Xenharmonic Wiki

Schismatic family: Difference between revisions

← Older editNewer edit →
VisualWikitext
Lériendil (talk | contribs)
8,046 edits
make it more clear -> non-linind comma
→Subgroup extensions: cleanup; sort by split of generator
Line 2,403: Line 2,403:


== Subgroup extensions ==
== Subgroup extensions ==
=== Taylor (2.3.5.13) ===
=== Photia (2.3.5.17) ===
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.
{{See also| No-elevens subgroup temperaments #Garibaldia }}


Subgroup: 2.3.5.13
[[Subgroup]]: 2.3.5.17


[[Comma list]]: 676/675, 32805/32768
[[Comma list]]: 256/255, 1458/1445


[[Gencom]]: [2 15/13; 676/675 32805/32768]
{{Mapping|legend=2| 1 2 -1 7 | 0 -1 8 -7 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 1}}, {{val|0 -2 16 0 0 13}}]
{{Mapping|legend=3| 1 2 -1 0 0 0 7 | 0 -1 8 0 0 0 -7 }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 1}}, {{val|0 -2 16 13}}]
: [[gencom]]: [2 4/3; 256/255 1458/1445]


[[Tp tuning|POL2 generator]]: ~15/13 = 249.145
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.491


{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents


=== Photia (2.3.5.17) ===
==== 2.3.5.17.19 ====
: <small>''See also: [[No-elevens subgroup temperaments #Garibaldia]]''</small>
[[Subgroup]]: 2.3.5.17.19


Subgroup: 2.3.5.17
[[Comma list]]: 171/170, 256/255, 324/323
 
[[Comma list]]: 256/255, 1458/1445


[[Gencom]]: [2 4/3; 256/255 1458/1445]
{{Mapping|legend=2| 1 2 -1 7 3 | 0 -1 8 -7 3 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7}}, {{val|0 -1 8 0 0 0 -7}}]
{{Mapping|legend=3| 1 2 -1 0 0 0 7 3 | 0 -1 8 0 0 0 -7 3 }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
: [[gencom]]: [2 4/3; 171/170 256/255 324/323]


[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.470


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
 
=== Nestoria (2.3.5.19) ===
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''


==== 2.3.5.17.19 ====
The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20]]'')}.  
Subgroup: 2.3.5.17.19


[[Comma list]]: 171/170, 256/255, 324/323
[[Subgroup]]: 2.3.5.19


[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
[[Comma list]]: 361/360, 513/512


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
: mapping generators: ~2, ~3


[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
{{Mapping|legend=3| 1 2 -1 0 0 0 0 3 | 0 -1 8 0 0 0 0 3 }}


{{Optimal ET sequence|legend=1| 12, 41, 53, 65 }}
: [[gencom]]: [2 4/3; 361/360 513/512]


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.746


=== Quintilischis (2.3.5.17) ===
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
: <small>''For full 17- and 19-limit extensions, see [[Schismatic family#Quintilipyth|Quintilipyth]] or [[Schismatic family#Quintaschis|Quintaschis]].''</small>


Subgroup: 2.3.5.17
[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents


[[Comma list]]: 32805/32768, 1419857/1417176
=== Taylor (2.3.5.13) ===
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.


[[Gencom]]: [2 18/17; 32805/32768 1419857/1417176]
[[Subgroup]]: 2.3.5.13


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5}}, {{val|0 -5 40 0 0 0 -11}}]
[[Comma list]]: 676/675, 32805/32768


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5}}, {{val|0 -5 40 -11}}]
{{Mapping|legend=2| 1 2 -1 1 | 0 -2 16 13 }}


[[Tp tuning|POL2 generator]]: ~18/17 = 99.649
{{Mapping|legend=3| 1 2 -1 0 0 1 | 0 -2 16 0 0 13 }}


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
: [[gencom]]: [2 15/13; 676/675 32805/32768]


[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~15/13 = 249.145


==== 2.3.5.17.19 ====
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236 }}
Subgroup: 2.3.5.17.19


[[Comma list]]: 4624/4617, 6144/6137, 6885/6859
[[Tp tuning #T2 tuning|RMS error]]: 0.1485 cents


[[Gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]
=== Quintilischis (2.3.5.17) ===
: ''For full 17- and 19-limit extensions, see [[#Quintilipyth]] or [[#Quintaschis]].''


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 5 4}}, {{val|0 -5 40 0 0 0 -11 3}}]
[[Subgroup]]: 2.3.5.17


[[Mapping|Sval mapping]]: [{{val|1 2 -1 5 4}}, {{val|0 -5 40 -11 3}}]
[[Comma list]]: 32805/32768, 1419857/1417176


[[Tp tuning|POL2 generator]]: ~18/17 = 99.652
{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }}


{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}
{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1636 cents
: [[gencom]]: [2 18/17; 32805/32768 1419857/1417176]


=== Nestoria (2.3.5.19) ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~18/17 = 99.649
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]''


The [[S-expression]]-based comma list of this temperament is {[[1216/1215|S16/S18]], [[361/360|S19]] (, ''[[513/512|S15/S20]]'')}.
{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}


[[Subgroup]]: 2.3.5.19
[[Tp tuning #T2 tuning|RMS error]]: 0.0719 cents


[[Comma list]]: 361/360, 513/512
==== 2.3.5.17.19 ====
[[Subgroup]]: 2.3.5.17.19


{{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }}
[[Comma list]]: 4624/4617, 6144/6137, 6885/6859


: mapping generators: ~2, ~3
{{Mapping|legend=2| 1 2 -1 5 4 | 0 -5 40 -11 3 }}


{{Mapping|legend=3| 1 2 -1 0 0 0 0 3 | 0 -1 8 0 0 0 0 3 }}
{{Mapping|legend=3| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }}


: [[gencom]]: [2 4/3; 361/360 513/512]
: [[gencom]]: [2 18/17; 4624/4617 6144/6137 6885/6859]


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.746
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~18/17 = 99.652


{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171 }}
{{Optimal ET sequence|legend=1| 12, 253, 265, 277, 289 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1636 cents


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Schismatic_family"