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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 no-fives subgroup temperaments]].
* Trader has 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.


[[Comma]]: 4194304/4117715
== Temperaments with a 2.5.7 gene ==
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]]


[[Gencom]]: [2 8/7; 4194304/4117715]
=== Rainy ===
In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), 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 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.
 
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]]
 
{{Mapping|legend=3| 1 2 3 | 0 5 -3 }}
 
{{Mapping|legend=5| 1 0 2 3 | 0 0 5 -3 }}
: mapping generators: ~2, ~256/245
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}
: [[error map]]: {{val| +0.094 -0.072 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}
: error map: {{val| 0.000 -0.267 -0.454 }}
 
{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}
 
[[Badness]] (Sintel): 0.156
 
=== Augment ===
{{See also| Chromatic pairs #Augment }}
 
Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.
 
[[Subgroup]]: 2.5.7
 
[[Comma list]]: 128/125
 
{{Mapping|legend=3| 3 7 0 | 0 0 1 }}
 
{{Mapping|legend=5| 3 0 7 0 | 0 0 0 1 }}
: mapping generators: ~5/4, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}
: [[error map]]: {{val| -2.962 +6.776 -0.040 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 974.3418{{c}}
: error map: {{val| 0.000 +13.686 +5.516 }}
 
{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}
 
[[Badness]] (Sintel): 0.296
 
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11
 
Comma list: 56/55, 128/125
 
{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
 
{{Mapping|legend=4| 3 0 7 0 2 | 0 0 0 1 1 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]
Optimal tunings:  
* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}


[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
Badness (Sintel): 0.196


{{Val list|legend=1| 16, 37 }}
=== Frostburn ===
Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].


[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
[[Subgroup]]: 2.5.7


=== Didacus ===
[[Comma list]]: 78125/76832
{{See also| Hemimean clan #Didacus }}


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
{{Mapping|legend=3| 1 3 4 | 0 -4 -7 }}
: mapping generators: ~2, ~28/25


Subgroup: 2.5.7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}
: [[error map]]: {{val| +0.346 -2.630 +2.189 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 204.2027{{c}}
: error map: {{val| 0.000 -3.125 +1.755 }}


[[Comma]]: 3136/3125
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}


[[Gencom]]: [2 28/25; 3136/3125]
[[Badness]] (Sintel): 0.886


[[Gencom|Gencom mapping]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
==== 2.5.7.11 subgroup ====
Subgroup: 2.5.7.11


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
Comma list: 245/242, 625/616


[[Tp tuning|POL2 generator]]: ~28/25 = 93.772
{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
: mapping generators: ~2, ~28/25


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
Optimal tunings:
* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


=== Rainy ===
Badness (Sintel): 0.463


Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
=== Mabilic ===
{{Main| Mabilic and trismegistus }}
{{See also| Chromatic pairs #Mabilic }}


Subgroup: 2.5.7
Mabilic is the no-3 [[restriction]] 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]]: [[2100875/2097152]]
[[Subgroup]]: 2.5.7


[[Gencom]]: [2 256/245; 2100875/2097152]
[[Comma list]]: 1071875/1048576


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


[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]
{{Mapping|legend=5| 1 0 1 5 | 0 0 3 -5 }}
: mapping generators: ~2, ~175/128


[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}
: [[error map]]: {{val| +1.254 -1.698 -1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}
: error map: {{val| 0.000 -4.701 -4.846 }}


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
[[Badness]] (Sintel): 1.70


=== Mercy ===
=== Huntington ===
{{See also| Quince clan #Mercy }}
{{See also| Chromatic pairs #Huntington }}


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]].
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  


Subgroup: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 823543/819200
[[Comma list]]: 40960000/40353607


[[Gencom]]: [2 2744/2560; 823543/819200]
{{Mapping|legend=3| 1 -4 0 | 0 9 4 }}


[[Gencom|Gencom mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
{{Mapping|legend=5| 1 0 -4 0 | 0 0 9 4 }}
: mapping generators: ~2, ~80/49


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}
: [[error map]]: {{val| -0.422 -0.569 +1.866 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}
: error map: {{val| 0.000 -0.091 +2.828 }}


[[Tp tuning|POL2 generator]]: ~343/320 = 116.291
{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}


