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{{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]].  
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 6: Line 7:
{{Main| Parapyth }}
{{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
[[Subgroup]]: 2.3.7.11
Line 12: Line 13:
[[Comma list]]: 896/891
[[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:  
 
* [[WE]]: ~2 = 1199.3774{{c}}, ~3/2 = 703.4693{{c}}, ~7/4 = 969.3690{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8345, ~7/4 = 969.8722
: [[error map]]: {{val| -0.623 +0.892 -0.702 +1.061 }}
* [[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 }}


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


[[Badness]]: 0.0205 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.299


=== Overview to extensions ===
=== Overview to extensions ===
==== Subgroup 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 ====
==== Full 11-limit extensions ====
Line 31: 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.  
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:  
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]]
* ''[[Apollo]]'' → [[Marvel family #Apollo|Marvel family]]
* [[Sensamagic]] → [[Sensamagic family #Undecimal sensamagic|Sensamagic 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]]
* ''[[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]]
* ''[[Canta]]'' → [[Canou family #Canta|Canou family]]


Line 52: Line 56:
Comma list: 352/351, 364/363
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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.8563, ~7/4 = 969.9074
Optimal tunings:  
* 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}}


{{Optimal ET sequence|legend=1| 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: 0.101 × 10<sup>-3</sup>
Badness (Sintel): 0.266


==== Etypyth ====
==== Etypyth ====
Line 65: Line 71:
Comma list: 352/351, 364/363, 442/441
Comma list: 352/351, 364/363, 442/441


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


{{Optimal ET sequence|legend=1| 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: 0.325 × 10<sup>-3</sup>
Badness (Sintel): 0.536


== Terrapyth ==
== 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
[[Subgroup]]: 2.3.5.7.11
Line 80: Line 88:
[[Comma list]]: 896/891, 585640/583443
[[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:  
* [[WE]]: ~2 = 1199.3126{{c}}, ~3/2 = 703.7780{{c}}, ~7/4 = 970.0657{{c}}
: [[error map]]: {{val| -0.687 +1.136 -0.101 -0.135 +0.199 }}
* [[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217
{{Optimal ET sequence|legend=1| 17c, 29, 46, 92de, 121, 167, 288be, 455bcde }}


{{Optimal ET sequence|legend=1| 17c, 29, 46, 92de, 121, 167, 288be }}
[[Badness]] (Sintel): 6.43
 
[[Badness]]: 5.35 × 10<sup>-3</sup>


=== 13-limit ===
=== 13-limit ===
Line 97: Line 107:
Mapping: {{mapping| 1 0 -31 0 7 12 | 0 1 0 21 0 4 -7 | 0 0 0 1 1 1 }}
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:
* 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}}


{{Optimal ET sequence|legend=1| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}
{{Optimal ET sequence|legend=0| 17c, 29, 46, 75e, 92def, 121, 167, 288be }}


Badness: 2.48 × 10<sup>-3</sup>
Badness (Sintel): 2.32


=== 17-limit ===
=== 17-limit ===
Line 110: Line 122:
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: {{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:
* 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}}


{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}


Badness: 1.52 × 10<sup>-3</sup>
Badness (Sintel): 1.45


== Pentafrost ==
== 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
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: [[245/242]], [[896/891]]
[[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|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:  
 
* [[WE]]: ~2 = 1200.1251{{c}}, ~3/2 = 701.9850{{c}}, ~7/4 = 964.6139{{c}}
[[Optimal tuning]]s:
: [[error map]]: {{val| +0.125 +0.155 -1.318 -3.962 +5.982 }}
* [[CTE]]: ~2 = 1\1, ~3/2 = 701.883, ~7/4 = 964.864
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9034{{c}}, ~7/4 = 964.6143{{c}}
* [[CWE]]: ~2 = 1\1, ~3/2 = 701.903, ~7/4 = 964.614
: error map: {{val| 0.000 -0.052 -1.541 -4.212 +5.683 }}
* [[CE]]: ~2 = 1\1, ~3/2 = 702.006, ~7/4 = 964.085


{{Optimal ET sequence|legend=1| 12, 24, 29, 36, 41, 106d }}
{{Optimal ET sequence|legend=1| 12, 24, 29, 36, 41, 106d }}


