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Subgroup temperaments: Difference between revisions

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For temperaments that omit various prime harmonics, see:  
For temperaments that omit various prime harmonics, see:  
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
Line 72: Line 73:
=== Baldy ===
=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|Schismatic family #Garibaldi}}
{{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 135: 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 272: 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 516: 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 787: 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 805: 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 812: 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 }}


Magicaltet is related to [[supermagic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.  
Magicaltet is related to [[keemic]], [[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 823: 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 835: 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 854: 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 877: 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 885: 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 902: 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 919: Line 970:
{{See also | Chromatic pairs #Gariberttet }}
{{See also | Chromatic pairs #Gariberttet }}


Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup.  
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].


[[Subgroup]]: 2.5/3.7/3.13/11
[[Subgroup]]: 2.5/3.7/3.13/11
Line 928: 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 935: 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 952: 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 960: 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 970: Line 1,016:
{{See also| Chromatic pairs #Ammon }}
{{See also| Chromatic pairs #Ammon }}


Ammon can be described as the {{nowrap| 8 & 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.
Ammon can be described as the {{nowrap| 8 & 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 old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.
 
It was formerly known as "semidim" but renamed to avoid confusion with another temperament of the same name.


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


{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}
[[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


{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
{{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=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
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].  
[[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 }}
 
Edson is related to [[pele]] and [[andromeda]].  


[[Subgroup]]: 2.3.7/5.11/5.13/5
[[Subgroup]]: 2.3.7/5.11/5.13/5
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 ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
[[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* }}
: <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 }}


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910
Comma list: [[323/322]], [[392/391]]


{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}
Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}


Scales: [[Oceanfront scales]]
{{Optimal ET sequence|legend=0| 7, 18, 25 }}


== 2.….49/5… subgroups ==
Badness (Sintel): 0.029
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]


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


[[Comma list]]: 2401/2400
Subgroup: 2.25/7.17/7.19/7.23/7


{{Mapping|legend=2| 1 1 3 | 0 2 1 }}
Comma list: [[323/322]], [[392/391]], [[476/475]]


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}


{{Optimal ET sequence|legend=1|?}}
Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* 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.'''
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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 ===
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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
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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 =
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* [[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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