{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Badness (Sintel): 1.87


==== 2.5.7.13 ====
==== 2.5.7.13 subgroup ====
Subgroup: 2.5.7.13
Subgroup: 2.5.7.13


[[Comma list]]: 343/338, 640/637
Comma list: 640/637, 10985/10976
 
{{Mapping|legend=2| 1 -4 0 3 | 0 9 4 1 }}
 
{{Mapping|legend=4| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~13/8


[[Gencom]]: [2 14/13; 343/338 640/637]
Optimal tunings:  
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


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


[[Mapping|Sval mapping]]: [{{val|1 3 3 4}}, {{val|0 -7 -2 -3}}]
Badness (Sintel): 0.319


[[Tp tuning|POL2 generator]]: ~14/13 = 116.094
==== Silver ====
{{See also| Chromatic pairs #Silver }}


{{Val list|legend=1| 10, 21, 31}}
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.


==== 2.5.7.13.17 ====
Subgroup: 2.5.7.13.17
Subgroup: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
Comma list: 170/169, 640/637, 5525/5488
 
{{Mapping|legend=2| 1 -4 0 3 9 | 0 9 4 1 -7 }}
 
{{Mapping|legend=4| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
 
Optimal tunings:
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}
 
{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}
 
Badness (Sintel): 0.504
 
=== Ostara ===
Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.
 
Subgroup: 2.5.7.11
 
Comma list: 8589934592/8544921875, 30691800524/30517578125
 
{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773
 
Optimal tunings:
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}
 
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}
 
Badness (Sintel): 11.7
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
 
{{Mapping|legend=2| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
 
Optimal tunings:
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}
 
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
 
Badness (Sintel): 3.42
 
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
 
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
 
{{Mapping|legend=2| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
 
Optimal tunings:
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}
 
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.99
 
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19
 
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
 
{{Mapping|legend=2| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
 
Optimal tunings:
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}
 
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.29
 
=== Bastille ===
{{Main| Bastille }}
 
Bastille is 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
 
[[Comma list]]: {{monzo| 1426 -596 -15 }}
 
{{Mapping|legend=3| 1 -4 254 | 0 15 -596 }}
: mapping generators: ~2, ~{{monzo| -380 159 4 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
: error map: {{val| 0.000 +0.041 +0.018 }}
 
{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}
 
[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
 
=== No-threes naiad (rank-3) ===
{{Todo|inline=1| review | comment = Devise the permanent name for this temperament; determine its status with respect to its extensions linked below. }}
{{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=3| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~77/50
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{c}}
: [[error map]]: {{val| +0.080 +0.507 -1.318 -0.643 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.7404{{c}}, ~77/50 = 745.4102{{c}}
: error map: {{val| 0.000 +0.427 -0.445 -0.808 }}
 
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}
 
[[Badness]] (Sintel): 1.86
 
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
 
Comma list: 847/845, 1001/1000
 
{{Mapping|legend=2| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}
 
Optimal tunings:
* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}
 
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}
 
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
 
{{Mapping|legend=2| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}
 
Optimal tunings:
* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}
 
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}
 
Badness (Sintel): 0.438
 
== Temperaments with a higher 2.5.''p'' gene ==
Temperaments discussed elsewhere include:
* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
 
=== Wizz ===
{{See also| Chromatic pairs #Wizz }}
 
Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[astrology]] and [[wizard]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 15625/15488
 
{{Mapping|legend=3| 2 0 -7 | 0 1 3 }}
 
{{Mapping|legend=5| 2 0 4 0 5 | 0 0 1 0 3 }}
: mapping generators: ~125/88, ~5/4
 
[[Optimal tuning]]s:
* [[WE]]: ~125/88 = 600.1831{{c}}, ~5/4 = 383.8848{{c}}
: [[error map]]: {{val| +0.366 -1.697 +1.252 }}
* [[CWE]]: ~125/88 = 600.0000{{c}}, ~5/4 = 383.9977{{c}}
: error map: {{val| 0.000 -2.316 +0.675 }}
 
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}
 
[[Badness]] (Sintel): 0.266
 
=== Insect ===
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 33275/32768
 
{{Mapping|legend=3| 1 0 5 | 0 3 -2 }}
: mapping generators, ~2, ~55/32
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1238{{c}}, ~55/32 = 928.5003{{c}}
: [[error map]]: {{val| +1.124 -0.813 -2.700 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~55/32 = 927.7384{{c}}
: error map: {{val| 0.000 -3.099 -6.975 }}
 
