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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 ==
== Overview by mapping of 5 ==
Subgroup: 2.5.7
Classified by focusing on the mapping of 5th harmonic, similar to [[Rank-2 temperaments by mapping of 3]].
 
* 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 }}


[[Comma]]: 4194304/4117715
[[Subgroup]]: 2.5.7


[[Gencom]]: [2 8/7; 4194304/4117715]
[[Comma list]]: 78125/76832


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


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
: Sval mapping generators: ~2, ~28/25


[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389


{{Val list|legend=1| 16, 37 }}
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
[[Badness]] (Sintel): 0.886


=== 2.5.7.11 ===
==== 2.5.7.11 ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma]]: 176/175, 1310720/1294139
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
 
[[Comma list]]: [[2100875/2097152]]


[[Gencom]]: [2 8/7; 176/175 1310720/1294139]
[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1}}, {{val|0 0 7 -1 13}}]
[[Gencom]]: [2 256/245; 2100875/2097152]


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


[[Tp tuning|POL2 generator]]: ~8/7 = 227.114
Optimal tuning ([[POTE]]): ~256/245 = 77.205


{{Val list|legend=1| 16, 21, 37 }}
{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}


=== 2.5.7.11.13 ===
[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Subgroup: 2.5.7.11
 
=== 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.


[[Comma]]: 176/175, 640/639, 1308672/1294139
Subgroup: 2.5.7


[[Gencom]]: [2 8/7; 176/175 640/639, 1308672/1294139]
Comma basis: {{monzo|372 -159 -1}}


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2}}, {{val|0 0 7 -1 13 9}}]
Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]


[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2}}, {{val|0 7 -1 13 9}}]
Optimal tuning (CTE): ~5/4 = 386.360


[[Tp tuning|POL2 generator]]: ~8/7 = 227.108
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...


{{Val list|legend=1| 16, 21, 37 }}
[[Badness]] (Sintel): 148.6


=== 2.5.7.11.13.17 ===
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma]]: 176/175, 221/200, 640/639, 833/832
Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}


[[Gencom]]: [2 8/7; 176/175 221/200, 640/639, 833/832]
Sval mapping: [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2 2}}, {{val|0 0 7 -1 13 9 11}}]
Optimal tuning (CTE): ~5/4 = 386.361


[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2 2}}, {{val|0 7 -1 13 9 11}}]
{{Optimal ET sequence|legend=0|264, 733}}, ...


[[Tp tuning|POL2 generator]]: ~8/7 = 227.242
Badness (Sintel): 52.150


{{Val list|legend=1| 16, 21, 37 }}
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13


== Didacus ==
Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
{{See also| Hemimean clan #Didacus }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
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}}, ...
 
Badness (Sintel): 10.518
 
=== Bastille ===
{{Main| Bastille }}
 
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]].


Subgroup: 2.5.7
Subgroup: 2.5.7


[[Comma]]: 3136/3125
Comma list: {{Monzo|1426 -596 -15}}
 
Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
 
Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
 
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
 
[[Badness]] (Sintel): 7224.3
 
=== Augment ===
{{See also| Chromatic pairs #Augment }}
 
Augment is related to [[augmented]].
 
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 56/55, 128/125
 
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
 
{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}
 
: [[gencom]]: [5/4 8/7; 56/55 128/125]
 
[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275
 
{{Optimal ET sequence|legend=1| 3, 6, 9, 15, 21 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents
 
=== 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.
 
Ostara can also refer to a collection of temperaments which temper out 16807/16796.
 
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
 
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢
 
{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
 
[[Badness]] (Sintel): 11.731
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
 
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
 
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
 
Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
 
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
 
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
 
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.985
 
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19
 
Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]
 
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
 
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢
 
{{Optimal ET sequence|legend=0| 93, 338, 431 }}
 
Badness (Sintel): 1.285
 
=== 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.
 
Subgroup: 2.5.7.11.19.29.31
 
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
 
Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
 
Optimal tuning (CTE): ~58/31 = 1084.628
 
{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
 
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}
 
Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.
 
[[Subgroup]]: 2.5.7.13


[[Gencom]]: [2 28/25; 3136/3125]
[[Comma list]]: [[640/637]], [[10985/10976]]


[[Gencom|Gencom mapping]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}


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


[[Tp tuning|POL2 generator]]: ~28/25 = 93.772
: [[gencom]]: [2 16/13; 640/637 10985/10976]


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}


== Rainy ==
[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents
Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].


Subgroup: 2.5.7
==== Silver ====
{{See also| Chromatic pairs #Silver }}
 
Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.
 
