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


For the rank-4 pentacircle temperament, see [[Rank-4 temperament #Pentacircle (896/891)]].  
For the rank-4 pentacircle temperament, see [[Rank-4 temperament #Pentacircle (896/891)]].  


== Parapyth ==
== Parapythic ==
{{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


[[Sval]] [[mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 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]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.8345, ~7/4 = 969.8722
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3774{{c}}, ~3/2 = 703.4693{{c}}, ~7/4 = 969.3690{{c}}
: [[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 }}


{{Val list|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 ====
The second comma in the comma list determines how we extend parapyth to include the harmonic 5.  
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]]. 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.  
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]]
* ''[[Julius]]'' or ''[[varda]]'' → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
* ''[[Parahemif]]'' → [[Rastmic rank three clan #Parahemif|Rastmic rank-3 clan]]
* [[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]]
* ''[[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]]'' → [[Canou family #Canta|Canou family]]


Considered below are tolerant, kujuku, and terrapyth.  
Considered below are tolerant, kujuku, and terrapyth.  


=== 2.3.7.11.13 subgroup ===
=== Parapyth ===
{{Main| Parapyth }}
 
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


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


Sval mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 0 0 -4 -7 }}, {{val| 0 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 GPV sequence: {{Val list| 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 63: Line 71:
Comma list: 352/351, 364/363, 442/441
Comma list: 352/351, 364/363, 442/441


Sval mapping: [{{val| 1 0 0 0 7 12 -13 }}, {{val| 0 1 0 0 -4 -7 9 }}, {{val| 0 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 GPV sequence: {{Val list| 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 78: Line 88:
[[Comma list]]: 896/891, 585640/583443
[[Comma list]]: 896/891, 585640/583443


[[Mapping]]: [{{val| 1 0 -31 0 7 }}, {{val| 0 1 21 0 -4 }}, {{val| 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]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217
[[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 }}


{{Val list|legend=1| 17c, 29, 46, 92de, 121, 167, 288be }}
{{Optimal ET sequence|legend=1| 17c, 29, 46, 92de, 121, 167, 288be, 455bcde }}


[[Badness]]: 5.35 × 10<sup>-3</sup>
[[Badness]] (Sintel): 6.43


=== 13-limit ===
=== 13-limit ===
Line 93: Line 105:
Comma list: 352/351, 364/363, 9295/9261
Comma list: 352/351, 364/363, 9295/9261


Mapping: [{{val| 1 0 -31 0 7 12 }}, {{val| 0 1 0 21 0 4 -7 }}, {{val| 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 GPV sequence: {{Val list| 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 106: Line 120:
Comma list: 352/351, 364/363, 442/441, 715/714
Comma list: 352/351, 364/363, 442/441, 715/714


Mapping: [{{val| 1 0 -31 0 7 12 -13 }}, {{val| 0 1 0 21 0 4 -7 9 }}, {{val| 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 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=0| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
 
Badness (Sintel): 1.45
 
== 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 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
 
[[Comma list]]: 245/242, 896/891
 
{{Mapping|legend=1| 1 0 15 0 7 | 0 1 -8 0 -4 | 0 0 0 1 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1251{{c}}, ~3/2 = 701.9850{{c}}, ~7/4 = 964.6139{{c}}
: [[error map]]: {{val| +0.125 +0.155 -1.318 -3.962 +5.982 }}
* [[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 }}
 
{{Optimal ET sequence|legend=1| 12, 24, 29, 36, 41, 106d }}
 
[[Badness]] (Sintel): 1.90
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 245/242, 352/351


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620
Mapping: {{mapping| 1 0 15 0 7 12 | 0 1 -8 0 -4 -7 | 0 0 0 1 1 1 }}


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


Badness: 1.52 × 10<sup>-3</sup>
{{Optimal ET sequence|legend=0| 12f, 24, 29, 41 }}


== Tolerant ==
Badness (Sintel): 1.49
=== 7-limit ===
 
[[Subgroup]]: 2.3.5.7
=== Permafrost ===
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 179200/177147
Comma list: 144/143, 245/242, 896/891


