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+{{Technical data page}}
 This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3.
 == Overview by mapping of 5 ==
 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]].
+* 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.
-* For no-fives, see [[#No-threes-or-fives subgroup temperaments]].
+== Temperaments with a 2.5.7 gene ==
-* French decimal and trader are with ~2/1 period and ~5/4 generator. There is one-to-one correspondence between 2.5 subgroup and mapped intervals.
+Temperaments discussed elsewhere include
-* Didacus is with a ~28/25 generator, two of which give the ~5/4.
+* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
-* Ostara, movila and vengeance are with variety expressed generator, three of which give the ~5/2.
+* Didacus ([[3136/3125]]) → [[Hemimean clan #Didacus|Hemimean clan]]
-* Insect is with a ~55/32 generator, three of which give the ~5/1.
+* Mercy ([[823543/819200]]) → [[Quince clan #Mercy|Quince clan]]
-* Frostburn is with a ~28/25 generator, four of which give the ~8/5.
+* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]
-Others are more far.
+* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
-Temperaments discussed elsewhere include
+=== Rainy ===
-* Jubilic → [[Jubilismic clan #Jubilic]]
+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.
-== Llywelyn aka shoe ==
+A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.
-{{See also| Chromatic pairs #Shoe }}
-{{See also| Llywelyn clan #Llywelyn aka shoe }}
 [[Subgroup]]: 2.5.7
-[[Comma list]]: 4194304/4117715
+[[Comma list]]: [[2100875/2097152]]
+{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
+{{Mapping|legend=3| 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
-[[Sval]] [[mapping]]: [{{Val| 1 1 3 }}, {{Val| 0 7 -1 }}]
+[[Comma list]]: 128/125
-Mapping generators: 2, ~8/7
+{{Mapping|legend=2| 3 7 0 | 0 0 1 }}
-[[Gencom]] [[mapping]]: [{{Val| 1 0 1 3 }}, {{Val| 0 0 7 -1 }}]
+{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
+: mapping generators: ~5/4, ~7
-[[Gencom]]: [2 8/7; 4194304/4117715]
+[[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 tuning]] ([[POTE]]): ~8/7 = 226.910
+{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}
-{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37 }}
+[[Badness]] (Sintel): 0.296
-=== 2.5.7.11 subgroup ===
+==== 2.5.7.11 subgroup ====
 Subgroup: 2.5.7.11
-Comma list: 176/175, 1310720/1294139
+Comma list: 56/55, 128/125
-Sval mapping: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
+Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}
-Gencom: [2 8/7; 176/175 1310720/1294139]
+Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}
-Gencom mapping: [{{val| 1 0 1 3 1 }}, {{val| 0 0 7 -1 13 }}]
+Optimal tunings:
+* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}
+* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}
-Optimal tuning (POTE): ~8/7 = 227.114
+{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}
-{{Optimal ET sequence|legend=1| 16, 21, 37 }}
+Badness (Sintel): 0.196
-=== 2.5.7.11.13 subgroup ===
+=== Frostburn ===
-Subgroup: 2.5.7.11.13
+Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].
-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]]: 78125/76832
-Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
+{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
+: mapping generators: ~2, ~28/25
-Gencom mapping: [{{val| 1 0 1 3 1 2 }}, {{val| 0 0 7 -1 13 9 }}]
+[[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 }}
-Optimal tuning (POTE): ~8/7 = 227.108
+{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
-{{Optimal ET sequence|legend=1| 16, 21, 37 }}
+[[Badness]] (Sintel): 0.886
-=== 2.5.7.11.13.17 subgroup ===
+==== 2.5.7.11 subgroup ====
-Subgroup: 2.5.7.11.13.17
+Subgroup: 2.5.7.11
-Comma list: 176/175, 221/200, 640/637, 833/832
+Comma list: 245/242, 625/616
-Sval mapping: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]
-Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
+Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
+: mapping generators: ~2, ~28/25
-Gencom mapping: [{{val| 1 0 1 3 1 2 2 }}, {{val| 0 0 7 -1 13 9 11 }}]
+Optimal tunings:
+* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}
-Optimal tuning (POTE): ~8/7 = 227.242
+{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
-{{Optimal ET sequence|legend=1| 16, 21, 37 }}
+Badness (Sintel): 0.463
-== Didacus ==
+=== Mabilic ===
-{{See also| Hemimean clan #Didacus }}
+{{Main| Mabilic and trismegistus }}
+{{See also| Chromatic pairs #Mabilic }}
-Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
+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.