[[Subgroup]]: 2.5.7.13.17
 
[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]
 
{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}
 
{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
 
: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711
 
{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
 
=== Pakkanen ===
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 625/616
 
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
 
: mapping generators: ~2, ~5, ~11
 
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
 
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
 
[[Badness]] (Sintel): 0.573
 
=== 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}}]
 
[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
 
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}
 
Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: [[15625/15488]]
 
{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
 
{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
 
: [[gencom]]: [125/88 5/4; 15625/15488]
 
[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768
 
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3997
 
=== Insect ===
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 33275/32768
 
{{Mapping|legend=2|1 0 5|0 3 -2}}
 
: Mapping generators, ~2, ~[[55/32]]
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
 
{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
 
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
 
Sephiroth is the no-7 restriction of [[muggles]].
 
[[Subgroup]]: 2.5.11.13.17
 
[[Comma list]]: 65/64, 170/169, 221/220
 
{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
 
{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}
 
: [[gencom]]: [2 5/4; 65/64 170/169 221/220]
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236
 
{{Optimal ET sequence|legend=1| 10, 13, 16, 29 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents
 
=== Trader ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: [[26/25]]
 
{{Mapping|legend=2|1 2 3|0 1 2}}
 
: Mapping generators, ~2, ~[[5/4]]
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
 
{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
 
=== Superquintal ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 64000000/62748517
 
{{Mapping|legend=2|1 5 6|0 -7 -6}}
 
: Mapping generators, ~2, ~13/10
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
 
{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
 
== No-threes-or-fives subgroup temperaments ==
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 }}
 
{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
 
: [[gencom]]: [2, 16/13; 28672/28561]
 
[[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 ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents
 
=== Ultrakleismic ===
[[Subgroup]]: 2.7.17
 
[[Comma list]]: 4913/4802
 
{{Mapping|legend=2|1 2 3|0 3 4}}
 
: Mapping generators, ~2, ~[[17/14]]
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446
 
{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}
 
=== Counterultrakleismic ===
[[Subgroup]]: 2.7.17
 
[[Comma list]]: 2024782584832/2015993900449
 
{{Mapping|legend=2|1 0 1|0 10 11}}
 
: Mapping generators, ~2, ~[[17/14]]
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858
 
{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}
 
=== Shipwreck ===
 
[[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


[[Comma]]: [[2100875/2097152]]
=== Blackbirds ===
{{See also | Chromatic pairs #Blackbirds }}


[[Gencom]]: [2 256/245; 2100875/2097152]
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.


[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
[[Subgroup]]: 2.11.13


[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]
[[Comma list]]: [[29282/28561]]


[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
{{Mapping|legend=2| 4 0 1 | 0 1 1 }}


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
: [[gencom]]: [13/11 11/8; 29282/28561]


== Mercy ==
[[Optimal tuning]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660
{{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 ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}


Subgroup: 2.5.7
[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents


[[Comma list]]: 823543/819200
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}


[[Gencom]]: [2 2744/2560; 823543/819200]
[[Subgroup]]: 2.11.13


[[Gencom|Gencom mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
[[Comma list]]: [[265837/262144]]


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}


[[Tp tuning|POL2 generator]]: ~343/320 = 116.291
{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}


{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
: [[gencom]]: [2 143/128; 265837/262144]


=== 2.5.7.13 ===
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368
Subgroup: 2.5.7.13


[[Comma list]]: 343/338, 640/637
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}


[[Gencom]]: [2 14/13; 343/338 640/637]
[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4}}, {{val|0 0 -7 -2 -3}}]
=== 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]].


[[Mapping|Sval mapping]]: [{{val|1 3 3 4}}, {{val|0 -7 -2 -3}}]
[[Subgroup]]: 2.11.13.17.19


[[Tp tuning|POL2 generator]]: ~14/13 = 116.094
[[Comma list]]: 209/208, 2057/2048, 83521/83486


{{Val list|legend=1| 10, 21, 31}}
[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]


=== 2.5.7.13.17 ===
Optimal tuning ([[POTE]]): ~17/13 = 462.9606
Subgroup: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4}}, {{val|0 0 -7 -2 -3 1}}]
=== Mavericks ===


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4}}, {{val|0 -7 -2 -3 1}}]
[[Subgroup]]: 2.13.19


[[Tp tuning|POL2 generator]]: ~14/13 = 115.769
[[Comma list]]: 47525504/47045881


{{Val list|legend=1| 10, 21, 31}}
[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]


=== 2.5.7.13.17.19 ===
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
Subgroup: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4 3}}, {{val|0 0 -7 -2 -3 1 13}}]
[[Comma list]]: 209/208, 2057/2048


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4 3}}, {{val|0 -7 -2 -3 1 13}}]
[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}


[[Tp tuning|POL2 generator]]: ~14/13 = 115.716
[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638


{{Val list|legend=1| 10, 21, 31, 52f}}
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
{{Todo| cleanup | review }}
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