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No-threes subgroup temperaments: Difference between revisions

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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
 
[[Comma list]]: 78125/76832
 
{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
 
: Sval mapping generators: ~2, ~28/25
 
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
 
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
 
[[Badness]] (Sintel): 0.886
 
==== 2.5.7.11 ====
Subgroup: 2.5.7.11
 
Comma list: 245/242, 625/616
 
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
 
: Sval mapping generators: ~2, ~28/25
 
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
 
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
 
Badness (Sintel): 0.463
 
=== Mabilic ===
{{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.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 1071875/1048576
 
{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
 
{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
 
: [[gencom]]: [2 175/128; 1071875/1048576]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents
 
=== 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]].)
 
A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.


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


[[Comma list]]: 4194304/4117715
[[Comma list]]: [[2100875/2097152]]


[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]


Mapping generators: 2, ~8/7
[[Gencom]]: [2 256/245; 2100875/2097152]


[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]


[[Gencom]]: [2 8/7; 4194304/4117715]
Optimal tuning ([[POTE]]): ~256/245 = 77.205


[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910
{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}


{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents


=== 2.5.7.11 subgroup ===
=== 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.
 
Subgroup: 2.5.7
 
Comma basis: {{monzo|372 -159 -1}}
 
Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]
 
Optimal tuning (CTE): ~5/4 = 386.360
 
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
 
[[Badness]] (Sintel): 148.6
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


Comma list: 176/175, 1310720/1294139
Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
 
Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
 
Optimal tuning (CTE): ~5/4 = 386.361
 
{{Optimal ET sequence|legend=0|264, 733}}, ...
 
Badness (Sintel): 52.150
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
 
Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
 
Optimal tuning (CTE): ~5/4 = 386.361
 
{{Optimal ET sequence|legend=0|1525, 1789}}, ...


Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
Badness (Sintel): 10.518


Gencom: [2 8/7; 176/175 1310720/1294139]
=== Bastille ===
{{Main| Bastille }}


Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
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]].


Optimal tuning (POTE): ~8/7 = 227.114
Subgroup: 2.5.7


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
Comma list: {{Monzo|1426 -596 -15}}


=== 2.5.7.11.13 subgroup ===
Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
Subgroup: 2.5.7.11.13


Comma list: 176/175, 640/637, 847/845
Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243


Sval mapping: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...


Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
[[Badness]] (Sintel): 7224.3


Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
=== Augment ===
{{See also| Chromatic pairs #Augment }}


Optimal tuning (POTE): ~8/7 = 227.108
Augment is related to [[augmented]].  


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Subgroup]]: 2.5.7.11


=== 2.5.7.11.13.17 subgroup ===
[[Comma list]]: 56/55, 128/125
Subgroup: 2.5.7.11.13.17


Comma list: 176/175, 221/200, 640/637, 833/832
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}


Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}


Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
: [[gencom]]: [5/4 8/7; 56/55 128/125]


Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275


Optimal tuning (POTE): ~8/7 = 227.242
{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}


{{Optimal ET sequence|legend=1| 16, 21, 37 }}
[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents


== Didacus ==
=== Ostara ===
{{See also| Hemimean clan #Didacus }}
'''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.


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
Ostara can also refer to a collection of temperaments which temper out 16807/16796.


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 3136/3125
[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200


[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]


[[Gencom]]: [2 28/25; 3136/3125]
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢


[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}


Optimal tuning ([[POTE]]): ~28/25 = 93.772
[[Badness]] (Sintel): 11.731


{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


== Rainy ==
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


[[Subgroup]]: 2.5.7
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
 
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
 
Badness (Sintel): 3.415
 
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17


[[Comma list]]: [[2100875/2097152]]
Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]


[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


[[Gencom]]: [2 256/245; 2100875/2097152]
Optimal tunings:  
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢


[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


Optimal tuning ([[POTE]]): ~256/245 = 77.205
Badness (Sintel): 1.985


{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]


== Mercy ==
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
{{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 tunings:
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢


[[Subgroup]]: 2.5.7
{{Optimal ET sequence|legend=0| 93, 338, 431 }}


[[Comma list]]: 823543/819200
Badness (Sintel): 1.285


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


[[Gencom]]: [2 2744/2560; 823543/819200]
Subgroup: 2.5.7.11.19.29.31


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688


Optimal tuning ([[POTE]]): ~343/320 = 116.291
Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]


{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Optimal tuning (CTE): ~58/31 = 1084.628


=== 2.5.7.13 ===
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.


[[Gencom]]: [2 14/13; 343/338 640/637]
[[Subgroup]]: 2.5.7.13


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
[[Comma list]]: [[640/637]], [[10985/10976]]


Optimal tuning ([[POTE]]): ~14/13 = 116.094
{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}


{{Optimal ET sequence|legend=1| 10, 21, 31}}
{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}


=== 2.5.7.13.17 ===
: [[gencom]]: [2 16/13; 640/637 10985/10976]
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002


[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
==== Silver ====
{{See also| Chromatic pairs #Silver }}


Optimal tuning ([[POTE]]): ~14/13 = 115.769
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.


