Pentacircle clan: Difference between revisions

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The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. This has the effect of identifying [[14/11]] at the [[81/64|Pythagorean major third]].

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

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~3/2 = 703.~~8345~~, ~7/4 = 969.~~8722~~

[[Optimal tuning]]s:

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

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

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

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

{{Optimal ET sequence|legend=1| 12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee }}

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The second comma in the comma list determines how we extend parapyth to include the harmonic 5.

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.

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.

Temperaments discussed elsewhere are:

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

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

* ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]

* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]

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

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

* ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]]

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

* ''[[Varda]]'' → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]

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

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

* ''[[Canta]]'' → [[Canou family #Canta|Canou family]]

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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 tunings:

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

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

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

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

Badness: 0.101 × 10-3

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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 tunings:

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

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

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

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

Badness: 0.325 × 10-3

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

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 704.~~1814~~, ~7/4 = 970.~~6217~~

[[Optimal tuning]]s:

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

: [[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

: 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 }}

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Mapping: {{mapping| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}

Optimal ~~tuning (~~POTE): ~2 = ~~1\1~~, ~3/2 = 704.~~1691~~, ~7/4 = 970.~~8432~~

Optimal tunings:

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

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

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

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

Badness: 2.48 × 10-3

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Mapping: {{mapping| 1 0 -31 0 7 12 -13 | 0 1 0 21 0 4 -7 9 | 0 0 0 1 1 1 1 1 }}

Optimal ~~tuning (~~POTE): ~2 = ~~1\1~~, ~3/2 = 704.~~1628~~, ~7/4 = 970.~~6620~~

Optimal tunings:

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

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

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

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

Badness: 1.52 × 10-3

== Pentafrost ==

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.

[[Subgroup]]: 2.3.5.7.11

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

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

[[Optimal tuning]]s:

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

: [[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

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

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

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

[[Badness]]: 1.58 × 10-3

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

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

Badness: 1.59 × 10-3

=== Permafrost ===

Subgroup: 2.3.5.7.11.13

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

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

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

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

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

Badness: 2.62 × 10-3

== Tolerant ==

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

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~3/2 = 703.~~9571~~, ~5/4 = 386.~~8863~~

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.608, ~5/4 = 385.669

: [[error map]]: {{val| 0.000 +1.653 -0.645 -0.472 }}

* [[POTE]]: ~2 = 1200.000, ~3/2 = 703.957, ~5/4 = 386.886

: error map: {{val| 0.000 +2.002 +0.573 +0.930 }}

{{Optimal ET sequence|legend=1| 41, 80, 87, 121, 167, 208, 329b, 375b, ~~537b, 583b, 704bd~~ }}

{{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 329b, 375b, 496bd }}

[[Badness]]: 0.165 × 10-3

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[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~3/2 = 704.~~0412~~, ~5/4 = 387.~~2927~~

[[Optimal tuning]]s:

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

: [[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

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

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

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

[[Badness]]: 1.039 × 10-3

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Mapping: {{mapping| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}

Optimal ~~tuning (~~POTE): ~2 = ~~1\1~~, ~3/2 = 703.~~9605~~, ~5/4 = 386.~~9831~~

Optimal tunings:

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

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

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

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

Badness: 1.~~021~~ × 10-3

Badness: 1.02 × 10-3

=== 17-limit ===

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Mapping: {{mapping| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}

Optimal ~~tuning (~~POTE): ~2 = ~~1\1~~, ~3/2 = 704.~~0831~~, ~5/4 = 387.~~3269~~

Optimal tunings:

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

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

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

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

Badness: 0.982 × 10-3

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.~~4956~~, ~5/4 = ~~386~~.~~7868~~

[[Optimal tuning]]s:

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

: [[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

: 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 }}

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Mapping: {{mapping| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~26/15 = 951.~~8367~~, ~5/4 = 386.~~4048~~

Optimal tunings:

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

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

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

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

Badness: 1.06 × 10-3

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Mapping: {{mapping| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~26/15 = 951.~~8015~~, ~5/4 = 386.~~9912~~

