Pentacircle clan: Difference between revisions

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* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.7426{{c}}, ~7/4 = 969.0476{{c}}

: error map: {{val| 0.000 +1.788 +0.222 +2.759 }}

~~<!-- * [[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 }} -->~~

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

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

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

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

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

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

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

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

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

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

Considered below are tolerant, kujuku, and terrapyth.

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* WE: ~2 = 1199.3706{{c}}, ~3/2 = 703.4872{{c}}, ~7/4 = 969.3987{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8328{{c}}, ~7/4 = 969.1612{{c}}

~~<!-- * 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}} -->

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

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* WE: ~2 = 1199.3607{{c}}, ~3/2 = 703.6564{{c}}, ~7/4 = 970.0880{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.0139{{c}}, ~7/4 = 969.8715{{c}}

~~<!-- * 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}} -->

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

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1544{{c}}, ~7/4 = 969.8575{{c}}

: error map: {{val| 0.000 +2.199 +0.928 +1.032 +1.922 }}

~~<!-- * [[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, 455bcde }}

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Comma list: 352/351, 364/363, 9295/9261

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

{{Mapping|legend=0| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}

Optimal tunings:

* WE: ~2 = 1199.3695{{c}}, ~3/2 = 703.7992{{c}}, ~7/4 = 970.3331{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1459{{c}}, ~7/4 = 970.0967{{c}}

~~<!-- * 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}} -->

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

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Comma list: 352/351, 364/363, 442/441, 715/714

~~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 }}

{{Mapping|legend=0| 1 0 -31 0 7 12 -13 | 0 1 0 21 0 4 -7 9 | 0 0 0 1 1 1 1 1 }}

Optimal tunings:

* WE: ~2 = 1199.3783{{c}}, ~3/2 = 703.7980{{c}}, ~7/4 = 970.1592{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1406{{c}}, ~7/4 = 969.9458{{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}} -->

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

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== 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~~.

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 eight [[4/3|perfect fourths]] minus an octave.

It was named by [[Tristan Bay]] in 2024 as a portmanteau of ''pentacircle'' and ''frost''.

[[Subgroup]]: 2.3.5.7.11

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9034{{c}}, ~7/4 = 964.6143{{c}}

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

~~<!-- * [[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 }}~~

* [[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 }} -->~~

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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|legend=0| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }}

Optimal tunings:

* WE: ~2 = 1200.2502{{c}}, ~3/2 = 702.3077{{c}}, ~7/4 = 962.1832{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}, ~7/4 = 962.1748{{c}}

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

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

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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|legend=0| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }}

Optimal tunings:

* WE: 2 = 1199.6241{{c}}, ~3/2 = 701.5280{{c}}, ~7/4 = 966.2056{{c}}

* CWE: 2 = 1200.000{{c}}, ~3/2 = 701.7534{{c}}, ~7/4 = 966.4455{{c}}

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

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

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9092{{c}}, ~5/4 = 386.9306{{c}}

: error map: {{val| 0.000 +1.951 +0.617 +0.281 +2.164 }}

~~<!-- * [[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 }} -->~~

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

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Comma list: 325/324, 352/351, 364/363

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

{{Mapping|legend=0| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}

Optimal tunings:

* WE: ~2 = 1199.5161{{c}}, ~3/2 = 703.6767{{c}}, ~5/4 = 386.8270{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8968{{c}}, ~5/4 = 386.8916{{c}}

~~<!-- * 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}} -->

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

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Comma list: 256/255, 325/324, 352/351, 364/363

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

{{Mapping|legend=0| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}

Optimal tunings:

* WE: ~2 = 1199.3929{{c}}, ~3/2 = 703.7268{{c}}, ~5/4 = 387.1310{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.0472{{c}}, ~5/4 = 387.3450{{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}} -->

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

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~121/70 = 951.8708{{c}}, ~5/4 = 387.2432{{c}}

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

~~<!-- * [[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 }} -->~~

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

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Comma list: 352/351, 364/363, 676/675

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

{{Mapping|legend=0| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}

Optimal tunings:

