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{{Technical data page}}
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.  
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.  


== Llywelyn aka shoe ==
== Overview by mapping of 5 ==
{{See also| Chromatic pairs #Shoe }}
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
{{See also| Llywelyn clan #Llywelyn aka shoe }}
 
* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
* French decimal and trader have a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.
* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.
* Insect has a ~55/32 generator, three of which give the ~5/1.
* Frostburn has a ~28/25 generator, four of which give the ~8/5.
Others have a more complex mapping of 5.
 
== 2.5.7 temperaments ==
 
Temperaments discussed elsewhere include
* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
* Sidewalk ([[823543/800000]]) → [[2023/2000#Sidewalk]]
=== Frostburn ===
{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 4194304/4117715
[[Comma list]]: 78125/76832


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}


Mapping generators: 2, ~8/7
: Sval mapping generators: ~2, ~28/25


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389


[[Gencom]]: [2 8/7; 4194304/4117715]
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
[[Badness]] (Sintel): 0.886


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
==== 2.5.7.11 ====
 
=== 2.5.7.11 subgroup ===
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


Comma list: 176/175, 1310720/1294139
Comma list: 245/242, 625/616


Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}


Gencom: [2 8/7; 176/175 1310720/1294139]
: Sval mapping generators: ~2, ~28/25


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745


Optimal tuning (POTE): ~8/7 = 227.114
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
Badness (Sintel): 0.463


=== 2.5.7.11.13 subgroup ===
=== Mabilic ===
Subgroup: 2.5.7.11.13
{{See also| Chromatic pairs #Mabilic }}{{Main|Mabilic and trismegistus}}Given below is the no-three version of [[Mavila family#Armodue|armodue]], [[Mabila family#Semabila|semabila]], and [[Magic family#Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.


Comma list: 176/175, 640/637, 847/845
[[Subgroup]]: 2.5.7


Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
[[Comma list]]: 1071875/1048576


Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}


Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}


Optimal tuning (POTE): ~8/7 = 227.108
: [[gencom]]: [2 175/128; 1071875/1048576]


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236


=== 2.5.7.11.13.17 subgroup ===
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}
Subgroup: 2.5.7.11.13.17


Comma list: 176/175, 221/200, 640/637, 833/832
[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents


Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
=== Rainy ===
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]] (and [[valentine]]). Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5s and stacked 7s become easier to navigate, using the general-purpose diesis to simplify clusters. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)


Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.


Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
[[Subgroup]]: 2.5.7


Optimal tuning (POTE): ~8/7 = 227.242
[[Comma list]]: [[2100875/2097152]]


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]


== Didacus ==
[[Gencom]]: [2 256/245; 2100875/2097152]
{{See also| Hemimean clan #Didacus }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]


[[Subgroup]]: 2.5.7
Optimal tuning ([[POTE]]): ~256/245 = 77.205


[[Comma list]]: 3136/3125
{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}


[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents


[[Gencom]]: [2 28/25; 3136/3125]
=== French decimal ===
Conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
Subgroup: 2.5.7


Optimal tuning ([[POTE]]): ~28/25 = 93.772
Comma basis: {{monzo|372 -159 -1}}


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
Optimal tuning (CTE): ~5/4 = 386.360


== Rainy ==
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


[[Subgroup]]: 2.5.7
[[Badness]] (Sintel): 148.6
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


[[Comma list]]: [[2100875/2097152]]
Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]


[[Gencom]]: [2 256/245; 2100875/2097152]
Optimal tuning (CTE): ~5/4 = 386.361


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
{{Optimal ET sequence|legend=0|264, 733}}, ...


Optimal tuning ([[POTE]]): ~256/245 = 77.205
Badness (Sintel): 52.150


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625


== Mercy ==
Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
{{See also| Quince clan #Mercy }}


Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
Optimal tuning (CTE): ~5/4 = 386.361


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=0|1525, 1789}}, ...


[[Comma list]]: 823543/819200
Badness (Sintel): 10.518


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
=== Bastille ===
{{Main| Bastille }}


[[Gencom]]: [2 2744/2560; 823543/819200]
Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an [[wikipedia:Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
Subgroup: 2.5.7


Optimal tuning ([[POTE]]): ~343/320 = 116.291
Comma list: {{Monzo|1426 -596 -15}}


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]


=== 2.5.7.13 ===
Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
[[Badness]] (Sintel): 7224.3


[[Gencom]]: [2 14/13; 343/338 640/637]
=== Augment ===
{{See also| Chromatic pairs #Augment }}


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
Augment is related to [[augmented]].