{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}
 
[[Badness]] (Sintel): 0.564
 
=== 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|legend=3| 1 1 3 | 0 3 1 }}
: mapping generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.0339{{c}}, ~11/8 = 528.4296{{c}}
: [[error map]]: {{val| +3.034 +2.009 -13.787 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 528.1575{{c}}
: error map: {{val| 0.000 -1.841 -23.160 }}
 
{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
 
[[Badness]] (Sintel): 0.718
 
=== Sephiroth ===
{{See also| Chromatic pairs #Sephiroth }}
 
Sephiroth is the common [[restriction]] of [[muggles]] and [[submerged]].
 
[[Subgroup]]: 2.5.11
 
[[Comma list]]: 34375/32768
 
{{Mapping|legend=3| 1 0 15 | 0 1 -5 }}
 
{{Mapping|legend=5| 1 0 0 0 15 | 0 0 1 0 -5 }}
: mapping generators: ~2, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3290{{c}}, ~5/4 = 373.6097{{c}}
: [[error map]]: {{val| +3.329 -6.046 -2.722 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 372.1586{{c}}
: error map: {{val| 0.000 -14.155 -12.111 }}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
 
[[Badness]] (Sintel): 1.85
 
==== 2.5.11.13 subgroup ====
Subgroup: 2.5.11.13
 
Comma list: 65/64, 6875/6656
 
{{Mapping|legend=2| 1 0 15 6 | 0 1 -5 -1 }}
 
{{Mapping|legend=4| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}
 
Badness (Sintel): 0.410
 
==== 2.5.11.13.17 subgroup ====
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=4| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}
 
{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}
 
Badness (Sintel): 0.299
 
=== Trader ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 26/25
 
{{Mapping|legend=3| 1 2 3 | 0 1 2 }}
: mapping generators, ~2, ~5/4
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0216{{c}}, ~5/4 = 410.2152{{c}}
: [[error map]]: {{val| -1.978 +19.945 -26.033 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 408.9029{{c}}
: error map: {{val| 0.000 +22.589 -22.722 }}
 
{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}
 
[[Badness]] (Sintel): 0.138
 
=== Superquintal ===
[[Subgroup]]: 2.5.13
 
[[Comma list]]: 64000000/62748517
 
{{Mapping|legend=3| 1 -2 0 | 0 7 6 }}
: mapping generators, ~2, ~20/13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5925{{c}}, ~20/13 = 740.6286{{c}}
: [[error map]]: {{val| -0.408 -1.098 +3.244 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 740.8058{{c}}
: error map: {{val| 0.000 -0.673 +4.307 }}
 
{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}
 
[[Badness]] (Sintel): 1.93
 
== No-threes no-fives subgroup temperaments ==
Temperaments discussed elsewhere include
* Orgone → [[Orgonia #Orgone|Orgonia]]
* 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=3| 1 0 -19 | 0 1 8 }}
: mapping generators: ~2, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9846{{c}}, ~7/4 = 968.9078{{c}}
: [[error map]]: {{val| -0.015 +0.051 -0.010 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 968.9174{{c}}
: error map: {{val| 0.000 +0.091 +0.021 }}
 
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}
 
Badness (Sintel): 0.0309
 
=== Argument ===
Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 1372/1331
 
{{Mapping|legend=3| 3 0 2 | 0 1 1 }}
: mapping generators: ~14/11, ~7
 
[[Optimal tuning]]s:
* [[WE]]: ~14/11 = 399.8041{{c}}, ~7/4 = 963.1666{{c}}
: [[error map]]: {{val| -0.588 -6.835 +10.281 }}
* [[CWE]]: ~14/11 = 400.0000{{c}}, ~4/4 = 962.7466{{c}}
: error map: {{val| 0.000 -6.079 +11.429 }}
 
{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}
 
[[Badness]] (Sintel): 0.475
 
=== Score ===
{{See also| Chromatic pairs #Score }}
 
Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.
 
