Xenharmonic Wiki

Subgroup temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
m →2.….9/7… subgroups: oops, I noticed I had patent val on
Line 1,129: Line 1,129:
{{See also | Chromatic pairs #Guanyintet }}
{{See also | Chromatic pairs #Guanyintet }}


Guanyintet, the 4 & 9 temperament in the 2.5.7/3.11/3 subgroup, is related to [[guanyin]] as well as to [[orwell]].  
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is related to [[guanyin]] as well as to [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  


[[Subgroup]]: 2.5.7/3.11/3
[[Subgroup]]: 2.5.7/3.11/3


[[Comma list]]: [[176/175]], [[540/539]]
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})


{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
: mapping generators: ~2, ~12/7


{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
Line 1,141: Line 1,143:
: [[gencom]]: [2 7/6; 176/175 540/539]
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 270.093
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907
 
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 89, 191bc, 227bc, 231bc, 271bc, 311bc, 316bcd }}
<nowiki/>* wart for 7/3


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
Line 1,154: Line 1,160:
[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Subgroup]]: 2.5.7/3.11/3.13/3


[[Comma list]]: [[144/143]], [[176/175]], [[196/195]]
[[Comma list]]: [[144/143]] ({{monzo| 4 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}


{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
Line 1,162: Line 1,168:
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 269.300
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700
 
{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}


{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 58, 156bde, 205bde }}
<nowiki/>* wart for 7/3
 
† wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
Line 1,175: Line 1,187:
[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Subgroup]]: 2.5.7/3.11/3.13/3


[[Comma list]]: 56/55, 78/77, 91/90
[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})


{{Mapping|legend=2| 1 2 1 2 2 | 0 -3 -2 1 -1 }}
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
 
: mapping generators: ~2, ~13/12


{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
Line 1,183: Line 1,197:
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/12 = 132.428
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428


{{Optimal ET sequence|legend=1| 9, 63, 82bd, 91bde }}
{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents
Line 1,196: Line 1,212:
[[Subgroup]]: 2.9/5.7/3.11/9
[[Subgroup]]: 2.9/5.7/3.11/9


[[Comma list]]: 100/99, 245/242
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})


{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20


{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
Line 1,204: Line 1,222:
: [[gencom]]: [2 21/20; 100/99 245/242]
: [[gencom]]: [2 21/20; 100/99 245/242]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~21/20 = 87.892
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892


{{Optimal ET sequence|legend=1| 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd }}
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents
Line 1,213: Line 1,233:
{{See also| Chromatic pairs #Mothwelltri }}
{{See also| Chromatic pairs #Mothwelltri }}


Mothwelltri, the 1 &amp; 4 temperament in the 2.7/3.11 subgroup, is related to [[orwell]].  
Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.  


[[Subgroup]]: 2.7/3.11
[[Subgroup]]: 2.7/3.11


[[Comma list]]: [[99/98]]
[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})


{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3


{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
Line 1,225: Line 1,247:
: [[gencom]]: [2 7/6; 99/98]
: [[gencom]]: [2 7/6; 99/98]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~7/6 = 273.174
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174


{{Optimal ET sequence|legend=1| 9, 22, 40, 49c, 58c, 67c, 76c, 79, 101b, 123bc }}
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
Retrieved from "https://en.xen.wiki/w/Subgroup_temperaments"