Pentacircle clan: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
→Trienparapyth: comma bases
Preliminary cleanup on the intros. - redundant category
 
(31 intermediate revisions by 7 users not shown)
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]].  
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 ====
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 (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.  
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=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
 
Mapping: {{mapping| 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}}


{{Optimal ET sequence|legend=1| 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg }}
{{Optimal ET sequence|legend=0| 12f, 24, 29, 41 }}


Badness: 1.52 × 10<sup>-3</sup>
Badness (Sintel): 1.49


== Tolerant ==
=== Permafrost ===
=== 7-limit ===
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7


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


{{Mapping|legend=1| 1 0 0 -10 | 0 1 0 11 | 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 }}


{{Optimal ET sequence|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


Line 138: 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 152: 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 165: 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 179: 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 195: 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 210: 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 into three ~[[11/10]]'s by tempering out [[4000/3993]] in the 11-limit and it interprets (11/10)<sup>2</sup> accurately as [[23/19]] in its full subgroup, tempering out [[2300/2299]], 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]]. Next, for 8 gens, 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 8 gens is 8 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 231: Line 293:


{{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 ===
Line 248: Line 311:


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


=== 2.3.5.7.11.13.19 subgroup ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 267: Line 329:


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


=== 2.3.5.7.11.13.19.23 subgroup ===
=== 2.3.5.7.11.13.19.23 subgroup ===
Line 284: Line 345:


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]]

Latest revision as of 13:44, 16 March 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The pentacircle clan of rank-3 temperaments tempers out the pentacircle comma, 896/891. This has the effect of identifying 14/11 at the Pythagorean major third.

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

Parapythic

Main article: 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.

Subgroup: 2.3.7.11

Comma list: 896/891

Subgroup-val mapping: [⟨1 0 0 7], ⟨0 1 0 -4], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

  • WE: ~2 = 1199.3774 ¢, ~3/2 = 703.4693 ¢, ~7/4 = 969.3690 ¢
error map: ⟨-0.623 +0.892 -0.702 +1.061]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 703.7426 ¢, ~7/4 = 969.0476 ¢
error map: ⟨0.000 +1.788 +0.222 +2.759]

Optimal ET sequence: 12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee

Badness (Sintel): 0.299

Overview to extensions

Subgroup extensions

By tempering out 896/891, we have mapped 14/11 to the major third, suggesting a slightly sharp fifth. This makes the minor third very close to the flat-of-Pythagorean 13/11, and extending the temperament to include harmonic 13 this way implies we temper out 352/351. In fact, 896/891 = (352/351)⋅(364/363), so it is a very natural interpretation, giving rise to the 2.3.7.11.13 subgroup temperament shown below.

Full 11-limit extensions

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 syntonic comma (81/80) with the 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.

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:

Considered below are tolerant, kujuku, and terrapyth.

Parapyth

Main article: Parapyth

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

Subgroup-val mapping: [⟨1 0 0 7 12], ⟨0 1 0 -4 -7], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3706 ¢, ~3/2 = 703.4872 ¢, ~7/4 = 969.3987 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8328 ¢, ~7/4 = 969.1612 ¢

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

Badness (Sintel): 0.266

Etypyth

Subgroup: 2.3.7.11.13.17

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

Subgroup-val mapping: [⟨1 0 0 7 12 -13], ⟨0 1 0 -4 -7 9], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3607 ¢, ~3/2 = 703.6564 ¢, ~7/4 = 970.0880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.0139 ¢, ~7/4 = 969.8715 ¢

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

Badness (Sintel): 0.536

Terrapyth

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

Subgroup: 2.3.5.7.11

Comma list: 896/891, 585640/583443

Mapping: [⟨1 0 -31 0 7], ⟨0 1 21 0 -4], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3126 ¢, ~3/2 = 703.7780 ¢, ~7/4 = 970.0657 ¢
error map: ⟨-0.687 +1.136 -0.101 -0.135 +0.199]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1544 ¢, ~7/4 = 969.8575 ¢
error map: ⟨0.000 +2.199 +0.928 +1.032 +1.922]

Optimal ET sequence: 17c, 29, 46, 92de, 121, 167, 288be, 455bcde

Badness (Sintel): 6.43

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨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 ¢, ~3/2 = 703.7992 ¢, ~7/4 = 970.3331 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1459 ¢, ~7/4 = 970.0967 ¢

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

Badness (Sintel): 2.32

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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 ¢, ~3/2 = 703.7980 ¢, ~7/4 = 970.1592 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1406 ¢, ~7/4 = 969.9458 ¢

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

Badness (Sintel): 1.45

Pentafrost

Pentafrost tempers out the frostma in addition to 896/891 which also means that the schisma is tempered out, mapping prime 5 to eight 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: [⟨1 0 15 0 7], ⟨0 1 -8 0 -4], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.1251 ¢, ~3/2 = 701.9850 ¢, ~7/4 = 964.6139 ¢
error map: ⟨+0.125 +0.155 -1.318 -3.962 +5.982]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9034 ¢, ~7/4 = 964.6143 ¢
error map: ⟨0.000 -0.052 -1.541 -4.212 +5.683]

Optimal ET sequence: 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

Mapping: [⟨1 0 15 0 7 12], ⟨0 1 -8 0 -4 -7], ⟨0 0 0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.2502 ¢, ~3/2 = 702.3077 ¢, ~7/4 = 962.1832 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1455 ¢, ~7/4 = 962.1748 ¢

Optimal ET sequence: 12f, 24, 29, 41

Badness (Sintel): 1.49

Permafrost

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 15 0 7 -3], ⟨0 1 -8 0 -4 6], ⟨0 0 0 1 1 -1]]

Optimal tunings:

