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Subgroup temperaments: Difference between revisions

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{{See also|No-threes subgroup temperaments #Frostburn}}
{{See also|No-threes subgroup temperaments #Frostburn}}


Baldy results from taking every other generator of the [[garibaldi temperament]]. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.
Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.


[[Subgroup]]: 2.9.5.7
[[Subgroup]]: 2.9.5.7
Line 137: Line 137:


{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}
== 2.3.25 subgroup ==
=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.
Subgroup: 2.3.25
Edo join: 17 & 12
Comma list: [[2048/2025]]
{{Mapping|legend=2| 1 1 7| 0 1 -4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136
==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''
Edo join: 17 & 12
Comma list: 2048/2025, 576/575, 82/81
{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264
===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.
Subgroup: 2.3.7.23.25.41.59
Edo join: 17 & 12
Comma list: 64/63, 225/224, 162/161, 82/81, 177/175
{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}
Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387


== 2.9.5.11 subgroup ==
== 2.9.5.11 subgroup ==
Line 274: Line 313:


Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
== 2.9.7.13.17 subgroup ==
=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].
[[Subgroup]]: 2.9.7.13.17
[[Comma list]]: 729/728, 442/441, 833/832
{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}
[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
Badness (Dirichlet): 0.142


== 2.9.11 subgroup ==
== 2.9.11 subgroup ==
Line 518: Line 572:


Fourwar is named after the closely related [[hemiwar]] temperament.
Fourwar is named after the closely related [[hemiwar]] temperament.
{{Todo|inline=1|cleanup}}


<pre>  
<pre>  
Line 789: Line 845:
Mapping generators: ~4, ~9/64
Mapping generators: ~4, ~9/64


[[Optimal tuning]] ([[CTE]]): ~[[9/8]] = 219.190
[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190


[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49
Line 807: Line 863:
: [[gencom]]: [8 9/8; 64/63]
: [[gencom]]: [8 9/8; 64/63]


[[Optimal tuning]] ([[CTE]]): 1\[[3ed8]] = 1600.0, ~9/8 = 219.1898
[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898


[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}
Line 814: Line 870:


= 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 }}
Line 825: Line 881:


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


{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]
: [[gencom]]: [2 6/5; 100/99 385/384]


Line 837: Line 891:


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
 
: <nowiki/>* wart for 5/3
<nowiki/>* Wart for 5/3


[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents
Line 856: Line 909:


{{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 }}
: [[gencom]]: [2 6/5; 126/125]
: [[gencom]]: [2 6/5; 126/125]


Line 879: Line 931:


{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]
: [[gencom]]: [2 11/10; 121/120 126/125]


Line 887: Line 938:


{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
 
: <nowiki/>* wart for 11/3
<nowiki/>* Wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents
Line 904: Line 954:


{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]


Line 930: Line 979:


{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]
: [[gencom]]: [2 13/11; 275/273 847/845]


Line 937: Line 985:


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


[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents
Line 954: Line 1,001:


{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]


Line 962: Line 1,008:


{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*<sup>†</sup> }}
 
: <nowiki/>* wart for 7/3
<nowiki/>* Wart for 7/3
: <sup>†</sup> wart for 11/3
 
<sup>†</sup> Wart for 11/3


[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents
Line 981: Line 1,025:


{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]


Line 1,004: Line 1,047:


{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]
: [[gencom]]: [2 9/7; 245/243]


Line 1,011: Line 1,053:


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


[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents
Line 1,021: Line 1,061:
{{See also| Chromatic pairs #Marveltwintri }}
{{See also| Chromatic pairs #Marveltwintri }}


Marveltwintri can be described as the {{nowrap| 3 & 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.  
Marveltwintri can be described as the {{nowrap| 3 & 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. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].


[[Subgroup]]: 2.5/3.13/9
[[Subgroup]]: 2.5/3.13/9
Line 1,030: Line 1,070:


{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]
: [[gencom]]: [2 6/5; 325/324]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.886
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.861
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861


{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}
Line 1,041: Line 1,080:
[[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 }}


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.  
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. 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
Line 1,051: Line 1,090:
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})
[[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 1 3 | 0 -3 1 -5 }}
 
: mapping generators: ~2, ~7/6
: 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 }}
: [[gencom]]: [2 7/6; 176/175 540/539]
: [[gencom]]: [2 7/6; 176/175 540/539]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093


{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
 
: <nowiki/>* wart for 7/3
<nowiki/>* wart for 7/3


[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents


==== Laz ====
==== Tridecimal guanyintet ====
{{See also | Chromatic pairs #Laz }}
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])<sup>3</sup> = [[13/10]] and ([[15/14]])<sup>3</sup> = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.