[[Badness]]: 1.58 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.90


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: [[105/104]], [[245/242]], [[352/351]]
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: {{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:
Optimal tunings:
* CTE: ~2 = 1\1, ~3/2 = 702.106, ~7/4 = 962.655
* WE: ~2 = 1200.2502{{c}}, ~3/2 = 702.3077{{c}}, ~7/4 = 962.1832{{c}}
* CWE: ~2 = 1\1, ~3/2 = 702.145, ~7/4 = 962.175
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}}, ~7/4 = 962.1748{{c}}
* CE: ~2 = 1\1, ~3/2 = 702.360, ~7/4 = 962.210


Optimal ET sequence: {{Optimal ET sequence| 12f, 24, 29, 41 }}
{{Optimal ET sequence|legend=0| 12f, 24, 29, 41 }}


Badness: 1.59 × 10<sup>-3</sup>
Badness (Sintel): 1.49


=== Permafrost ===
=== Permafrost ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: [[144/143]], [[245/242]], [[896/891]]
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: {{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:
Optimal tunings:
* CTE: 2 = 1\1, ~3/2 = 701.783, ~7/4 = 966.113
* WE: 2 = 1199.6241{{c}}, ~3/2 = 701.5280{{c}}, ~7/4 = 966.2056{{c}}
* CWE: 2 = 1\1, ~3/2 = 701.753, ~7/4 = 966.445
* CWE: 2 = 1200.000{{c}}, ~3/2 = 701.7534{{c}}, ~7/4 = 966.4455{{c}}
* CE: 2 = 1\1, ~3/2 = 701.770, ~7/4 = 965.771


Optimal ET sequence: {{Optimal ET sequence| 12, 17, 24, 36, 41, 77e }}
{{Optimal ET sequence|legend=0| 12, 17, 24, 36, 41, 77e }}


Badness: 2.62 × 10<sup>-3</sup>
Badness (Sintel): 2.45


== Tolerant ==
== Tolerant ==
=== 7-limit ===
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Tolerant]].''
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 179200/177147
 
{{Mapping|legend=1| 1 0 0 -10 | 0 1 0 11 | 0 0 1 -2 }}
 
: mapping generators: ~2, ~3, ~5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.9571, ~5/4 = 386.8863
{{Optimal ET sequence|legend=1| 41, 80, 87, 121, 167, 208, 329b, 375b, 537b, 583b, 704bd }}
[[Badness]]: 0.165 × 10<sup>-3</sup>
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 194: Line 189:


{{Mapping|legend=1| 1 0 0 -10 -3 | 0 1 0 11 7 | 0 0 1 -2 -2 }}
{{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]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.0412, ~5/4 = 387.2927
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4396{{c}}, ~3/2 = 703.7124{{c}}, ~5/4 = 387.1118{{c}}
: [[error map]]: {{val| -0.560 +1.197 -0.323 -0.532 +0.445 }}
* [[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 }}


{{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>
[[Badness]] (Sintel): 1.25


=== 13-limit ===
=== 13-limit ===
Line 208: Line 208:
Mapping: {{mapping| 1 0 0 -10 -3 2 | 0 1 0 11 7 4 | 0 0 1 -2 -2 -2 }}
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:  
* 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}}


{{Optimal ET sequence|legend=1| 41, 46, 80, 87, 121, 167, 208, 375be, 583bef }}
{{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be }}


Badness: 1.021 × 10<sup>-3</sup>
Badness (Sintel): 0.955


=== 17-limit ===
=== 17-limit ===
Line 221: Line 223:
Mapping: {{mapping| 1 0 0 -10 -3 2 8 | 0 1 0 11 7 4 -1 | 0 0 1 -2 -2 -2 -1 }}
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:
* 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}}


{{Optimal ET sequence|legend=1| 41, 46, 75e, 80, 87, 121, 167, 288beg }}
{{Optimal ET sequence|legend=0| 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg }}


Badness: 0.982 × 10<sup>-3</sup>
Badness (Sintel): 0.934


== Kujuku ==
== Kujuku ==
Line 235: Line 239:


{{Mapping|legend=1| 1 0 0 -13 -6 | 0 2 0 17 9 | 0 0 1 1 1 }}
{{Mapping|legend=1| 1 0 0 -13 -6 | 0 2 0 17 9 | 0 0 1 1 1 }}
: mapping generators: ~2, ~121/70, ~5
: mapping generators: ~2, ~121/70, ~5