Optimal tunings:

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

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

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

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

Badness: 1.24 × 10-3

~~[[Category:Temperament clans]]~~

~~[[Category:Pentacircle clan| ]] ~~

~~[[Category:Rank 3]]~~

~~[[Category:Tolerant]]~~

== Trienparapyth ==

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[[Optimal tuning]]s:

* [[CTE]]: ~2 = ~~1\1~~, ~11/10 = 165.~~4134~~, ~5/4 = 386.~~8872~~

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

* [[CWE]]: ~2 = ~~1\1~~, ~11/10 = 165.~~3593~~, ~5/4 = 387.~~8093~~

: [[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

: 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 }}

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~11/10 = 165.~~3975~~, ~5/4 = 386.~~7908~~

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

* CWE: ~2 = ~~1\1~~, ~11/10 = 165.~~3802~~, ~5/4 = 387.~~8759~~

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

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

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Badness: 1.23 × 10-3

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

=== 2.3.5.7.11.13.19 subgroup ===

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

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~11/10 = 165.~~2990~~, ~5/4 = 386.~~3154~~

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

* CWE: ~2 = ~~1\1~~, ~11/10 = 165.~~2976~~, ~5/4 = 387.~~7451~~

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

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

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Badness: 1.22 × 10-3

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

=== 2.3.5.7.11.13.19.23 subgroup ===

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

Subgroup: 2.3.5.7.11.13.19.23

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Optimal tunings

* CTE: ~2 = ~~1\1~~, ~11/10 = 165.~~2579~~, ~5/4 = 386.~~1446~~

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

* CWE: ~2 = ~~1\1~~, ~11/10 = 165.~~2679~~, ~5/4 = 387.~~7240~~

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

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

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Badness: 1.04 × 10-3

~~== Pentafrost ==~~

[[Category:Temperament clans]]

~~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.~~

[[Category:Pages with mostly numerical content]]

[[Category:Pentacircle clan| ]]

~~Subgroup: 2.3.5.7.11~~

[[Category:Rank 3]]

[[Category:Tolerant]]

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

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

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

~~Optimal tunings:~~

* CTE: 2 = 1\1, ~3 = 1901.883, ~7 = 3364.864

* CWE: 2 = 1\1, ~3 = 1901.903, ~7 = 3364.614

* CE: 2 = 1\1, ~3 = 1902.006, ~7 = 3364.085

~~=== 13~~-~~limit ===~~

~~Subgroup: 2.3.5.7.11.13~~

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

~~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 = 1\1, ~3 = 1902.106, ~7 = 3362.655

* CWE: 2 = 1\1, ~3 ~~= 1902.145, ~7 = 3362.175~~

* CE: 2 = 1\1, ~3 = 1902.360, ~7 = 3362.210

~~ ~~

~~=== Permafrost ===~~

~~Subgroup: 2.3.5.7.11.13~~

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

~~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 = 1\1, ~3 = 1901.783, ~7 = 3366.113