Laz is related to [[avalokita]] as well as to [[winston]].  
[[Subgroup]]: 2.5.7/3.11/3.13


[[Subgroup]]: 2.5.7/3.11/3.13/3
[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})


[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}
{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7
 
[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218
 
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} <small> using subgroup TE </small>
: <nowiki/>* wart for 7/3
 
Badness (Sintel): 0.329
 
==== Laz ====
{{See also | Chromatic pairs #Laz }}
 
Laz is related to [[avalokita]] as well as to [[winston]].
 
[[Subgroup]]: 2.5.7/3.11/3.13/3
 
[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -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 }}


{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]


Line 1,089: Line 1,143:


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


[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents
Line 1,106: Line 1,158:


{{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 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 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]


Line 1,131: Line 1,181:


{{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 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 }}
: [[gencom]]: [2 21/20; 100/99 245/242]
: [[gencom]]: [2 21/20; 100/99 245/242]


Line 1,156: Line 1,204:


{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3
: 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 }}
: [[gencom]]: [2 7/6; 99/98]
: [[gencom]]: [2 7/6; 99/98]


Line 1,171: Line 1,217:
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


== 2.….9/7… subgroups ==
== 2.….9/7.… subgroups ==
=== Marveltri ===
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}
{{See also| Chromatic pairs #Marveltri }}
Line 1,182: Line 1,228:


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


{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]
: [[gencom]]: [2 5; 225/224]


Line 1,194: Line 1,238:


{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
 
: <nowiki/>* wart for 9/7
<nowiki/>* Wart for 9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents
Line 1,216: Line 1,259:
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


== 2.….7/5… subgroups ==
== 2.….7/5.… subgroups ==
 
=== Hydrothermal ===
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
Line 1,231: Line 1,273:
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}


=== Edson ===
=== Argentic ===
Edson is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  


==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
[[Subgroup]]: 2.3.7/5
 
[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}
 
{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830
 
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>
 
Badness (Sintel): 0.119
 
==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}
{{See also| Chromatic pairs #Edson }}


Line 1,244: Line 1,302:


{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]


Line 1,271: Line 1,327:


{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]


Line 1,311: Line 1,366:


{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]
: [[gencom]]: [63/50 10/9; 250047/250000]


Line 1,333: Line 1,387:


{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]
: [[gencom]]: [2 13/10; 847/845 1001/1000]


Line 1,342: Line 1,395:
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents


== 2.….11/5… subgroups ==
==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5


[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]
{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}
{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95<sup>t</sup>, 124<sup>t</sup> }}
: <sup>t</sup> wart for 17/5
[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents
== 2.….11/5.… subgroups ==
=== Petrtri ===
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| Chromatic pairs #Petrtri }}
Line 1,357: Line 1,426:


{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]
: [[gencom]]: [2 13/10; 2200/2197]


Line 1,378: Line 1,446:


{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]
: [[gencom]]: [2 15/13; 352/351 676/675]


Line 1,388: Line 1,455:


=== Hypnosis ===
=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]


[[Subgroup]]: 2.3.7.11/5.13
[[Subgroup]]: 2.3.7.11/5.13
Line 1,402: Line 1,469:
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


== 2.….11/7… subgroups ==
=== Trisect ===
=== Pepperoni ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


Pepperoni is generated by a fifth and can be described as the 5 &amp; 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].


[[Subgroup]]: 2.3.11/7.13/7
[[Subgroup]]: 2.3.7.11/5


[[Comma list]]: 352/351, 364/363
[[Comma list]]: 1029/1024, 4000/3993


{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}
{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}


{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742


: [[gencom]]: [2 3/2; 352/351 364/363]
{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
[[Tp tuning #T2 tuning|RMS error]]: ???


{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13


<nowiki />* Wart for 11/7
[[Comma list]]: 1029/1024, 1575/1573, 2080/2079


<sup>†</sup> Wart for 13/7
{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918


== 2.….13/5… subgroups ==
{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.


[[Subgroup]]: 2.3.13/5
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17


[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}


{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820


[[Badness]]: 0.002335
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}


; Music
[[Tp tuning #T2 tuning|RMS error]]: ???
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]


==== Tobago ====
===== Trisector =====
{{See also| Chromatic pairs #Tobago }}
[[Subgroup]]: 2.3.7.11/5.13.17.19


Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079


[[Subgroup]]: 2.3.11.13/5
{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}


[[Comma list]]: [[243/242]], [[676/675]]
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894


{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}
{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}


{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
[[Tp tuning #T2 tuning|RMS error]]: ???


: [[gencom]]: [55/39 15/13; 243/242 676/675]
===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079


{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038
 
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???
 
===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29
 
[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079
 
{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102
 
{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}
 
[[Tp tuning #T2 tuning|RMS error]]: ???


[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


==== Pakkanian hemipyth ====
Pepperoni is generated by a fifth and can be described as the 5 &amp; 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.


