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Biyatismic clan: Difference between revisions

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== Protomere ==
== Protomere ==
The head of this clan is protomere, named by [[Aura]] as part of a series of names ending in ''mere'' (Greek [[wiktionary: %CE%BC%CE%AD%CF%81%CE%BF%CF%82 #Ancient Greek|μέρος]], "part") where a number of [[rastma]]s are equated with a [[syntonic comma]]; in this case, it is a single rastma, hence ''proto-''.
The systematically derived name of the temperament from ''biyatisma'' would be ''biyatic''.
[[Subgroup]]: 2.3.5.11
[[Subgroup]]: 2.3.5.11


[[Comma list]]: 121/120
[[Comma list]]: 121/120


{{Mapping|legend=2| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
{{Mapping|legend=3| 1 0 1 2 | 0 1 1 1 | 0 0 -2 -1 }}
: mapping generators: ~2, ~3, ~11/10
: mapping generators: ~2, ~3, ~11/10


Line 75: Line 79:
Comma list: 121/120, 176/175, 351/350
Comma list: 121/120, 176/175, 351/350


Mapping: {{mapping| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}
{{Mapping|legend=0| 1 0 1 4 2 7 | 0 1 1 -1 1 -2 | 0 0 -2 3 -1 -1 }}


Mapping to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]
Mapping to lattice: [{{val| 0 1 -1 2 0 -3 }}, {{val| 0 1 1 -1 1 -2 }}]
Line 109: Line 113:
Comma list: 66/65, 121/120, 176/175
Comma list: 66/65, 121/120, 176/175


Mapping: {{mapping| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}
{{Mapping|legend=0| 1 0 1 4 2 2 | 0 1 1 -1 1 1 | 0 0 -2 3 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 143: Line 147:
Comma list: 105/104, 121/120, 196/195
Comma list: 105/104, 121/120, 196/195


Mapping: {{mapping| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}
{{Mapping|legend=0| 1 0 1 -3 2 -5 | 0 1 1 4 1 6 | 0 0 -2 -4 -1 -6 }}


Optimal tunings:  
Optimal tunings:  
Line 158: Line 162:
Comma list: 121/120, 225/224, 275/273
Comma list: 121/120, 225/224, 275/273


Mapping: {{mapping| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}
{{Mapping|legend=0| 1 0 1 -3 2 7 | 0 1 1 4 1 -2 | 0 0 -2 -4 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 190: Line 194:
Comma list: 66/65, 121/120, 126/125
Comma list: 66/65, 121/120, 126/125


Mapping: {{mapping| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}
{{Mapping|legend=0| 1 0 1 2 2 2 | 0 1 1 1 1 1 | 0 0 -2 -6 -1 1 }}


Optimal tunings:  
Optimal tunings:  
Line 224: Line 228:
Comma list: 121/120, 169/168, 352/351
Comma list: 121/120, 169/168, 352/351


Mapping: {{mapping| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}
{{Mapping|legend=0| 1 0 1 11 2 7 | 0 1 1 -5 1 -2 | 0 0 -2 -2 -1 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 261: Line 265:
Comma list: 66/65, 99/98, 121/120
Comma list: 66/65, 99/98, 121/120


Mapping: {{mapping| 1 0 3 1 3 1 | 0 1 3 2 2 0 | 0 0 -4 -1 -2 2 }}
{{Mapping|legend=0| 1 0 3 1 3 1 | 0 1 3 2 2 0 | 0 0 -4 -1 -2 2 }}


Optimal tunings:  
Optimal tunings:  
Line 291: Line 295:
[[Badness]] (Sintel): 0.701
[[Badness]] (Sintel): 0.701


==== 13-limit ====
=== Eros ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 351/350, 441/440
 
Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
 
Optimal tunings:
* WE: ~2 = 1201.0656{{c}}, ~3/2 = 700.7374{{c}}, ~22/21 = 78.5908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7005{{c}}, ~22/21 = 78.3253{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
 
Badness (Sintel): 1.36
 
==== Eros ====
Eros fairs impressively into the 23-limit as a rank-3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 [[~]][[23/22]]'s will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.  
Eros fairs impressively into the 23-limit as a rank-3 temperament; not only is it fairly simple (considering this is a subgroup as complex as the full 23-limit, with many challenges) but has the pleasing property that all the harmonics are on the negative side of the last generator. Specifically, -3 to 2 fifths and -5 to 0 [[~]][[23/22]]'s will provide odd harmonics 1–23 up to octave equivalence; you can think of this as a 6×6 grid, which is a recommendable place to start looking at its structure.  