[[Subgroup]]: 2.7.11.13
 
[[Comma list]]: 343/338, 847/832
 
{{Mapping|legend=3| 1 1 3 1 | 0 4 1 6 }}
 
{{Mapping|legend=5| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
: mapping generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}
 
{{Optimal ET sequence|legend=1| 9, 11, 20 }}
 
[[Badness]] (Sintel): 0.368
 
=== Bossier ===
{{See also| Chromatic pairs #Bossier }}
 
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].
 
[[Subgroup]]: 2.7.11
 
[[Comma list]]: 214358881/210827008
 
{{Mapping|legend=3| 1 0 1 | 0 8 7 }}
 
{{Mapping|legend=5| 1 0 0 0 1 | 0 0 0 8 7 }}
: mapping generators: ~2, ~14/11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1886{{c}}, ~14/11 = 421.2661{{c}}
: [[error map]]: {{val| +0.189 +1.303 -2.266 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2365{{c}}
: error map: {{val| 0.000 +1.066 -2.662 }}
 
{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}
 
[[Badness]] (Sintel): 1.73
 
==== 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 }}
 
Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}
 
{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}
 
Badness (Sintel): 0.307
 
=== Voltage ===
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 7<sup>1/4</sup> of 842.2065{{c}}, which is also the CTC (constrained Tenney–Chebyshevian) tuning.
 
[[Subgroup]]: 2.7.13
 
[[Comma list]]: 28672/28561
 
{{Mapping|legend=3| 1 0 3 | 0 4 1 }}
 
{{Mapping|legend=5| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
: mapping generators: ~2, ~13
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}
: [[error map]]: {{val| -0.217 -0.143 +0.991 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/8 = 842.2568{{c}}
: error map: {{val| 0.000 +0.201 +1.729 }}
 
{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}
 
[[Badness]] (Sintel): 0.115
 
=== Ultrakleismic ===
[[Subgroup]]: 2.7.17
 
[[Comma list]]: 4913/4802
 
{{Mapping|legend=3| 1 2 3 | 0 3 4 }}
: mapping generators, ~2, ~17/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1379{{c}}, ~17/14 = 324.3440{{c}}
: [[error map]]: {{val| +0.138 +4.482 -7.166 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 324.3738{{c}}
: error map: {{val| 0.000 +4.295 -7.460 }}
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}
 
[[Badness]] (Sintel): 0.460
 
=== Counterultrakleismic ===
[[Subgroup]]: 2.7.17
 
[[Comma list]]: 2024782584832/2015993900449
 
{{Mapping|legend=3| 1 0 1 | 0 10 11 }}
: mapping generators, ~2, ~17/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9723{{c}}, ~17/14 = 336.8586{{c}}
: [[error map]]: {{val| -0.028 -0.240 +0.462 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 336.8621{{c}}
: error map: {{val| 0.000 -0.205 +0.528 }}
 
{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}
 
[[Badness]] (Sintel): 0.860
 
=== Shipwreck ===
[[Subgroup]]: 2.7.53
 
[[Comma list]]: 1048576/1042139
 
{{Mapping|legend=3| 1 0 6 | 0 3 -1 }}]
: mapping generators, ~2, ~64/53
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6967{{c}}, ~64/53 = 323.1839{{c}}
: [[error map]]: {{val| -0.303 +0.119 +1.491 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/53 = 323.1959{{c}}
: error map: {{val| 0.000 +0.762 +3.300 }}
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
 
[[Badness]] (Sintel): 0.224
 
=== Lovecraft ===
{{See also | Chromatic pairs #Lovecraft }}
 
Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, tempers out [[1352/1331]], and 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=3| 1 3 3 | 0 2 3 }}
 
{{Mapping|legend=5| 1 0 0 0 3 3 | 0 0 0 0 2 3 }}
: mapping generators, ~2, ~13/11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5223{{c}}, ~13/11 = 279.2064{{c}}
: [[error map]]: {{val| -0.478 +5.662 -4.341 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/11 = 278.9918{{c}}
: error map: {{val| 0.000 +6.666 -3.552 }}
 
{{Optimal ET sequence|legend=1| 4, 9, 13, 30, 43, 73, 116e }}
 
[[Badness]] (Sintel): 0.175
 
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}
 
[[Subgroup]]: 2.11.13
 
[[Comma list]]: 265837/262144
 
{{Mapping|legend=3| 1 0 6 | 0 3 -2 }}
 
{{Mapping|legend=5| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
: mapping generators, ~2, ~143/128
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8795{{c}}, ~143/128 = 182.5017{{c}}
: [[error map]]: {{val| +0.880 -1.174 -2.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~143/128 = 182.4386{{c}}
: error map: {{val| 0.000 -4.002 -5.405 }}
 