* WE: ~2 = 1199.3660{{c}}, ~26/15 = 951.3934{{c}}, ~5/4 = 387.4050{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.8815{{c}}, ~5/4 = 387.1043{{c}}

~~~~

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

Badness (Sintel): 0.991

~~Complexity spectrum: 15/13, 4/3, 13/10, 9/8, 13/11, 15/11, 12/11, 11/9, 11/8, 14/11, 16/13, 16/15, 11/10, 13/12, 9/7, 5/4, 18/13, 7/6, 6/5, 8/7, 10/9, 14/13, 15/14, 7/5~~

=== 17-limit ===

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Comma list: 256/255, 352/351, 364/363, 676/675

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

{{Mapping|legend=0| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}

Optimal tunings:

* WE: ~2 = 1199.2826{{c}}, ~26/15 = 951.3284{{c}}, ~5/4 = 387.6639{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.8791{{c}}, ~5/4 = 387.7230{{c}}

~~~~

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

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== Trienparapyth ==

~~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.

Named by [[Godtone]] in 2024, trienparapyth can be described as the {{nowrap| 58 & 80 & 87 }} temperament, with an extension to the no-17's 23-limit. It splits the ~4/3 generator of parapythic into three [[~]][[11/10]]'s by tempering out [[4000/3993]] ([[S-expression|S10/S11]]) in the 11-limit. It further interprets (11/10)2 accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]] ([[S-expression|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]] ({{S|39}}). 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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* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/10 = 165.3593{{c}}, ~5/4 = 387.8093{{c}}

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

~~<!-- * [[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 }} -->~~

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

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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|legend=0| 1 2 0 2 1 0 | 0 -3 0 -11 1 10 | 0 0 1 1 1 1 }}

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

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* WE: ~2 = 1199.4286{{c}}, ~11/10 = 165.2932{{c}}, ~5/4 = 388.2127{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.3802{{c}}, ~5/4 = 387.8759{{c}}

~~~~

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

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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|legend=0| 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

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* WE: ~2 = 1199.3123{{c}}, ~11/10 = 165.2022{{c}}, ~5/4 = 388.1654{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.2976{{c}}, ~5/4 = 387.7451{{c}}

~~~~

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

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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|legend=0| 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