Line 317: Line 306:
Comma list: 121/120, 196/195, 352/351
Comma list: 121/120, 196/195, 352/351


Mapping: {{mapping| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}
{{Mapping|legend=0| 1 0 1 3 2 7 | 0 1 1 0 1 -2 | 0 0 -4 -3 -2 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 327: Line 316:
Badness (Sintel): 1.08
Badness (Sintel): 1.08


===== 17-limit =====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 121/120, 154/153, 196/195, 352/351
Comma list: 121/120, 154/153, 196/195, 352/351


Mapping: {{mapping| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}
{{Mapping|legend=0| 1 0 1 3 2 7 6 | 0 1 1 0 1 -2 -1 | 0 0 -4 -3 -2 -2 -5 }}


Optimal tunings:  
Optimal tunings:  
Line 342: Line 331:
Badness (Sintel): 0.931
Badness (Sintel): 0.931


===== 19-limit =====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 121/120, 154/153, 196/195, 286/285, 352/351
Comma list: 121/120, 154/153, 196/195, 286/285, 352/351


Mapping: {{mapping| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}
{{Mapping|legend=0| 1 0 1 3 2 7 6 9 | 0 1 1 0 1 -2 -1 -3 | 0 0 -4 -3 -2 -2 -5 0 }}


Optimal tunings:  
Optimal tunings:  
Line 357: Line 346:
Badness (Sintel): 1.16
Badness (Sintel): 1.16


===== 23-limit =====
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351
Comma list: 121/120, 154/153, 161/160, 196/195, 286/285, 352/351


Mapping: {{mapping| 1 0 1 3 2 7 6 9 3 | 0 1 1 0 1 -2 -1 -3 1 | 0 0 -4 -3 -2 -2 -5 0 -1 }}
{{Mapping|legend=0| 1 0 1 3 2 7 6 9 3 | 0 1 1 0 1 -2 -1 -3 1 | 0 0 -4 -3 -2 -2 -5 0 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 372: Line 361:
Badness (Sintel): 1.08
Badness (Sintel): 1.08


==== Inanna ====
=== Astarte ===
This extension is catalogued as tridecimal aphrodite in [[Graham Breed]]'s temperament finder.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 351/350, 441/440
 
{{Mapping|legend=0| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 -4 -3 -2 -11 }}
 
Optimal tunings:
* WE: ~2 = 1201.0656{{c}}, ~3/2 = 700.7374{{c}}, ~22/21 = 78.5908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.7005{{c}}, ~22/21 = 78.3253{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 29ff, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
 
Badness (Sintel): 1.36
 
=== Inanna ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 121/120, 275/273
Comma list: 105/104, 121/120, 275/273


Mapping: {{mapping| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}
{{Mapping|legend=0| 1 0 1 3 2 1 | 0 1 1 0 1 2 | 0 0 -4 -3 -2 -7 }}


Optimal tunings:  
Optimal tunings:  
Line 387: Line 393:
Badness (Sintel): 1.01
Badness (Sintel): 1.01


==== Ishtar ====
=== Ishtar ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 121/120, 441/440
Comma list: 91/90, 121/120, 441/440


Mapping: {{mapping| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}
{{Mapping|legend=0| 1 0 1 3 2 -1 | 0 1 1 0 1 3 | 0 0 -4 -3 -2 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 404: Line 410:
== References ==
== References ==


[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic| ]] <!-- key article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Biyatismic_clan"