  • WE: 2 = 1199.6241 ¢, ~3/2 = 701.5280 ¢, ~7/4 = 966.2056 ¢
  • CWE: 2 = 1200.000 ¢, ~3/2 = 701.7534 ¢, ~7/4 = 966.4455 ¢

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

Badness (Sintel): 2.45

Tolerant

For the 7-limit version, see Miscellaneous 7-limit temperaments #Tolerant.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187

Mapping: [⟨1 0 0 -10 -3], ⟨0 1 0 11 7], ⟨0 0 1 -2 -2]]

mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1199.4396 ¢, ~3/2 = 703.7124 ¢, ~5/4 = 387.1118 ¢
error map: ⟨-0.560 +1.197 -0.323 -0.532 +0.445]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9092 ¢, ~5/4 = 386.9306 ¢
error map: ⟨0.000 +1.951 +0.617 +0.281 +2.164]

Optimal ET sequence: 34d, 39d, 41, 80, 87, 121, 167, 208, 288be, 375be

Badness (Sintel): 1.25

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -10 -3 2], ⟨0 1 0 11 7 4], ⟨0 0 1 -2 -2 -2]]

Optimal tunings:

  • WE: ~2 = 1199.5161 ¢, ~3/2 = 703.6767 ¢, ~5/4 = 386.8270 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8968 ¢, ~5/4 = 386.8916 ¢

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

Badness (Sintel): 0.955

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨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 ¢, ~3/2 = 703.7268 ¢, ~5/4 = 387.1310 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.0472 ¢, ~5/4 = 387.3450 ¢

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

Badness (Sintel): 0.934

Kujuku

Kujuku splits the perfect twelfth into two. Scott Dakota has aliased this temperament SQPP (for semiquartal parapyth).

Subgroup: 2.3.5.7.11

Comma list: 896/891, 14700/14641

Mapping: [⟨1 0 0 -13 -6], ⟨0 2 0 17 9], ⟨0 0 1 1 1]]

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

Optimal tunings:

  • WE: ~2 = 1199.3881 ¢, ~121/70 = 951.4033 ¢, ~5/4 = 387.4865 ¢
error map: ⟨-0.612 +0.852 -0.051 -0.752 +1.246]
  • CWE: ~2 = 1200.0000 ¢, ~121/70 = 951.8708 ¢, ~5/4 = 387.2432 ¢
error map: ⟨0.000 +1.787 +0.930 +0.220 +2.762]

Optimal ET sequence: 24, 29, 34d, 53d, 58, 87, 121, 145, 179e, 208, 266e

Badness (Sintel): 2.72

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 0 -13 -6 -1], ⟨0 2 0 17 9 3], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3660 ¢, ~26/15 = 951.3934 ¢, ~5/4 = 387.4050 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.8815 ¢, ~5/4 = 387.1043 ¢

Optimal ET sequence: 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

Subgroup: 2.3.5.7.11.13.17

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

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 ¢, ~26/15 = 951.3284 ¢, ~5/4 = 387.6639 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.8791 ¢, ~5/4 = 387.7230 ¢

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

Badness (Sintel): 1.18

Trienparapyth

Named by Godtone in 2024, trienparapyth can be described as the 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 (S10/S11) in the 11-limit. It further interprets (11/10)2 accurately as 23/19 in its full subgroup, tempering out 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 (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 (S39). Next, for eight generator steps, observe that (11/10)9/(11/10)/2 = (4/3)3/(11/10)/2 = (32/27)/(11/10) = 320/297 is sharp of 15/14 by 896/891, which is reasonable to equate it with because in an optimal tuning 11/10 will be very slightly sharp so that the interval of eight generator steps is eight times as sharp. Thus, tempering out 896/891 and 4000/3993 defines trienparapyth in the 11-limit, which also tempers out 3388/3375, the 13-limit adds 352/351, the no-17's 19-limit equates 40/39 with 39/38 and the no-17's 23-limit equates 23/19 with (11/10)2 as already mentioned.

Structurally, trienparapyth is three copies of parapyth with the independent generator of 7 connected to an equivalent independent generator for 5 through the ~15/7 reached at (11/10)8 so that ~20/7 is reached at (11/10)11, and this is how the last generator can be either 5 or 7.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 3388/3375

Mapping: [⟨1 2 0 2 1], ⟨0 -3 0 -11 1], ⟨0 0 1 1 1]]

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

Optimal tunings:

  • WE: ~2 = 1199.3706 ¢, ~11/10 = 165.2428 ¢, ~5/4 = 388.1147 ¢
error map: ⟨-0.629 +1.058 +0.542 -0.899 +0.151]
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.3593 ¢, ~5/4 = 387.8093 ¢
error map: ⟨0.000 +1.967 +1.496 +0.031 +1.851]

Optimal ET sequence: 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: [⟨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 ¢, ~11/10 = 165.2932 ¢, ~5/4 = 388.2127 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.3802 ¢, ~5/4 = 387.8759 ¢

Optimal ET sequence: 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: [⟨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 ¢, ~11/10 = 165.2022 ¢, ~5/4 = 388.1654 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.2976 ¢, ~5/4 = 387.7451 ¢

Optimal ET sequence: 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: [⟨1 2 0 2 1 0 0 0], ⟨0 -3 0 -11 1 10 14 16], ⟨0 0 1 1 1 1 1 1]]

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

Optimal tunings:

  • WE: ~2 = 1199.2714 ¢, ~11/10 = 165.1718 ¢, ~5/4 = 388.1729 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.2679 ¢, ~5/4 = 387.7240 ¢

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

Badness (Sintel): 1.14

Retrieved from "https://en.xen.wiki/index.php?title=Pentacircle_clan&oldid=226184"