Optimal tuning ([[POTE]]): ~14/13 = 116.094
[[Subgroup]]: 2.5.7.11


{{Optimal ET sequence|legend=1| 10, 21, 31}}
[[Comma list]]: 56/55, 128/125


=== 2.5.7.13.17 ===
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
: [[gencom]]: [5/4 8/7; 56/55 128/125]


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}


Optimal tuning ([[POTE]]): ~14/13 = 115.769
[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents


{{Optimal ET sequence|legend=1| 10, 21, 31}}
=== Ostara ===
'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.


=== 2.5.7.13.17.19 ===
Ostara can also refer to a collection of temperaments which temper out 16807/16796.
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
[[Subgroup]]: 2.5.7.11


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢


Optimal tuning ([[POTE]]): ~14/13 = 115.716
{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
[[Badness]] (Sintel): 11.731


== Pakkanen (rank 3) ==
==== 2.5.7.11.13 subgroup ====
[[Subgroup]]: 2.5.7.11
Subgroup: 2.5.7.11.13


[[Comma list]]: 625/616
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]


: mapping generators: ~2, ~5, ~11
Optimal tunings:  
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
Badness (Sintel): 3.415


== Frostburn ==
==== 2.5.7.11.13.17 subgroup ====
{{See also| Magic family #Quadrimage }}
Subgroup: 2.5.7.11.13.17


[[Subgroup]]: 2.5.7
Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]


[[Comma list]]: 78125/76832
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢


: Sval mapping generators: ~2, ~28/25
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
Badness (Sintel): 1.985


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19


=== 2.5.7.11 ===
Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢


: Sval mapping generators: ~2, ~28/25
{{Optimal ET sequence|legend=0| 93, 338, 431 }}


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Badness (Sintel): 1.285


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
=== Tricesimoprimal miracloid ===
{{See also|Tricesimoprimal miracloid/Eliora's approach|l1=Eliora's approach to tricesimoprimal miracloid}}
Described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.


== Yer (rank 3) ==
Subgroup: 2.5.7.11.19.29.31
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688


[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
Optimal tuning (CTE): ~58/31 = 1084.628


{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...


== Yamablu ==
=== Huntington ===
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].
{{See also| Chromatic pairs #Huntington }}


[[Subgroup]]: 2.11.13.17.19
Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.  


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Subgroup]]: 2.5.7.13


[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
[[Comma list]]: [[640/637]], [[10985/10976]]


Optimal tuning ([[POTE]]): ~17/13 = 462.9606
{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}


{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
: [[gencom]]: [2 16/13; 640/637 10985/10976]


== Ostara ==
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002
'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.  


Ostara can also refer to a collection of temperaments which temper out 16807/16796.
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}


[[Subgroup]]: 2.5.7.11
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
==== Silver ====
{{See also| Chromatic pairs #Silver }}


[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
[[Subgroup]]: 2.5.7.13.17


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]


=== 2.5.7.11.13 subgroup ===
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}
Subgroup: 2.5.7.11.13


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]


Optimal tuning (POTE): ~1664/1225 = 529.003¢
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}


=== 2.5.7.11.13.17 subgroup ===
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
Subgroup: 2.5.7.11.13.17


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
=== Pakkanen ===
[[Subgroup]]: 2.5.7.11


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
[[Comma list]]: 625/616


Optimal tuning (POTE): ~1664/1225 = 529.003¢
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
: mapping generators: ~2, ~5, ~11


=== 2.5.7.11.13.17.19 subgroup ===
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
Subgroup: 2.5.7.11.13.17.19


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
[[Badness]] (Sintel): 0.573


Optimal tuning (POTE): ~19/14 = 529.003¢
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}


== Pure onzonic ==
This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands [[Subgroup temperaments #Tridec|tridec]] and [[Subgroup temperaments #Naiadec|naiadec]].
The 2.5.11.13 subgroup primarily contains temperaments developed for 1789edo, since it tempers out the jacobin comma 6656/6655, for which 2.5.11.13 is the subgroup, and the year 1789 is hallmark for the French revolution.