* CWE: 2 = 1\1, ~3 = 1901.753, ~7 = 3366.445

* CE: 2 = 1\1, ~3 = 1901.770, ~7 = 3365.771

@@ Line 1: / Line 1: @@
+{{Technical data page}}
 The '''pentacircle clan''' of [[rank-3 temperament]]s tempers out the pentacircle comma, [[896/891]]. This has the effect of identifying [[14/11]] at the [[81/64|Pythagorean major third]].
@@ Line 16: / Line 17: @@
 : sval mapping generators: ~2, ~3, ~7
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8345, ~7/4 = 969.8722
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.576, ~7/4 = 967.554
+: [[error map]]: {{val| 0.000 +1.621 -1.272 +1.937 }}
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 703.834, ~7/4 = 969.872
+: error map: {{val| 0.000 +1.879 +1.046 +3.216 }}
 {{Optimal ET sequence|legend=1| 12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee }}
@@ Line 29: / Line 34: @@
 The second comma in the comma list determines how we extend parapyth to include the harmonic 5.
-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.
+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.
 Temperaments discussed elsewhere are:
-* ''[[Melpomene]]'' → [[Didymus rank three family #Melpomene|Didymus rank-3 family]]
+* ''[[Melpomene]]'' → [[Didymus rank-3 family #Melpomene|Didymus rank-3 family]]
 * ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]
 * [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
-* ''[[Pele]]'' → [[Hemifamity family #Pele|Hemifamity family]]
+* [[Pele]] → [[Hemifamity family #Pele|Hemifamity family]]
 * ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]]
-* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
+* ''[[Varda]]'' → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
-* ''[[Parahemif]]'' → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]
+* ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif|Rastmic rank-3 clan]]
 * ''[[Canta]]'' → [[Canou family #Canta|Canou family]]
@@ Line 54: / Line 59: @@
 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 tunings:
+* CTE: ~2 = 1200.000, ~3/2 = 703.786, ~7/4 = 967.665
+* POTE: ~2 = 1200.000, ~3/2 = 703.856, ~7/4 = 969.907
-{{Optimal ET sequence|legend=1| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}
+Optimal ET sequence: {{Optimal ET sequence| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}
 Badness: 0.101 × 10<sup>-3</sup>
@@ Line 67: / Line 74: @@
 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 tunings:
+* CTE: ~2 = 1200.000, ~3/2 = 703.978, ~7/4 = 968.399
+* POTE: ~2 = 1200.000, ~3/2 = 704.032, ~7/4 = 970.605
-{{Optimal ET sequence|legend=1| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}
+Optimal ET sequence: {{Optimal ET sequence| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}
 Badness: 0.325 × 10<sup>-3</sup>
@@ Line 84: / Line 93: @@
 : mapping generators: ~2, ~3, ~7
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 704.102, ~7/4 = 968.390
+: [[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
+: 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 }}
@@ Line 97: / Line 110: @@
 Mapping: {{mapping| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~3/2 = 704.099, ~7/4 = 968.601
+* POTE: ~2 = 1200.000, ~3/2 = 704.169, ~7/4 = 970.843
-{{Optimal ET sequence|legend=1| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
+Optimal ET sequence: {{Optimal ET sequence| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
 Badness: 2.48 × 10<sup>-3</sup>
@@ Line 110: / Line 125: @@
 Mapping: {{mapping| 1 0 -31 0 7 12 -13 | 0 1 0 21 0 4 -7 9 | 0 0 0 1 1 1 1 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~3/2 = 704.096, ~7/4 = 968.504
+* POTE: ~2 = 1200.000, ~3/2 = 704.163, ~7/4 = 970.662
-{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
+Optimal ET sequence: {{Optimal ET sequence| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
 Badness: 1.52 × 10<sup>-3</sup>
+== Pentafrost ==
+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.