[[Subgroup]]: 2.3.11.13/5.17
[[Subgroup]]: 2.3.11/7.13/7


[[Comma list]]: 221/220, 243/242, 289/288
[[Comma list]]: 352/351, 364/363


{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}


[[Optimal tuning]]s:  
{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
: [[gencom]]: [2 3/2; 352/351 364/363]
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
 
{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
: <nowiki />* wart for 11/7
: <sup>†</sup> wart for 13/7
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
 
== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
 
[[Subgroup]]: 2.3.13/5
 
[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}
 
[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
 
{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
 
[[Badness]]: 0.002335
 
; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]
 
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}
 
Tobago, the 10 &amp; 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].
 
[[Subgroup]]: 2.3.11.13/5
 
[[Comma list]]: [[243/242]], [[676/675]]
 
{{Mapping|legend=2| 2 0 -1 -2 | 0 2 5 3 }}
 
{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]
 
[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312
 
{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents
 
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17
 
[[Comma list]]: 221/220, 243/242, 289/288
 
{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
 
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)


{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5
=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
[[Subgroup]]: 2.3.7.13/5
[[Comma list]]: 64/63, 91/90
{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
Scales: [[Oceanfront scales]]
== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]
[[Subgroup]]: 2.3.49/5
[[Comma list]]: 2401/2400
{{Mapping|legend=2| 1 1 3 | 0 2 1 }}
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
{{Optimal ET sequence|legend=1|7, 10, 17}}
[[Tp tuning #T2 tuning|RMS error]]: ?
== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.
[[Subgroup]]: 2.3.17/5
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.
[[Subgroup]]: 2.3.19/7
[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
<nowiki/>* wart for 19/7
[[Badness]] (Sintel): 0.082
=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].
[[Subgroup]]: 2.17/7.19/7
[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})
{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}


<nowiki />* Wart for 13/5
{{Optimal ET sequence|legend=1| 7, 18, 25 }}


=== Oceanfront ===
[[Badness]] (Sintel): 0.005
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]


[[Subgroup]]: 2.3.7.13/5
==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.


[[Comma list]]: 64/63, 91/90
Subgroup: 2.17/7.19/7.23/7


{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}
Comma list: [[323/322]], [[392/391]]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}


{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
{{Optimal ET sequence|legend=0| 7, 18, 25 }}


Scales: [[Oceanfront scales]]
Badness (Sintel): 0.029


== 2.….49/5… subgroups ==
==== 2.25/7.17/7.19/7.23/7 subgroup ====
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]


[[Subgroup]]: 2.3.49/5
Subgroup: 2.25/7.17/7.19/7.23/7


[[Comma list]]: 2401/2400
Comma list: [[323/322]], [[392/391]], [[476/475]]


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


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
Optimal tunings:
 
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
{{Optimal ET sequence|legend=1|?}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}


[[Tp tuning #T2 tuning|RMS error]]: ?
{{Optimal ET sequence|legend=0| 7, 18, 25 }}


Badness (Sintel): 0.053


== 3/2.5/2… subgroups ==
== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}
{{Main|Half-prime subgroup}}


Line 1,541: Line 1,772:


{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14
: sval mapping generators: ~3/2, ~15/14


Line 1,547: Line 1,777:


Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


==== 3/2.5/2.7/2.11/2 ====
==== 3/2.5/2.7/2.11/2 ====
Line 1,556: Line 1,785:


{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14
: sval mapping generators: ~3/2, ~15/14


Line 1,562: Line 1,790:


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


==== 3/2.5/2.7/2.11/2.13/2 ====
==== 3/2.5/2.7/2.11/2.13/2 ====
Line 1,575: Line 1,802:


[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


=== Semiwolf ===
=== Semiwolf ===
Line 1,613: Line 1,839:
[[Optimal ET sequence]]: [[8edf]], [[11edf]]
[[Optimal ET sequence]]: [[8edf]], [[11edf]]


== 3/2.5/4… subgroups ==
== 3/2.5/4.… subgroups ==
=== Poseidon ===
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''
'''This temperament will be subjected to renaming due to a conflict.'''
Line 1,642: Line 1,868:


Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


=== Doubleton ===
=== Doubleton ===
Line 1,657: Line 1,882:


Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
 
: <nowiki />* wart for 3/2
<nowiki />* Wart for 3/2


== 5/2-equave subgroups ==
== 5/2-equave subgroups ==
=== Hyperion ===
=== Hyperion ===
[[Subgroup]]: 5/2.7.11
[[Subgroup]]: 5/2.7.11
Line 1,674: Line 1,897:


Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
 
: <nowiki />* wart for 5/2
<nowiki />* Wart for 5/2


= Related temperament collections =
= Related temperament collections =
Line 1,682: Line 1,904:
* [[Substitute harmonic]] temperaments
* [[Substitute harmonic]] temperaments


<!-- main article -->
[[Category:Subgroup temperaments| ]] <!-- main article -->
 
[[Category:Temperament collections]]
[[Category:Temperament collections]][[Category:Subgroup]]
{{Todo| review | cleanup }}
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