 [[Subgroup]]: 2.5.7
-[[Comma list]]: 3136/3125
+[[Comma list]]: 1071875/1048576
-[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]
+{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
-[[Gencom]]: [2 28/25; 3136/3125]
+{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
+: mapping generators: ~2, ~175/128
-[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]
+[[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 }}
-Optimal tuning ([[POTE]]): ~28/25 = 93.772
+{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}
-{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
+[[Badness]] (Sintel): 1.70
-[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
+=== Huntington ===
+{{See also| Chromatic pairs #Huntington }}
-== Rainy ==
+Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
-Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]]. Rainy is notable theoretically as it equates ([[2/1]])/([[5/4]])<sup>3</sup> = [[128/125]] (the 2.3.5 diesis) with ([[2/1]])/([[8/7]])<sup>5</sup> (the [[cloudy comma]]), which has a similar size and role to the 2.3.5 diesis so might be considered the 2.3.7 diesis, in that if you temper it, it imparts significant error on the tuning of 8/7 (so that it is now ~8.8{{cent}} sharp), similar to how tempering 128/125 imparts significant error on the tuning of 5/4 (so that it is ~13.7{{cent}} sharp). Combined with the small size of these "dieses", it implies that the locations of 5/4 and 8/7 are in some sense awkward to conceptualize due to the small but not insignificant size of the remnant with the octave, so it is might be favorable to equate 128/125 with the cloudy comma to create a general-purpose diesis. (Note that this analysis assumes a [[lattice]]-based conceptualization of [[JI]] which is often called "stacking-based"; see [[taxonomies of xen approaches]].)
 [[Subgroup]]: 2.5.7
-[[Comma list]]: [[2100875/2097152]]
+[[Comma list]]: 40960000/40353607
+{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}
+{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
+: mapping generators: ~2, ~80/49
+[[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 }}
+{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}
-[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
+Badness (Sintel): 1.87
-[[Gencom]]: [2 256/245; 2100875/2097152]
+==== 2.5.7.13 subgroup ====
+Subgroup: 2.5.7.13
-[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
+Comma list: 640/637, 10985/10976
-Optimal tuning ([[POTE]]): ~256/245 = 77.205
+Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
-{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
+Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
+: mapping generators: ~2, ~13/8
-[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
+Optimal tunings:
+* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}
-== Mercy ==
+{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}
-{{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]].
+Badness (Sintel): 0.319
-[[Subgroup]]: 2.5.7
+==== Silver ====
+{{See also| Chromatic pairs #Silver }}
-[[Comma list]]: 823543/819200
+Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.
-[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
+Subgroup: 2.5.7.13.17
-[[Gencom]]: [2 2744/2560; 823543/819200]
+Comma list: 170/169, 640/637, 5525/5488
-[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
+Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
-Optimal tuning ([[POTE]]): ~343/320 = 116.291
+Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
-{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
+Optimal tunings:
+* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}
-=== 2.5.7.13 ===
+{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}
-[[Subgroup]]: 2.5.7.13
-[[Comma list]]: 343/338, 640/637
+Badness (Sintel): 0.504
-[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]
+=== 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.