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[245/242]], [[896/891]]
+{{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
+: [[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
+: error map: {{val| 0.000 -0.052 -1.541 -4.212 +5.683 }}
+* [[CE]]: ~2 = 1200.000, ~3/2 = 702.006, ~7/4 = 964.085
+: 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]]: 1.58 × 10<sup>-3</sup>
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: [[105/104]], [[245/242]], [[352/351]]
+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
+* CWE: ~2 = 1200.000, ~3/2 = 702.145, ~7/4 = 962.175
+* CE: ~2 = 1200.000, ~3/2 = 702.360, ~7/4 = 962.210
+Optimal ET sequence: {{Optimal ET sequence| 12f, 24, 29, 41 }}
+Badness: 1.59 × 10<sup>-3</sup>
+=== Permafrost ===
+Subgroup: 2.3.5.7.11.13
+Comma list: [[144/143]], [[245/242]], [[896/891]]
+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
+* CWE: 2 = 1200.000, ~3/2 = 701.753, ~7/4 = 966.445
+* CE: 2 = 1200.000, ~3/2 = 701.770, ~7/4 = 965.771
+Optimal ET sequence: {{Optimal ET sequence| 12, 17, 24, 36, 41, 77e }}
+Badness: 2.62 × 10<sup>-3</sup>
 == Tolerant ==
@@ Line 126: / Line 202: @@
 : mapping generators: ~2, ~3, ~5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.9571, ~5/4 = 386.8863
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.608, ~5/4 = 385.669
+: [[error map]]: {{val| 0.000 +1.653 -0.645 -0.472 }}
+* [[POTE]]: ~2 = 1200.000, ~3/2 = 703.957, ~5/4 = 386.886
+: error map: {{val| 0.000 +2.002 +0.573 +0.930 }}
-{{Optimal ET sequence|legend=1| 41, 80, 87, 121, 167, 208, 329b, 375b, 537b, 583b, 704bd }}
+{{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 329b, 375b, 496bd }}
 [[Badness]]: 0.165 × 10<sup>-3</sup>
@@ Line 139: / Line 219: @@
 {{Mapping|legend=1| 1 0 0 -10 -3 | 0 1 0 11 7 | 0 0 1 -2 -2 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.0412, ~5/4 = 387.2927
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~3/2 = 703.642, ~5/4 = 386.223
+: [[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
+: error map: {{val| 0.000 +2.086 +0.979 +1.042 +2.385 }}
-{{Optimal ET sequence|legend=1| 41, 80, 87, 121, 167, 208, 334be, 375be, 542bce }}
+{{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 288be, 375be }}
 [[Badness]]: 1.039 × 10<sup>-3</sup>
@@ Line 152: / Line 236: @@
 Mapping: {{mapping| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9605, ~5/4 = 386.9831
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~3/2 = 703.705, ~5/4 = 386.616
+* POTE: ~2 = 1200.000, ~3/2 = 703.961, ~5/4 = 386.983
-{{Optimal ET sequence|legend=1| 41, 46, 80, 87, 121, 167, 208, 375be, 583bef }}
+Optimal ET sequence: {{Optimal ET sequence| 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be }}
-Badness: 1.021 × 10<sup>-3</sup>
+Badness: 1.02 × 10<sup>-3</sup>
 === 17-limit ===
@@ Line 165: / Line 251: @@
 Mapping: {{mapping| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0831, ~5/4 = 387.3269
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~3/2 = 703.891, ~5/4 = 387.424
+* POTE: ~2 = 1200.000, ~3/2 = 704.083, ~5/4 = 387.327
-{{Optimal ET sequence|legend=1| 41, 46, 75e, 80, 87, 121, 167, 288beg }}
+Optimal ET sequence: {{Optimal ET sequence| 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg }}
 Badness: 0.982 × 10<sup>-3</sup>
@@ Line 182: / Line 270: @@
 : mapping generators: ~2, ~121/70, ~5
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.4956, ~5/4 = 386.7868
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~121/70 = 951.837, ~5/4 = 386.405
+: [[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
+: 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 }}
@@ Line 195: / Line 287: @@
 Mapping: {{mapping| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8367, ~5/4 = 386.4048
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~26/15 = 951.852, ~5/4 = 386.089
+* CWE: ~2 = 1200.000, ~26/15 = 951.881, ~5/4 = 387.104