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~121/70 = 951.4956, ~5/4 = 386.7868
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3881{{c}}, ~121/70 = 951.4033{{c}}, ~5/4 = 387.4865{{c}}
: [[error map]]: {{val| -0.612 +0.852 -0.051 -0.752 +1.246 }}
* [[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 }}


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


[[Badness]]: 2.26 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.72


=== 13-limit ===
=== 13-limit ===
Line 251: Line 258:
Mapping: {{mapping| 1 0 0 -13 -6 -1 | 0 2 0 17 9 3 | 0 0 1 1 1 1 }}
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:
* 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=1| 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: 1.06 × 10<sup>-3</sup>
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
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
Line 266: Line 275:
Mapping: {{mapping| 1 0 0 -13 -6 -1 8 | 0 2 0 17 9 3 -2 | 0 0 1 1 1 1 -1 }}
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:
 
* WE: ~2 = 1199.2826{{c}}, ~26/15 = 951.3284{{c}}, ~5/4 = 387.6639{{c}}
{{Optimal ET sequence|legend=1| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.8791{{c}}, ~5/4 = 387.7230{{c}}


Badness: 1.24 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}


[[Category:Temperament clans]]
Badness (Sintel): 1.18
[[Category:Pentacircle clan| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Tolerant]]


== Trienparapyth ==
== 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.
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.
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 284: Line 290:
[[Subgroup]]: 2.3.5.7.11
[[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|legend=1| 1 2 0 2 1 | 0 -3 0 -11 1 | 0 0 1 1 1 }}
: mapping generators: ~2, ~11/10, ~5
: mapping generators: ~2, ~11/10, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~11/10 = 165.4134, ~5/4 = 386.8872
* [[WE]]: ~2 = 1199.3706{{c}}, ~11/10 = 165.2428{{c}}, ~5/4 = 388.1147{{c}}
* [[CWE]]: ~2 = 1\1, ~11/10 = 165.3593, ~5/4 = 387.8093
: [[error map]]: {{val| -0.629 +1.058 +0.542 -0.899 +0.151 }}
* [[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 }}


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


[[Badness]]: 1.26 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.52


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
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: {{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
: mapping generators: ~2, ~11/10, ~5


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 165.3975, ~5/4 = 386.7908
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 165.2932{{c}}, ~5/4 = 388.2127{{c}}
* CWE: ~2 = 1\1, ~11/10 = 165.3802, ~5/4 = 387.8759
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.3802{{c}}, ~5/4 = 387.8759{{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| 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce }}


Badness: 1.23 × 10<sup>-3</sup>
Badness (Sintel): 1.15


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


Subgroup: 2.3.5.7.11.13.19
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: {{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
: mapping generators: ~2, ~11/10, ~5


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/10 = 165.2990, ~5/4 = 386.3154
* WE: ~2 = 1199.3123{{c}}, ~11/10 = 165.2022{{c}}, ~5/4 = 388.1654{{c}}
* CWE: ~2 = 1\1, ~11/10 = 165.2976, ~5/4 = 387.7451
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.2976{{c}}, ~5/4 = 387.7451{{c}}


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


Badness: 1.22 × 10<sup>-3</sup>
Badness (Sintel): 1.20
 
=== no-17's 23-limit ===
Note [[109edo]] is a good patent val tuning not listed in the optimal ET sequence here.


=== 2.3.5.7.11.13.19.23 subgroup ===
Subgroup: 2.3.5.7.11.13.19.23
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: {{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


: mapping generators: ~2, ~11/10, ~5
Optimal tunings:  
* 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 tunings
{{Optimal ET sequence|legend=0| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
* CTE: ~2 = 1\1, ~11/10 = 165.2579, ~5/4 = 386.1446
* CWE: ~2 = 1\1, ~11/10 = 165.2679, ~5/4 = 387.7240


Optimal ET sequence: {{Optimal ET sequence| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}
Badness (Sintel): 1.14


Badness: 1.04 × 10<sup>-3</sup>
[[Category:Temperament clans]]
[[Category:Pentacircle clan| ]] <!-- main article -->
[[Category:Rank 3]]
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