-[[Gencom]]: [2 14/13; 343/338 640/637]
+Subgroup: 2.5.7.11
-[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]
+Comma list: 8589934592/8544921875, 30691800524/30517578125
-Optimal tuning ([[POTE]]): ~14/13 = 116.094
+Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
+: mapping generators: ~2, ~5120/3773
-{{Optimal ET sequence|legend=1| 10, 21, 31}}
+Optimal tunings:
+* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}
-=== 2.5.7.13.17 ===
+{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}
-[[Subgroup]]: 2.5.7.13.17
-[[Comma list]]: 170/169, 224/221, 640/637
+Badness (Sintel): 11.7
-[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]
+==== 2.5.7.11.13 subgroup ====
+Subgroup: 2.5.7.11.13
-[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
+Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
-[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]
+Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
-Optimal tuning ([[POTE]]): ~14/13 = 115.769
+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=1| 10, 21, 31}}
+{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
-=== 2.5.7.13.17.19 ===
+Badness (Sintel): 3.42
-[[Subgroup]]: 2.5.7.13.17.19
-[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
+==== 2.5.7.11.13.17 subgroup ====
+Subgroup: 2.5.7.11.13.17
-[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]
+Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
-[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]
+Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
-[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
+Optimal tunings:
+* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}
-Optimal tuning ([[POTE]]): ~14/13 = 115.716
+{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
-{{Optimal ET sequence|legend=1| 10, 21, 31, 52f }}
+Badness (Sintel): 1.99
-== Pakkanen (rank 3) ==
+==== 2.5.7.11.13.17.19 subgroup ====
-[[Subgroup]]: 2.5.7.11
+Subgroup: 2.5.7.11.13.17.19
-[[Comma list]]: 625/616
+Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
-{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
+Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
-: mapping generators: ~2, ~5, ~11
+Optimal tunings:
+* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}
-[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
+{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
-{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
+Badness (Sintel): 1.29
-== Frostburn ==
+=== French decimal ===
-{{See also| Magic family #Quadrimage }}
+French decimal is 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 list]]: 78125/76832
+[[Comma list]]: {{monzo| 372 -159 -1 }}
-{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
+{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
+: mapping generators: ~2, ~5
-: Sval mapping generators: ~2, ~28/25
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
+: [[error map]]: {{val| -0.010 +0.023 +0.000 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}
+: error map: {{val| 0.000 +0.046 +0.019 }}
-[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
+{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
-{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
+[[Badness]] (Sintel): 148
-=== 2.5.7.11 ===
+==== 2.5.7.11 subgroup ====
 Subgroup: 2.5.7.11
-Comma list: 245/242, 625/616
+Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
+Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}
+Optimal tunings:
+* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
+{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}
+Badness (Sintel): 52.2
+==== 2.5.7.11.13 subgroup ====
+Subgroup: 2.5.7.11.13
+Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
+Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
+Optimal tunings:
+* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
+{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}
+Badness (Sintel): 10.5
+=== 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=2| 1 3 4 5 | 0 -4 -7 -9 }}
+{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
+: mapping generators: ~2, ~{{monzo| -380 159 4 }}
-: Sval mapping generators: ~2, ~28/25
+[[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 tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
+{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}
-{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
+[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
-== Ostara ==
+=== No-threes naiad (rank-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.
+{{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 }}
-Ostara can also refer to a collection of temperaments which temper out 16807/16796.
+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]]: 8589934592/8544921875, 53710650917/53687091200
+[[Comma list]]: 5021863/5000000
+{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
+: mapping generators: ~2, ~5, ~77/50
-[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
+[[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 tuning]] ([[POTE]]): ~5120/3773 = 529.003¢
+{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}
-{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
+[[Badness]] (Sintel): 1.86
-=== 2.5.7.11.13 subgroup ===
+==== 2.5.7.11.13 subgroup ====
 Subgroup: 2.5.7.11.13
-Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
+Comma list: 847/845, 1001/1000
+Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}
-Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
+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 tuning (POTE): ~1664/1225 = 529.003¢
+{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}
-{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
+Badness (Sintel): 0.179
-=== 2.5.7.11.13.17 subgroup ===
+==== 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: 170/169, 221/220, 847/845
-Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
+Subgroup-val mapping: {{mapping| 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 }}
-Optimal tuning (POTE): ~1664/1225 = 529.003¢
+Badness (Sintel): 0.438
-{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
+== Temperaments with a higher 2.5.''p'' gene ==
+Temperaments discussed elsewhere include:
+* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
-=== 2.5.7.11.13.17.19 subgroup ===
+=== Wizz ===
-Subgroup: 2.5.7.11.13.17.19
+{{See also| Chromatic pairs #Wizz }}
-Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
+Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[astrology]] and [[wizard]].
-Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
+[[Subgroup]]: 2.5.11
-Optimal tuning (POTE): ~19/14 = 529.003¢
+[[Comma list]]: 15625/15488
-== Pure onzonic ==
+{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
-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=3| 2 0 4 0 5 | 0 0 1 0 3 }}
+: mapping generators: ~125/88, ~5/4
-Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
+[[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 }}
-Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
+{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}
-Optimal tuning (POTE): ~11/8 = 551.370
+[[Badness]] (Sintel): 0.266
-{{Optimal ET sequence|legend=1|37, 1789}}
+=== Insect ===
+[[Subgroup]]: 2.5.11
-== Tricesimoprimal miracloid ==
+[[Comma list]]: 33275/32768
-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
+{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}
+: mapping generators, ~2, ~55/32
-Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
+[[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 }}
-Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
+{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}
-Optimal tuning (CTE): ~58/31 = 1084.628
+[[Badness]] (Sintel): 0.564
-{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
+=== 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]].
-== French decimal ==
+[[Subgroup]]: 2.5.11
-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 list]]: 1331/1280
-Comma basis: {{monzo|372 -159 -1}}
+{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
+: mapping generators: ~2, ~11/8
-Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
+[[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 tuning (CTE): ~5/4 = 386.360
+{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
-{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
+[[Badness]] (Sintel): 0.718
-=== 2.5.7.11 subgroup ===
+=== Sephiroth ===
-Subgroup: 2.5.7.11
+{{See also| Chromatic pairs #Sephiroth }}
-Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
+Sephiroth is the no-7 [[restriction]] of [[muggles]].
-Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
+[[Subgroup]]: 2.5.11
-Optimal tuning (CTE): ~5/4 = 386.361
+[[Comma list]]: 34375/32768
-{{Optimal ET sequence|legend=1|264, 733}}, ...
+{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
-=== 2.5.7.11.13 subgroup ===
+{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}
-Subgroup: 2.5.7.11.13
+: mapping generators: ~2, ~5
-Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
+[[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 }}
-Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
+{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}
-Optimal tuning (CTE): ~5/4 = 386.361
+[[Badness]] (Sintel): 1.85
-{{Optimal ET sequence|legend=1|1525, 1789}}, ...
+==== 2.5.11.13 subgroup ====
+Subgroup: 2.5.11.13
-== Bastille ==
+Comma list: 65/64, 6875/6656
-{{Main|Bastille}}
-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
+Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
-Comma list: {{Monzo|1426 -596 -15}}
+Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}
-Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
+Optimal tunings:
+* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}
-Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
+{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceeeff }}
-{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
+Badness (Sintel): 0.410
-== Movila ==
+==== 2.5.11.13.17 subgroup ====
-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: 2.5.11.13.17
-[[Subgroup]]: 2.5.11
+Comma list: 65/64, 170/169, 221/220
-[[Comma list]]: 1331/1280
+Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
-[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
+Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
-[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
+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=1| 7, 9, 16, 25, 41e, 66ee }}
+{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}
-== Vengeance ==
+Badness (Sintel): 0.299
-''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 generators, ~2, ~5/4
-{{Mapping|legend=2|1 1 1|0 3 7}}
+[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[34/25]] = 529.095
+{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}
-{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
+[[Badness]] (Sintel): 0.138
-== Superquintal ==
+=== Superquintal ===
 [[Subgroup]]: 2.5.13
 [[Comma list]]: 64000000/62748517
-{{Mapping|legend=2|1 5 6|0 -7 -6}}
+{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}
+: mapping generators, ~2, ~20/13
-: Mapping generators, ~2, ~13/10
+[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~13/10 = 459.281
+{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}
-{{Optimal ET sequence|legend=1|8, 13, 21, 34, 81, 115}}
+[[Badness]] (Sintel): 1.93
-== Insect ==
+== No-threes no-fives subgroup temperaments ==
-[[Subgroup]]: 2.5.11
+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=2| 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=2| 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=2| 1 1 3 1 | 0 4 1 6 }}
+{{Mapping|legend=3| 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=2| 1 0 1 | 0 8 7 }}
-[[Comma list]]: 33275/32768
+{{Mapping|legend=3| 1 0 0 0 1 | 0 0 0 8 7 }}
+: mapping generators: ~2, ~14/11
-{{Mapping|legend=2|1 0 5|0 3 -2}}
+[[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 }}
-: Mapping generators, ~2, ~[[55/32]]
+{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[55/32]] = 928.032
+[[Badness]] (Sintel): 1.73
-{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
+==== 2.7.11.13 subgroup ====
+Subgroup: 2.7.11.13
-== Trader ==
+Comma list: 1573/1568, 15488/15379
-[[Subgroup]]: 2.5.13
-[[Comma list]]: [[26/25]]
+Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}
-{{Mapping|legend=2|1 2 3|0 1 2}}
+Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
-: Mapping generators, ~2, ~[[5/4]]
+Optimal tunings:
+* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
+{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}
-{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
+Badness (Sintel): 0.307
-== No-threes-or-fives subgroup temperaments ==
+=== Voltage ===
-Temperaments discussed elsewhere include
+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.