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* WE: ~2 = 1199.2714{{c}}, ~11/10 = 165.1718{{c}}, ~5/4 = 388.1729{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.2679{{c}}, ~5/4 = 387.7240{{c}}

~~~~

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

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Badness (Sintel): 1.14

[[Category:Pentacircle clan| ]]

[[Category:Temperament clans]]

[[Category:~~Pentacircle clan| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-3 temperaments]]

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

~~[[Category:Tolerant~~]]

@@ Line 21: / Line 21: @@
 * [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.7426{{c}}, ~7/4 = 969.0476{{c}}
 : error map: {{val| 0.000 +1.788 +0.222 +2.759 }}
-<!-- * [[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 }} -->
 {{Optimal ET sequence|legend=1| 12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee }}
@@ Line 45: / Line 41: @@
 * ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]
 * [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic family]]
-* [[Pele]] → [[Hemifamity family #Pele|Hemifamity family]]
+* [[Pele]] → [[Aberschismic family #Pele|Aberschismic family]]
 * ''[[Uni]]'' → [[Hemimage family #Uni|Hemimage family]]
 * ''[[Varda]]'' → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
 * ''[[Parahemif]]'' → [[Rastmic rank-3 clan #Parahemif|Rastmic rank-3 clan]]
-* ''[[Canta]]'' → [[Canou family #Canta|Canou family]]
+* ''[[Canta]]'' → [[Cathartic family #Canta|Cathartic family]]
 Considered below are tolerant, kujuku, and terrapyth.
@@ Line 65: / Line 61: @@
 * WE: ~2 = 1199.3706{{c}}, ~3/2 = 703.4872{{c}}, ~7/4 = 969.3987{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8328{{c}}, ~7/4 = 969.1612{{c}}
-<!-- * 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}} -->
 {{Optimal ET sequence|legend=0| 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff }}
@@ Line 82: / Line 76: @@
 * WE: ~2 = 1199.3607{{c}}, ~3/2 = 703.6564{{c}}, ~7/4 = 970.0880{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.0139{{c}}, ~7/4 = 969.8715{{c}}
-<!-- * 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}} -->
 {{Optimal ET sequence|legend=0| 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e }}
@@ Line 103: / Line 95: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1544{{c}}, ~7/4 = 969.8575{{c}}
 : error map: {{val| 0.000 +2.199 +0.928 +1.032 +1.922 }}
-<!-- * [[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, 455bcde }}
@@ Line 117: / Line 105: @@
 Comma list: 352/351, 364/363, 9295/9261
-Mapping: {{mapping| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}
+{{Mapping|legend=0| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}
 Optimal tunings:
 * WE: ~2 = 1199.3695{{c}}, ~3/2 = 703.7992{{c}}, ~7/4 = 970.3331{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1459{{c}}, ~7/4 = 970.0967{{c}}
-<!-- * 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}} -->
 {{Optimal ET sequence|legend=0| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
@@ Line 134: / Line 120: @@
 Comma list: 352/351, 364/363, 442/441, 715/714
-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 }}
+{{Mapping|legend=0| 1 0 -31 0 7 12 -13 | 0 1 0 21 0 4 -7 9 | 0 0 0 1 1 1 1 1 }}
 Optimal tunings:
 * WE: ~2 = 1199.3783{{c}}, ~3/2 = 703.7980{{c}}, ~7/4 = 970.1592{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.1406{{c}}, ~7/4 = 969.9458{{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}} -->
 {{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
@@ Line 147: / Line 131: @@
 == 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.
+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 eight [[4/3|perfect fourths]] minus an octave.
+It was named by [[Tristan Bay]] in 2024 as a portmanteau of ''pentacircle'' and ''frost''.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 160: / Line 146: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9034{{c}}, ~7/4 = 964.6143{{c}}
 : error map: {{val| 0.000 -0.052 -1.541 -4.212 +5.683 }}
-<!-- * [[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 }}
-* [[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 }}
@@ Line 174: / Line 156: @@
 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|legend=0| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }}
 Optimal tunings:
 * WE: ~2 = 1200.2502{{c}}, ~3/2 = 702.3077{{c}}, ~7/4 = 962.1832{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}, ~7/4 = 962.1748{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.106{{c}}, ~7/4 = 962.655{{c}}
-* CEE: ~2 = 1200.000{{c}}, ~3/2 = 702.360{{c}}, ~7/4 = 962.210{{c}} -->
 {{Optimal ET sequence|legend=0| 12f, 24, 29, 41 }}
@@ Line 191: / Line 171: @@
 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|legend=0| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }}
 Optimal tunings:
 * WE: 2 = 1199.6241{{c}}, ~3/2 = 701.5280{{c}}, ~7/4 = 966.2056{{c}}
 * CWE: 2 = 1200.000{{c}}, ~3/2 = 701.7534{{c}}, ~7/4 = 966.4455{{c}}
-<!-- * CTE: 2 = 1200.000{{c}}, ~3/2 = 701.783{{c}}, ~7/4 = 966.113{{c}}
-* CEE: 2 = 1200.000{{c}}, ~3/2 = 701.770{{c}}, ~7/4 = 965.771{{c}} -->
 {{Optimal ET sequence|legend=0| 12, 17, 24, 36, 41, 77e }}
@@ Line 218: / Line 196: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.9092{{c}}, ~5/4 = 386.9306{{c}}
 : error map: {{val| 0.000 +1.951 +0.617 +0.281 +2.164 }}
-<!-- * [[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 }} -->
 {{Optimal ET sequence|legend=1| 34d, 39d, 41, 80, 87, 121, 167, 208, 288be, 375be }}