{{Optimal ET sequence|legend=1| 10, 21, 31}}
[[Subgroup]]: 2.5.7.13.17


=== 2.5.7.13.17.19 ===
[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}


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


[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711


Optimal tuning ([[POTE]]): ~14/13 = 115.716
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}


{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents


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


Line 183: Line 304:
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}


== Frostburn ==
[[Badness]] (Sintel): 0.573
{{See also| Magic family #Quadrimage }}


[[Subgroup]]: 2.5.7
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}
 
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]].
 
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 5021863/5000000
 
{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
 
: mapping generators: ~2, ~5, ~100/77
 
[[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¢
 
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
 
[[Badness]] (Sintel): 1.862
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma list: 847/845, 1001/1000
 
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢
 
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
 
Badness (Sintel): 0.179
 
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
 
Comma list: 170/169, 221/220, 847/845
 
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
 
Optimal tunings:
* WE: ~2 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢
 
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}
 
Badness (Sintel): 0.438
 
== Higher 2.5 temperaments ==
 
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
 
=== 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]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 1331/1280
 
[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]


[[Comma list]]: 78125/76832
[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846


{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}


: Sval mapping generators: ~2, ~28/25
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}


[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].  


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
[[Subgroup]]: 2.5.11


=== 2.5.7.11 ===
[[Comma list]]: [[15625/15488]]
Subgroup: 2.5.7.11


Comma list: 245/242, 625/616
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}


{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}


: Sval mapping generators: ~2, ~28/25
: [[gencom]]: [125/88 5/4; 15625/15488]


Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}


== Yer (rank 3) ==
[[Tp tuning #T2 tuning|RMS error]]: 0.3997
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
=== Insect ===
[[Subgroup]]: 2.5.11


[[Sval]] [[mapping]]: [{{val| 1 0 0 11 4 }}, {{val| 0 1 0 -2 -1 }}, {{val| 0 0 1 0 1 }}]
[[Comma list]]: 33275/32768


Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
{{Mapping|legend=2|1 0 5|0 3 -2}}


{{Optimal ET sequence|legend=1| 13, 24, 33, 37, 46, 57, 70, 127 }}
: Mapping generators, ~2, ~[[55/32]]


== Yamablu ==
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
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
{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}


[[Comma list]]: 209/208, 2057/2048, 83521/83486
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}


[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
Sephiroth is the no-7 restriction of [[muggles]].


Optimal tuning ([[POTE]]): ~17/13 = 462.9606
[[Subgroup]]: 2.5.11.13.17


{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
[[Comma list]]: 65/64, 170/169, 221/220


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}


== Ostara ==
{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}
'''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.
: [[gencom]]: [2 5/4; 65/64 170/169 221/220]


[[Subgroup]]: 2.5.7.11
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}


[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
=== Trader ===
[[Subgroup]]: 2.5.13


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
[[Comma list]]: [[26/25]]


=== 2.5.7.11.13 subgroup ===
{{Mapping|legend=2|1 2 3|0 1 2}}
Subgroup: 2.5.7.11.13


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
: Mapping generators, ~2, ~[[5/4]]


Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079


Optimal tuning (POTE): ~1664/1225 = 529.003¢
{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
=== Superquintal ===
[[Subgroup]]: 2.5.13


=== 2.5.7.11.13.17 subgroup ===
[[Comma list]]: 64000000/62748517
Subgroup: 2.5.7.11.13.17


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
{{Mapping|legend=2|1 5 6|0 -7 -6}}


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
: Mapping generators, ~2, ~13/10


Optimal tuning (POTE): ~1664/1225 = 529.003¢
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}


=== 2.5.7.11.13.17.19 subgroup ===
== No-threes-or-fives subgroup temperaments ==
Subgroup: 2.5.7.11.13.17.19
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.


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
=== Amaranthine ===
{{See also| No-fives subgroup temperaments #Chrysanthemum }}


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
Amaranthine is the high-accuracy 2.7.11 subgroup strong restriction of [[Gamelismic clan#11-limit 3|undecimal mothra]].


Optimal tuning (POTE): ~19/14 = 529.003¢
[[Subgroup]]: 2.7.11


== Pure onzonic ==
[[Comma list]]: 5767168/5764801
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
{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}


Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~7/4 = 968.913


Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}


Optimal tuning (POTE): ~11/8 = 551.370
Badness (Sintel): 0.031


{{Optimal ET sequence|legend=1|37, 1789}}
=== Score ===
{{See also| Chromatic pairs #Score }}


== Tricesimoprimal miracloid ==
[[Subgroup]]: 2.7.11.13
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
[[Comma list]]: 343/338, 847/832


Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}


Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
{{Mapping|legend=3| 1 0 0 1 3 1| 0 0 0 4 1 6 }}


Optimal tuning (CTE): ~58/31 = 1084.628
: [[gencom]]: [2 11/8; 343/338 847/832]


{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 540.099


== French decimal ==
{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
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
[[Tp tuning #T2 tuning|RMS error]]: 1.282 cents


Comma basis: {{monzo|372 -159 -1}}
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
Bossier can be described as the 3 &amp; 17 in the 2.7.11.13 subgroup.