[[Mapping]]: [{{val| 1 0 0 -10 }}, {{val| 0 1 0 11 }}, {{val| 0 0 1 -2 }}]
Mapping: {{mapping| 1 0 15 0 7 -3 | 0 1 -8 0 -4 6 | 0 0 0 1 1 -1 }}


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.9571, ~5/4 = 386.8863
{{Optimal ET sequence|legend=0| 12, 17, 24, 36, 41, 77e }}


{{Val list|legend=1| 41, 80, 87, 121, 167, 208, 329b, 375b, 537b, 583b, 704bd }}
Badness (Sintel): 2.45


[[Badness]]: 0.165 × 10<sup>-3</sup>
== Tolerant ==
: ''For the 7-limit version, see [[Miscellaneous 7-limit temperaments #Tolerant]].''


=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 896/891, 2200/2187
[[Comma list]]: 896/891, 2200/2187


[[Mapping]]: [{{val| 1 0 0 -10 -3 }}, {{val| 0 1 0 11 7 }}, {{val| 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 }}


{{Val list|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 148: Line 206:
Comma list: 325/324, 352/351, 364/363
Comma list: 325/324, 352/351, 364/363


Mapping: [{{val| 1 0 0 -10 -3 2 }}, {{val| 0 1 0 11 7 4 }}, {{val| 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 GPV sequence: {{Val list| 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 161: Line 221:
Comma list: 256/255, 325/324, 352/351, 364/363
Comma list: 256/255, 325/324, 352/351, 364/363


Mapping: [{{val| 1 0 0 -10 -3 2 8 }}, {{val| 0 1 0 11 7 4 -1 }}, {{val| 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 GPV sequence: {{Val list| 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 176: Line 238:
[[Comma list]]: 896/891, 14700/14641
[[Comma list]]: 896/891, 14700/14641


[[Mapping]]: [{{val| 1 0 0 -13 -6 }}, {{val| 0 2 0 17 9 }}, {{val| 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 = 951.8367, ~5/4 = 386.4048
[[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 }}


{{Val list|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 191: Line 256:
Comma list: 352/351, 364/363, 676/675
Comma list: 352/351, 364/363, 676/675


Mapping: [{{val| 1 0 0 -13 -6 -1 }}, {{val| 0 2 0 17 9 3 }}, {{val| 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, ~121/70 = 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 GPV sequence: {{Val list| 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 206: Line 273:
Comma list: 256/255, 352/351, 364/363, 676/675
Comma list: 256/255, 352/351, 364/363, 676/675


Mapping: [{{val| 1 0 0 -13 -6 -1 8 }}, {{val| 0 2 0 17 9 3 -2 }}, {{val| 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 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 }}
 
Badness (Sintel): 1.18
 
== Trienparapyth ==
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.
 
[[Subgroup]]: 2.3.5.7.11
 
[[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:
* [[WE]]: ~2 = 1199.3706{{c}}, ~11/10 = 165.2428{{c}}, ~5/4 = 388.1147{{c}}
: [[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| 22, 51, 58, 80, 87, 145, 167, 312ce, 479bce }}
 
[[Badness]] (Sintel): 1.52
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
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:
* 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 }}
 
Badness (Sintel): 1.15
 
=== 2.3.5.7.11.13.19 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
 
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:
* 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 }}
 
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
 
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 tuning (CTE): ~2 = 1\1, ~121/70 = 951.8015, ~5/4 = 386.9912
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 GPV sequence: {{Val list| 24, 34d, 58, 87, 121, 179ef, 208g, 266efg }}
{{Optimal ET sequence|legend=0| 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi }}


Badness: 1.24 × 10<sup>-3</sup>
Badness (Sintel): 1.14


[[Category:Temperament clans]]
[[Category:Temperament clans]]
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[[Category:Pentacircle clan| ]] <!-- main article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
[[Category:Tolerant]]
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