-{{Optimal ET sequence|legend=1| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}
+Optimal ET sequence: {{Optimal ET sequence| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}
 Badness: 1.06 × 10<sup>-3</sup>
@@ Line 210: / Line 304: @@
 Mapping: {{mapping| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~26/15 = 951.8015, ~5/4 = 386.9912
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~26/15 = 951.802, ~5/4 = 386.991
+* CWE: ~2 = 1200.000, ~26/15 = 951.879, ~5/4 = 387.723
-{{Optimal ET sequence|legend=1| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}
+Optimal ET sequence: {{Optimal ET sequence| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}
 Badness: 1.24 × 10<sup>-3</sup>
-[[Category:Temperament clans]]
-[[Category:Pentacircle clan| ]] <!-- main article -->
-[[Category:Rank 3]]
-[[Category:Tolerant]]
 == Trienparapyth ==
@@ Line 235: / Line 326: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~11/10 = 165.4134, ~5/4 = 386.8872
+* [[CTE]]: ~2 = 1200.000, ~11/10 = 165.413, ~5/4 = 386.887
-* [[CWE]]: ~2 = 1\1, ~11/10 = 165.3593, ~5/4 = 387.8093
+: [[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
+: 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 }}
@@ Line 252: / Line 345: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~11/10 = 165.3975, ~5/4 = 386.7908
+* CTE: ~2 = 1200.000, ~11/10 = 165.398, ~5/4 = 386.791
-* CWE: ~2 = 1\1, ~11/10 = 165.3802, ~5/4 = 387.8759
+* CWE: ~2 = 1200.000, ~11/10 = 165.380, ~5/4 = 387.876
 Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}
@@ Line 259: / Line 352: @@
 Badness: 1.23 × 10<sup>-3</sup>
-=== no-17's 19-limit ===
+=== 2.3.5.7.11.13.19 subgroup ===
 Note [[109edo]] is a good patent val tuning not listed in the optimal ET sequence here.
@@ Line 271: / Line 364: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~11/10 = 165.2990, ~5/4 = 386.3154
+* CTE: ~2 = 1200.000, ~11/10 = 165.299, ~5/4 = 386.315
-* CWE: ~2 = 1\1, ~11/10 = 165.2976, ~5/4 = 387.7451
+* CWE: ~2 = 1200.000, ~11/10 = 165.298, ~5/4 = 387.745
 Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}
@@ Line 278: / Line 371: @@
 Badness: 1.22 × 10<sup>-3</sup>
-=== no-17's 23-limit ===
+=== 2.3.5.7.11.13.19.23 subgroup ===
-Note [[109edo]] is a good patent val tuning not listed in the optimal ET sequence here.
 Subgroup: 2.3.5.7.11.13.19.23
@@ Line 290: / Line 381: @@
 Optimal tunings
-* CTE: ~2 = 1\1, ~11/10 = 165.2579, ~5/4 = 386.1446
+* CTE: ~2 = 1200.000, ~11/10 = 165.258, ~5/4 = 386.145
-* CWE: ~2 = 1\1, ~11/10 = 165.2679, ~5/4 = 387.7240
+* CWE: ~2 = 1200.000, ~11/10 = 165.268, ~5/4 = 387.724
 Optimal ET sequence: {{Optimal ET sequence| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
@@ Line 297: / Line 388: @@
 Badness: 1.04 × 10<sup>-3</sup>
-== Pentafrost ==
+[[Category:Temperament clans]]
-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.
+[[Category:Pages with mostly numerical content]]
+[[Category:Pentacircle clan| ]] <!-- main article -->
-Subgroup: 2.3.5.7.11
+[[Category:Rank 3]]
+[[Category:Tolerant]]
-Comma list: [[245/242]], [[896/891]]
-Mapping: {{mapping| 1 0 15 0 7 | 0 1 -8 0 -4 | 0 0 0 1 1 }}
-: mapping generators: 2, ~3, ~7
-Optimal tunings:
-* CTE: 2 = 1\1, ~3 = 1901.883, ~7 = 3364.864
-* CWE: 2 = 1\1, ~3 = 1901.903, ~7 = 3364.614
-* CE: 2 = 1\1, ~3 = 1902.006, ~7 = 3364.085
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: [[245/242]], [[105/104]], [[352/351]]
-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 = 1\1, ~3 = 1902.106, ~7 = 3362.655
-* CWE: 2 = 1\1, ~3 = 1902.145, ~7 = 3362.175
-* CE: 2 = 1\1, ~3 = 1902.360, ~7 = 3362.210
-<br>
-=== Permafrost ===
-Subgroup: 2.3.5.7.11.13
-Comma list: [[245/242]], [[896/891]], [[144/143]]
-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 = 1\1, ~3 = 1901.783, ~7 = 3366.113
-* CWE: 2 = 1\1, ~3 = 1901.753, ~7 = 3366.445
-* CE: 2 = 1\1, ~3 = 1901.770, ~7 = 3365.771