, ~26/15 = 951.881, ~5/4 = 387.104
+* CWE: ~2 = 1200.000{{c}}, ~26/15 = 951.881{{c}}, ~5/4 = 387.104{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}
+{{Optimal ET sequence|legend=0| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}
 Badness (Sintel): 0.991
@@ Line 289: / Line 282: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~26/15 = 951.802, ~5/4 = 386.991
+* CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.802{{c}}, ~5/4 = 386.991{{c}}
-* CWE: ~2 = 1200.000, ~26/15 = 951.879, ~5/4 = 387.723
+* CWE: ~2 = 1200.000{{c}}, ~26/15 = 951.879{{c}}, ~5/4 = 387.723{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}
+{{Optimal ET sequence|legend=0| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}
-Badness (Sintel): 1.181
+Badness (Sintel): 1.18
 == Trienparapyth ==
-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)<sup>2</sup> 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)<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|1521/1520 = S39]]. 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.
+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)<sup>2</sup> 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)<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|1521/1520 = S39]]. 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.
@@ Line 303: / Line 296: @@
 [[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: [[896/891]], [[3388/3375]]
+[[Comma list]]: 896/891, 3388/3375
 {{Mapping|legend=1| 1 2 0 2 1 | 0 -3 0 -11 1 | 0 0 1 1 1 }}
 : mapping generators: ~2, ~11/10, ~5
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~11/10 = 165.413, ~5/4 = 386.887
+* [[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, ~11/10 = 165.359, ~5/4 = 387.809
+* [[CWE]]: ~2 = 1200.000{{c}}, ~11/10 = 165.359{{c}}, ~5/4 = 387.809{{c}}
 : error map: {{val| 0.000 +1.967 +1.496 +0.031 +1.851 }}
 {{Optimal ET sequence|legend=1| 7d, 14e, 15d, 22, 51, 58, 80, 87, 145, 167, 312ce, 479bce }}
-[[Badness]] (Sintel): 1.515
+[[Badness]] (Sintel): 1.52
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: [[352/351]], [[364/363]], [[1001/1000]]
+Comma list: 352/351, 364/363, 1001/1000
 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 tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 165.398, ~5/4 = 386.791
+* CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.398{{c}}, ~5/4 = 386.791{{c}}
-* CWE: ~2 = 1200.000, ~11/10 = 165.380, ~5/4 = 387.876
+* CWE: ~2 = 1200.000{{c}}, ~11/10 = 165.380{{c}}, ~5/4 = 387.876{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}
+{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}
-Badness (Sintel): 1.154
+Badness (Sintel): 1.15
 === 2.3.5.7.11.13.19 subgroup ===
@@ Line 341: / Line 332: @@
 Subgroup: 2.3.5.7.11.13.19
-Comma list: [[286/285]], [[352/351]], [[364/363]], [[400/399]]
+Comma list: 286/285, 352/351, 364/363, 400/399
 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 tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 165.299, ~5/4 = 386.315
+* CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.299{{c}}, ~5/4 = 386.315{{c}}
-* CWE: ~2 = 1200.000, ~11/10 = 165.298, ~5/4 = 387.745
+* CWE: ~2 = 1200.000{{c}}, ~11/10 = 165.298{{c}}, ~5/4 = 387.745{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}
+{{Optimal ET sequence|legend=0| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}
-Badness (Sintel): 1.198
+Badness (Sintel): 1.20
 === 2.3.5.7.11.13.19.23 subgroup ===
 Subgroup: 2.3.5.7.11.13.19.23
-Comma list: [[208/207]], [[286/285]], [[352/351]], [[364/363]], [[400/399]]
+Comma list: 208/207, 286/285, 352/351, 364/363, 400/399
 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 tunings
-* CTE: ~2 = 1200.000, ~11/10 = 165.258, ~5/4 = 386.145
+* CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.258{{c}}, ~5/4 = 386.145{{c}}
-* CWE: ~2 = 1200.000, ~11/10 = 165.268, ~5/4 = 387.724
+* CWE: ~2 = 1200.000{{c}}, ~11/10 = 165.268{{c}}, ~5/4 = 387.724{{c}}
-Optimal ET sequence: {{Optimal ET sequence| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
+{{Optimal ET sequence|legend=0| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
-Badness (Sintel): 1.136
+Badness (Sintel): 1.14
 [[Category:Temperament clans]]