Subgroup: 2.5.11.13
[[Subgroup]]: 2.5.7.11


Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
[[Comma list]]: 5021863/5000000


Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}


Optimal tuning (POTE): ~11/8 = 551.370
: mapping generators: ~2, ~5, ~100/77


{{Optimal ET sequence|legend=1|37, 1789}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.080¢, ~5 = 2786.820¢, ~100/77 = 454.618¢
* [[CWE]]: ~2 = 1200.000¢, ~5 = 2786.740¢, ~100/77 = 454.590¢


== Tricesimoprimal miracloid ==
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
Described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.


Subgroup: 2.5.7.11.19.29.31
[[Badness]] (Sintel): 1.862


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
Comma list: 847/845, 1001/1000


Optimal tuning (CTE): ~58/31 = 1084.628
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}


{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
Optimal tunings:
* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢


== French decimal ==
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
Conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.


Subgroup: 2.5.7
Badness (Sintel): 0.179


Comma basis: {{monzo|372 -159 -1}}
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
Comma list: 170/169, 221/220, 847/845


Optimal tuning (CTE): ~5/4 = 386.360
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
Optimal tunings:
* WE: ~2 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢


=== 2.5.7.11 subgroup ===
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}
Subgroup: 2.5.7.11


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
Badness (Sintel): 0.438


Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
== Higher 2.5 temperaments ==


Optimal tuning (CTE): ~5/4 = 386.361
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]


{{Optimal ET sequence|legend=1|264, 733}}, ...
=== Movila ===
This temperament has a structure very similar to [[mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].


=== 2.5.7.11.13 subgroup ===
[[Subgroup]]: 2.5.11
Subgroup: 2.5.7.11.13


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
[[Comma list]]: 1331/1280


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]


Optimal tuning (CTE): ~5/4 = 386.361
[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


== Bastille ==
=== Wizz ===
{{Main|Bastille}}
{{See also| Chromatic pairs #Wizz }}
Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Pure Bastille|pure bastille]].


Subgroup: 2.5.7
Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].


Comma list: {{Monzo|1426 -596 -15}}
[[Subgroup]]: 2.5.11


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
[[Comma list]]: [[15625/15488]]


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}


== Shipwreck ==
: [[gencom]]: [125/88 5/4; 15625/15488]


[[Subgroup]]: 2.7.53
[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768


[[Comma list]]: 1048576/1042139
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}


[[Gencom]]: [2 64/53; 1048576/1042139]
[[Tp tuning #T2 tuning|RMS error]]: 0.3997


[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
=== Insect ===
[[Subgroup]]: 2.5.11


[[POTE generator]]: ~64/53 = 323.034
[[Comma list]]: 33275/32768


{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
{{Mapping|legend=2|1 0 5|0 3 -2}}


== Movila ==
: Mapping generators, ~2, ~[[55/32]]
This temperament has a structure very similar to [[mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].


[[Subgroup]]: 2.5.11
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032


[[Comma list]]: 1331/1280
{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}


[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
Sephiroth is the no-7 restriction of [[muggles]].  


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
[[Subgroup]]: 2.5.11.13.17


== Mavericks ==
[[Comma list]]: 65/64, 170/169, 221/220


[[Subgroup]]: 2.13.19
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}


[[Comma list]]: 47525504/47045881
{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}


[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
: [[gencom]]: [2 5/4; 65/64 170/169 221/220]


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}


== Vengeance ==
[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents
''Main article: [[vengeance]]''<br><br>


Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
=== Trader ===
[[Subgroup]]: 2.5.13


[[Subgroup]]: 2.5.17
[[Comma list]]: [[26/25]]


[[Comma list]]: 78608/78125
{{Mapping|legend=2|1 2 3|0 1 2}}


{{Mapping|legend=2|1 1 1|0 3 7}}
: Mapping generators, ~2, ~[[5/4]]


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079


{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}


== Superquintal ==
=== Superquintal ===
[[Subgroup]]: 2.5.13
[[Subgroup]]: 2.5.13


Line 425: Line 456:
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}


== Insect ==
== No-threes-or-fives subgroup temperaments ==
[[Subgroup]]: 2.5.11
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* Berylic → [[4th-octave temperaments #Berylic|4th-octave temperaments]]
* 21-23-commatic → [[21st-octave temperaments #21-23-commatic|21st-octave temperaments]]
* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]
* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)|37th-octave temperaments]]
* etc.
 