-* Berylic → [[4th-octave temperaments #Berylic]]
-* [[21st-octave temperaments #21-23-commatic]]
-* [[31st-octave temperaments #31-17/13-commatic]]
-* [[37th-octave temperaments #37-11-commatic (rank-1)]]
-* etc.
-=== Shipwreck ===
+[[Subgroup]]: 2.7.13
-[[Subgroup]]: 2.7.53
+[[Comma list]]: 28672/28561
-[[Comma list]]: 1048576/1042139
+{{Mapping|legend=2| 1 0 3 | 0 4 1 }}
-[[Gencom]]: [2 64/53; 1048576/1042139]
+{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
+: mapping generators: ~2, ~13
-[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
+[[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 }}
-[[POTE generator]]: ~64/53 = 323.034
+{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}
-{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
+[[Badness]] (Sintel): 0.115
 === Ultrakleismic ===
@@ Line 440: / Line 696: @@
 [[Comma list]]: 4913/4802
-{{Mapping|legend=2|1 2 3|0 3 4}}
+{{Mapping|legend=2| 1 2 3 | 0 3 4 }}
+: mapping generators, ~2, ~17/14
-: 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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 324.446
+{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}
-{{Optimal ET sequence|legend=1|4, 7, 11, 26, 37}}
+[[Badness]] (Sintel): 0.460
 === Counterultrakleismic ===
@@ Line 453: / Line 714: @@
 [[Comma list]]: 2024782584832/2015993900449
-{{Mapping|legend=2|1 0 1|0 10 11}}
+{{Mapping|legend=2| 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=2| 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=2| 1 3 3 | 0 2 3 }}
+{{Mapping|legend=3| 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=2| 1 0 6 | 0 3 -2 }}
+{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
+: mapping generators, ~2, ~143/128
-: Mapping generators, ~2, ~[[17/14]]
+[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[17/14]] = 336.858
+{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}
-{{Optimal ET sequence|legend=1|7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g}}
+[[Badness]] (Sintel): 0.451
-=== Yer (rank 3) ===
+=== Blackbirds ===
-[[Subgroup]]: 2.11.13.17.19
+{{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 }}
-[[Comma list]]: 209/208, 2057/2048
+{{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
+: mapping generators, ~13/11, ~11
-[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
+[[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 tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
+{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}
-{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
+[[Badness]] (Sintel): 0.668
 === 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]].
+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
@@ Line 479: / Line 823: @@
 [[Comma list]]: 209/208, 2057/2048, 83521/83486
-[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
+{{Mapping|legend=2| 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=2| 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
-Optimal tuning ([[POTE]]): ~17/13 = 462.9606
+{{Mapping|legend=2| 4 0 7 | 0 1 1 }}
+: mapping generators: ~44/37, ~11
-{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
+[[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 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
+{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}
+[[Badness]] (Sintel): 0.00188
 === Mavericks ===
 [[Subgroup]]: 2.13.19
 [[Comma list]]: 47525504/47045881
-[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
+{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
+: mapping generators: ~2, ~26/19
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
+[[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 }}
+{{Optimal ET sequence|legend=1| 9, 11, 20 }}
+[[Badness]] (Sintel): 0.559
+=== Yer (rank 3) ===
+[[Subgroup]]: 2.11.13.17.19
+[[Comma list]]: 209/208, 2057/2048
+{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
+: mapping generators: ~2, ~11, ~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}}
+{{Optimal ET sequence|legend=1| 11, 13, 24, 33, 37, 46, 57, 70, 127, 197eh }}
-{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
+[[Badness]] (Sintel): 0.106
 [[Category:Temperament collections]]
 [[Category:Subgroup temperaments]]

No-threes subgroup temperaments: Difference between revisions