{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}
 
[[Badness]] (Sintel): 0.451
 
=== 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=3| 4 0 1 | 0 1 1 }}
 
{{Mapping|legend=5| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
: mapping generators, ~13/11, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~13/11 = 299.9728{{c}}, ~11/8 = 546.6107{{c}}
: [[error map]]: {{val| -0.109 -5.033 +5.730 }}
* [[CWE]]: ~13/11 = 300.0000{{c}}, ~11/8 = 546.4664{{c}}
: error map: {{val| 0.000 -4.852 +5.939 }}
 
{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}
 
[[Badness]] (Sintel): 0.668
 
=== Yamablu ===
Yamablu, with a generator of ~26/17, 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). It extends the 2.11.13-subgroup temperament tempering out 556573090931/549755813888. The [[Kite's Genchain mode numbering|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
 
{{Mapping|legend=3| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
: mapping generators: ~2, ~26/17
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}
 
{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}
 
[[Badness]] (Sintel): 0.386
 
=== Arsenic ===
Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.
 
[[Subgroup]]: 2.11.13.17.19
 
[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432
 
{{Mapping|legend=3| 33 0 8 249 26 | 0 1 1 -1 1 }}
: mapping generators: ~247/242, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}
* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}
 
{{Optimal ET sequence|legend=1| 33, 165, 198, 231, 495h }}
 
[[Badness]] (Sintel): 2.30
 
=== Berylic ===
Berylic tempers out the [[berylisma]] in the 2.11.37 subgroup, representing the fact that [[44/37]] is a {{w|continued fraction}} convergent to 2<sup>1/4</sup> the fourth root of 2. Beryllic is an example of a temperament which has an astronomically low [[badness]], being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by edo systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within an obscure subgroup, 2.11.37.
 
If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.
 
[[Subgroup]]: 2.11.37
 
[[Comma list]]: 1874161/1874048
 
{{Mapping|legend=3| 4 0 7 | 0 1 1 }}
: mapping generators: ~44/37, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~44/37 = 300.0003{{c}}, ~11/8 = 551.3211{{c}}
: [[error map]]: {{val| +0.001 +0.007 -0.017 }}
* [[CWE]]: ~44/37 = 300.0000{{c}}, ~11/8 = 551.3237{{c}}
: error map: {{val| 0.000 +0.006 -0.020 }}
 
{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}
 
[[Badness]] (Sintel): 0.00188


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
=== Mavericks ===
[[Subgroup]]: 2.13.19


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4}}, {{val|0 0 -7 -2 -3 1}}]
[[Comma list]]: 47525504/47045881


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4}}, {{val|0 -7 -2 -3 1}}]
{{Mapping|legend=3| 1 1 2 | 0 6 5 }}
: mapping generators: ~2, ~26/19


[[Tp tuning|POL2 generator]]: ~14/13 = 115.769
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8817{{c}}, ~26/19 = 539.9150{{c}}
: [[error map]]: {{val| -0.118 -1.156 +1.825 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/19 = 539.9280{{c}}
: error map: {{val| 0.000 -0.960 +2.127 }}


{{Val list|legend=1| 10, 21, 31}}
{{Optimal ET sequence|legend=1| 9, 11, 20 }}


==== 2.5.7.13.17.19 ====
[[Badness]] (Sintel): 0.559
Subgroup: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
=== Yer (rank 3) ===
[[Subgroup]]: 2.11.13.17.19


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Comma list]]: 209/208, 2057/2048


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4 3}}, {{val|0 0 -7 -2 -3 1 13}}]
{{Mapping|legend=3| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
: mapping generators: ~2, ~11, ~13


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4 3}}, {{val|0 -7 -2 -3 1 13}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4447{{c}}, ~11/8 = 548.4929{{c}}, ~13/8 = 841.3613{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 548.2193{{c}}, ~13/8 = 841.4707{{c}}


[[Tp tuning|POL2 generator]]: ~14/13 = 115.716
{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}


{{Val list|legend=1| 10, 21, 31, 52f}}
[[Badness]] (Sintel): 0.106


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
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