=== Amaranthine ===
{{See also| No-fives subgroup temperaments #Chrysanthemum }}
 
Amaranthine is the high-accuracy 2.7.11 subgroup strong restriction of [[Gamelismic clan#11-limit 3|undecimal mothra]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 5767168/5764801
 
{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~7/4 = 968.913
 
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}
 
Badness (Sintel): 0.031
 
=== Score ===
{{See also| Chromatic pairs #Score }}
 
[[Subgroup]]: 2.7.11.13
 
[[Comma list]]: 343/338, 847/832
 
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}
 
{{Mapping|legend=3| 1 0 0 1 3 1| 0 0 0 4 1 6 }}
 
: [[gencom]]: [2 11/8; 343/338 847/832]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 540.099
 
{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents
 
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
 
Bossier can be described as the 3 &amp; 17 in the 2.7.11.13 subgroup.
 
[[Subgroup]]: 2.7.11.13
 
[[Comma list]]: [[1573/1568]], [[15488/15379]]
 
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
 
{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
 
: [[gencom]]: [2 14/11; 1573/1568 15488/15379]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/11 = 421.309
 
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
 
=== Voltage ===
Voltage is the 3 &amp; 7 temperament in the 2.7.13 subgroup.
 
[[Subgroup]]: 2.7.13
 
[[Comma list]]: [[28672/28561]]
 
{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}


[[Comma list]]: 33275/32768
{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}


{{Mapping|legend=2|1 0 5|0 3 -2}}
: [[gencom]]: [2, 16/13; 28672/28561]


: Mapping generators, ~2, ~[[55/32]]
[[Optimal tuning]]:
* [[POTE]]: ~2 = 1\1, ~16/13 = 357.677
* [[TOP tuning|POTT]]: ~2 = 1\1, ~16/13 = 357.794 (1200 - 300 log<sub>2</sub>(7))


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}


{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents


== Ultrakleismic ==
=== Ultrakleismic ===
[[Subgroup]]: 2.7.17
[[Subgroup]]: 2.7.17


Line 451: Line 556:
{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}
{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}


== Counterultrakleismic ==
=== Counterultrakleismic ===
[[Subgroup]]: 2.7.17
[[Subgroup]]: 2.7.17


Line 464: Line 569:
{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}
{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}


== Trader ==
=== Shipwreck ===
[[Subgroup]]: 2.5.13
 
[[Subgroup]]: 2.7.53
 
[[Comma list]]: 1048576/1042139
 
[[Gencom]]: [2 64/53; 1048576/1042139]
 
[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
 
[[POTE generator]]: ~64/53 = 323.034
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
 
=== Lovecraft ===
{{See also | Chromatic pairs #Lovecraft }}
 
Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: [[1352/1331]]
 
{{Mapping|legend=2| 1 3 3 | 0 2 3 }}
 
{{Mapping|legend=3| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
 
: [[gencom]]: [2 13/11; 1352/1331]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/11 = 279.318
 
{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents
 
=== Blackbirds ===
{{See also | Chromatic pairs #Blackbirds }}
 
Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: [[29282/28561]]
 
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}
 
{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
 
: [[gencom]]: [13/11 11/8; 29282/28561]
 
[[Optimal tuning]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660
 
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents
 
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: [[265837/262144]]
 
{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
 
{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
 
: [[gencom]]: [2 143/128; 265837/262144]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents
 
=== Yamablu ===
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].
 
[[Subgroup]]: 2.11.13.17.19
 
[[Comma list]]: 209/208, 2057/2048, 83521/83486
 
[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
 
Optimal tuning ([[POTE]]): ~17/13 = 462.9606
 
{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
 
=== Mavericks ===
 
[[Subgroup]]: 2.13.19
 
[[Comma list]]: 47525504/47045881
 
[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
 
{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}


[[Comma list]]: [[26/25]]
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19


{{Mapping|legend=2|1 2 3|0 1 2}}
[[Comma list]]: 209/208, 2057/2048


: Mapping generators, ~2, ~[[5/4]]
[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638


{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"