@@ Line 232: / Line 206: @@
 Comma list: 325/324, 352/351, 364/363
-Mapping: {{mapping| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}
+{{Mapping|legend=0| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}
 Optimal tunings:
 * WE: ~2 = 1199.5161{{c}}, ~3/2 = 703.6767{{c}}, ~5/4 = 386.8270{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8968{{c}}, ~5/4 = 386.8916{{c}}
-<!-- * 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}} -->
 {{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be }}
@@ Line 249: / Line 221: @@
 Comma list: 256/255, 325/324, 352/351, 364/363
-Mapping: {{mapping| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}
+{{Mapping|legend=0| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}
 Optimal tunings:
 * WE: ~2 = 1199.3929{{c}}, ~3/2 = 703.7268{{c}}, ~5/4 = 387.1310{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.0472{{c}}, ~5/4 = 387.3450{{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}} -->
 {{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg }}
@@ Line 276: / Line 246: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~121/70 = 951.8708{{c}}, ~5/4 = 387.2432{{c}}
 : error map: {{val| 0.000 +1.787 +0.930 +0.220 +2.762 }}
-<!-- * [[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 }} -->
 {{Optimal ET sequence|legend=1| 24, 29, 34d, 53d, 58, 87, 121, 145, 179e, 208, 266e }}
@@ Line 288: / Line 256: @@
 Comma list: 352/351, 364/363, 676/675
-Mapping: {{mapping| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}
+{{Mapping|legend=0| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}
 Optimal tunings:
 * WE: ~2 = 1199.3660{{c}}, ~26/15 = 951.3934{{c}}, ~5/4 = 387.4050{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.8815{{c}}, ~5/4 = 387.1043{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.852{{c}}, ~5/4 = 386.089{{c}} -->
 {{Optimal ET sequence|legend=0| 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef }}
 Badness (Sintel): 0.991
-Complexity spectrum: 15/13, 4/3, 13/10, 9/8, 13/11, 15/11, 12/11, 11/9, 11/8, 14/11, 16/13, 16/15, 11/10, 13/12, 9/7, 5/4, 18/13, 7/6, 6/5, 8/7, 10/9, 14/13, 15/14, 7/5
 === 17-limit ===
@@ Line 306: / Line 271: @@
 Comma list: 256/255, 352/351, 364/363, 676/675
-Mapping: {{mapping| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}
+{{Mapping|legend=0| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}
 Optimal tunings:
 * WE: ~2 = 1199.2826{{c}}, ~26/15 = 951.3284{{c}}, ~5/4 = 387.6639{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.8791{{c}}, ~5/4 = 387.7230{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~26/15 = 951.802{{c}}, ~5/4 = 386.991{{c}} -->
 {{Optimal ET sequence|legend=0| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}
@@ Line 318: / Line 282: @@
 == Trienparapyth ==
-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.
+Named by [[Godtone]] in 2024, trienparapyth can be described as the {{nowrap| 58 & 80 & 87 }} temperament, with an extension to the no-17's 23-limit. It splits the ~4/3 generator of parapythic into three [[~]][[11/10]]'s by tempering out [[4000/3993]] ([[S-expression|S10/S11]]) in the 11-limit. It further interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]] ([[S-expression|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]] ({{S|39}}). 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 334: / Line 298: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~11/10 = 165.3593{{c}}, ~5/4 = 387.8093{{c}}
 : error map: {{val| 0.000 +1.967 +1.496 +0.031 +1.851 }}
-<!-- * [[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 }} -->
 {{Optimal ET sequence|legend=1| 22, 51, 58, 80, 87, 145, 167, 312ce, 479bce }}
@@ Line 346: / Line 308: @@
 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|legend=0| 1 2 0 2 1 0 | 0 -3 0 -11 1 10 | 0 0 1 1 1 1 }}
 : mapping generators: ~2, ~11/10, ~5
@@ Line 352: / Line 314: @@
 * WE: ~2 = 1199.4286{{c}}, ~11/10 = 165.2932{{c}}, ~5/4 = 388.2127{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.3802{{c}}, ~5/4 = 387.8759{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.398{{c}}, ~5/4 = 386.791{{c}} -->
 {{Optimal ET sequence|legend=0| 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}
@@ Line 365: / Line 326: @@
 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|legend=0| 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
@@ Line 371: / Line 332: @@
 * WE: ~2 = 1199.3123{{c}}, ~11/10 = 165.2022{{c}}, ~5/4 = 388.1654{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.2976{{c}}, ~5/4 = 387.7451{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.299{{c}}, ~5/4 = 386.315{{c}} -->
 {{Optimal ET sequence|legend=0| 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h }}
@@ Line 382: / Line 342: @@
 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|legend=0| 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
@@ Line 388: / Line 348: @@
 * WE: ~2 = 1199.2714{{c}}, ~11/10 = 165.1718{{c}}, ~5/4 = 388.1729{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.2679{{c}}, ~5/4 = 387.7240{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~11/10 = 165.258{{c}}, ~5/4 = 386.145{{c}} -->
 {{Optimal ET sequence|legend=0| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
@@ Line 394: / Line 353: @@
 Badness (Sintel): 1.14
+[[Category:Pentacircle clan| ]] <!-- main article -->
 [[Category:Temperament clans]]
-[[Category:Pentacircle clan| ]] <!-- main article -->
+[[Category:Catalogs of rank-3 temperaments]]
-[[Category:Rank 3]]
-[[Category:Tolerant]]