Optimal tuning (CTE): ~5/4 = 386.360
[[Subgroup]]: 2.7.11.13


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
[[Comma list]]: [[1573/1568]], [[15488/15379]]


=== 2.5.7.11 subgroup ===
{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
Subgroup: 2.5.7.11


Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}


Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
: [[gencom]]: [2 14/11; 1573/1568 15488/15379]


Optimal tuning (CTE): ~5/4 = 386.361
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/11 = 421.309


{{Optimal ET sequence|legend=1|264, 733}}, ...
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}


=== 2.5.7.11.13 subgroup ===
[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
Subgroup: 2.5.7.11.13


Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
=== Voltage ===
Voltage is the 3 &amp; 7 temperament in the 2.7.13 subgroup.


Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
[[Subgroup]]: 2.7.13


Optimal tuning (CTE): ~5/4 = 386.361
[[Comma list]]: [[28672/28561]]


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}


== Mabon ==
{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.


Subgroup: 2.9.7
: [[gencom]]: [2, 16/13; 28672/28561]


Comma basis: 44957696/43046721
[[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))


Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}


Optimal tuning (CTE): ~729/448 = 870.792
[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents


{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...
=== Ultrakleismic ===
[[Subgroup]]: 2.7.17


=== 2.9.7.11 subgroup ===
[[Comma list]]: 4913/4802
Subgroup: 2.9.7.11


Comma basis: 896/891, 1331/1296
{{Mapping|legend=2|1 2 3|0 3 4}}


Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]
: Mapping generators, ~2, ~[[17/14]]


Optimal tuning (CTE): ~16/11 = 870.966
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446


{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}
{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}


== Bastille ==
=== Counterultrakleismic ===
{{Main|Bastille}}
[[Subgroup]]: 2.7.17
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
[[Comma list]]: 2024782584832/2015993900449


Comma list: {{Monzo|1426 -596 -15}}
{{Mapping|legend=2|1 0 1|0 10 11}}


Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
: Mapping generators, ~2, ~[[17/14]]


Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}


== Shipwreck ==
=== Shipwreck ===


[[Subgroup]]: 2.7.53
[[Subgroup]]: 2.7.53
Line 396: Line 583:
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}


== Movila ==
=== Lovecraft ===
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]].
{{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 }}


[[Subgroup]]: 2.5.11
: [[gencom]]: [2 13/11; 1352/1331]


[[Comma list]]: 1331/1280
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/11 = 279.318


[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}


[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents


{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
=== Blackbirds ===
{{See also | Chromatic pairs #Blackbirds }}


== Mavericks ==
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]]: 2.13.19
[[Subgroup]]: 2.11.13


[[Comma list]]: 47525504/47045881
[[Comma list]]: [[29282/28561]]


[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}


{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
: [[gencom]]: [13/11 11/8; 29282/28561]


== Vengeance ==
[[Optimal tuning]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660
Another lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].


[[Subgroup]]: 2.5.17
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}


[[Comma list]]: 78608/78125
[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents


{{Mapping|legend=2|1 1 1|0 3 7}}
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
[[Subgroup]]: 2.11.13


{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
[[Comma list]]: [[265837/262144]]


== Superquintal ==
{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
[[Subgroup]]: 2.5.13


[[Comma list]]: 64000000/62748517
{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}


{{Mapping|legend=2|1 5 6|0 -7 -6}}
: [[gencom]]: [2 143/128; 265837/262144]


: Mapping generators, ~2, ~13/10
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}


{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents


== Insect ==
=== Yamablu ===
[[Subgroup]]: 2.5.11
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]].


[[Comma list]]: 33275/32768
[[Subgroup]]: 2.11.13.17.19


{{Mapping|legend=2|1 0 5|0 3 -2}}
[[Comma list]]: 209/208, 2057/2048, 83521/83486


: Mapping generators, ~2, ~[[55/32]]
[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
Optimal tuning ([[POTE]]): ~17/13 = 462.9606


{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}


== Supraminor ==
[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Subgroup]]: 2.7.17


[[Comma list]]: 2024782584832/2015993900449
=== Mavericks ===


{{Mapping|legend=2|1 0 1|0 10 11}}
[[Subgroup]]: 2.13.19


: Mapping generators, ~2, ~[[17/14]]
[[Comma list]]: 47525504/47045881


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858
[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]


{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886


== Trader ==
{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
[[Subgroup]]: 2.5.13


[[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]]
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