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Subgroup temperaments: Difference between revisions

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→Petrtri: +see also
→Fractional subgroup temperaments: for 2.5/3 subgroups: factorize commas; +explanation on the names "magicaltet", "starlingtet", and "marveltwintri"; +CTE tunings; review optimal ET sequences
Line 869: Line 869:
= Fractional subgroup temperaments =
= Fractional subgroup temperaments =
== 2.5/3… subgroups ==
== 2.5/3… subgroups ==
=== Magicaltet ===
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}
{{See also| Chromatic pairs #Magicaltet }}


Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]].  
Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  


[[Subgroup]]: 2.5/3.7.11
[[Subgroup]]: 2.5/3.7.11
Line 888: Line 887:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~5/3 = 877.3426
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~5/3 = 877.351
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
Line 900: Line 899:
{{See also | Chromatic pairs #Starlingtet }}
{{See also | Chromatic pairs #Starlingtet }}


Starlingtet, the 4 & 15 temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]].  
Starlingtet, the 4 & 15 temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.  


[[Subgroup]]: 2.5/3.7/3
[[Subgroup]]: 2.5/3.7/3


[[Comma list]]: [[126/125]]
[[Comma list]]: [[126/125]] = {{monzo| 1 -3 1 }}


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


{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
Line 912: Line 913:
: [[gencom]]: [2 6/5; 126/125]
: [[gencom]]: [2 6/5; 126/125]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~6/5 = 311.154
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846


{{Optimal ET sequence|legend=1| 15, 19, 23, 27, 239b, 266b, 293b, 320b, 347b, 374b, 401b }}
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents
Line 921: Line 924:
{{See also| Chromatic pairs #Greeley }}
{{See also| Chromatic pairs #Greeley }}


Related temperaments: [[Porcupine family #Opossum|Opossum]], [[Starling temperaments #Nusecond|Nusecond]]
Greeley is related to [[opossum]] as well as to [[nusecond]].


[[Subgroup]]: 2.5/3.7/3.11/3
[[Subgroup]]: 2.5/3.7/3.11/3


[[Comma list]]: 121/120, 126/125
[[Comma list]]: 121/120 = {{monzo| -3 -1 0 2 }}, 126/125 = {{monzo| 1 -3 1 }}


{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}
Line 933: Line 936:
: [[gencom]]: [2 11/10; 121/120 126/125]
: [[gencom]]: [2 11/10; 121/120 126/125]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~11/10 = 155.776
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131†, 208*† }}
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}


<nowiki>*</nowiki> wart for 5/3
<nowiki>*</nowiki> wart for 11/3
 
<nowiki>†</nowiki> wart for 11/3


[[Tp tuning#T2 tuning|RMS error]]: 1.034 cents
[[Tp tuning#T2 tuning|RMS error]]: 1.034 cents
Line 950: Line 953:
[[Subgroup]]: 2.5/3.7/3.11.13/9
[[Subgroup]]: 2.5/3.7/3.11.13/9


[[Comma list]]: 56/55, 91/90, 100/99
[[Comma list]]: 56/55 = {{monzo| 3 -1 1 -1 }}, 91/90 = {{monzo| -1 -1 1 0 1 }}, 100/99 = {{monzo| 2 2 0 -1 }}


{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}
Line 958: Line 961:
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~6/5 = 313.842
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158


{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}
Line 974: Line 979:
[[Subgroup]]: 2.5/3.7/3.13/11
[[Subgroup]]: 2.5/3.7/3.13/11


[[Comma list]]: [[275/273]], [[847/845]]
[[Comma list]]: [[275/273]] = {{monzo| 0 2 -1 -1 }}, [[847/845]] = {{monzo| 0 -1 1 -2 }}


{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}
Line 982: Line 987:
: [[gencom]]: [2 13/11; 275/273 847/845]
: [[gencom]]: [2 13/11; 275/273 847/845]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/11 = 293.679
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679


{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143d, 237bd, 380bd }}
{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
 
<nowiki>*</nowiki> wart for 13/11


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
Line 995: Line 1,003:
[[Subgroup]]: 2.5/3.7/3.11/3
[[Subgroup]]: 2.5/3.7/3.11/3


[[Comma list]]: [[3025/3024]], [[3125/3087]]
[[Comma list]]: [[3025/3024]] = {{monzo| -4 2 -1 2 }}, [[3125/3087]] = {{monzo| 0 5 -3 }}


{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}
Line 1,003: Line 1,011:
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~12/11 = 147.010
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010
 
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
 
<nowiki>*</nowiki> wart for 7/3


{{Optimal ET sequence|legend=1| 33, 41, 49, 57, 106, 204, 253 }}
† wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
Line 1,012: Line 1,025:
{{See also| Chromatic pairs #Semidim }}
{{See also| Chromatic pairs #Semidim }}


Semidim can be described as the 8 &amp; 29 temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[#Tridec]].
Semidim can be described as the 8 &amp; 29 temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the name.  


[[Subgroup]]: 2.5/3.7/3.11/3.13/3
[[Subgroup]]: 2.5/3.7/3.11/3.13/3


[[Comma list]]: [[121/120]], [[169/168]], [[275/273]]
[[Comma list]]: [[121/120]] = {{monzo| -3 -1 0 2 }}, [[169/168]] = {{monzo| -3 0 -1 0 2 }}, [[275/273]] = {{monzo| 0 2 -1 1 -1 }}


{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}
Line 1,024: Line 1,037:
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 453.242
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242


{{Optimal ET sequence|legend=1| 5, 8, 13, 21, 29, 37, 44, 45, 188bde }}
{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents
Related temperament: [[Ammonite]]


=== Sentry ===
=== Sentry ===
Line 1,039: Line 1,052:
[[Subgroup]]: 2.5/3.9/7
[[Subgroup]]: 2.5/3.9/7


[[Comma list]]: [[245/243]]
[[Comma list]]: [[245/243]] = {{monzo| 0 1 -2 }}


{{Mapping|legend=2| 1 0 0 | 0 2 1 }}
{{Mapping|legend=2| 1 0 0 | 0 2 1 }}
Line 1,047: Line 1,060:
: [[gencom]]: [2 9/7; 245/243]
: [[gencom]]: [2 9/7; 245/243]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~9/7 = 440.902
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1\1, ~9/7 = 440.902
 
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}


{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145b, 166c, 197bc, 215c, 264bc }}
<nowiki>*</nowiki> wart for 5/3
 
† wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
Line 1,056: Line 1,074:
{{See also| Chromatic pairs #Marveltwintri }}
{{See also| Chromatic pairs #Marveltwintri }}


Marveltwintri can be described as the 3 &amp; 4 temperament in the 2.5/3.13/9 subgroup.  
Marveltwintri can be described as the 3 &amp; 4 temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name.  


[[Subgroup]]: 2.5/3.13/9
[[Subgroup]]: 2.5/3.13/9


[[Comma list]]: [[325/324]]
[[Comma list]]: [[325/324]] = {{monzo| -2 2 1 }}


{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}
{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}
Line 1,068: Line 1,086:
: [[gencom]]: [2 6/5; 325/324]
: [[gencom]]: [2 6/5; 325/324]


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~6/5 = 317.139
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861


{{Optimal ET sequence|legend=1| 11, 15, 19, 34, 53, 87, 140, 185, 219, 253, 287b, 321b }}
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents


== 2.….7/3… subgroups ==
== 2.….7/3… subgroups ==
=== Guanyintet ===
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}
{{See also | Chromatic pairs #Guanyintet }}
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