@@ Line 4: / Line 4: @@
 == 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]].
-* 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.
@@ Line 12: / Line 11: @@
 Others have a more complex mapping of 5.
-== 2.5.7 temperaments ==
+== Temperaments with a 2.5.7 gene ==
 Temperaments discussed elsewhere include
 * Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]
@@ Line 19: / Line 17: @@
 * 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]]
+* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]
-=== Frostburn ===
-{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}
+=== 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]]: 78125/76832
+[[Comma list]]: [[2100875/2097152]]
-{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
+{{Mapping|legend=2| 1 2 3 | 0 5 -3 }}
-: Sval mapping generators: ~2, ~28/25
+{{Mapping|legend=3| 1 0 2 3 | 0 0 5 -3 }}
+: mapping generators: ~2, ~256/245
-[[Optimal tuning]] ([[TE tuning|TE]]): ~2/1 = 1200.3479, ~28/25 = 204.3389
+[[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| 6, 29, 35, 41, 47 }}
+{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}
-[[Badness]] (Sintel): 0.886
+[[Badness]] (Sintel): 0.156
-==== 2.5.7.11 ====
+=== Augment ===
-Subgroup: 2.5.7.11
+{{See also| Chromatic pairs #Augment }}
-Comma list: 245/242, 625/616
+Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.
-{{Mapping|legend=2| 1 3 4 5 | 0 -4 -7 -9 }}
+[[Subgroup]]: 2.5.7
-: Sval mapping generators: ~2, ~28/25
+[[Comma list]]: 128/125
-Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
+{{Mapping|legend=2| 3 7 0 | 0 0 1 }}
-{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
+{{Mapping|legend=3| 3 0 7 0 | 0 0 0 1 }}
+: mapping generators: ~5/4, ~7
-Badness (Sintel): 0.463
+[[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 }}
-=== Mabilic ===
+{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}
-{{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 &amp; 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
-[[Subgroup]]: 2.5.7
+[[Badness]] (Sintel): 0.296
-[[Comma list]]: 1071875/1048576
+==== 2.5.7.11 subgroup ====
+Subgroup: 2.5.7.11
-{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
+Comma list: 56/55, 128/125
-{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
+Subgroup-val mapping: {{mapping| 3 7 0 2 | 0 0 1 1 }}
-: [[gencom]]: [2 175/128; 1071875/1048576]
+Gencom mapping: {{mapping| 3 0 7 0 2 | 0 0 0 1 1 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 527.236
+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 ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305bc }}
+{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.7729 cents
+Badness (Sintel): 0.196
-=== Rainy ===
+=== Frostburn ===
-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]].)
+Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].
-A highly notable tuning of rainy not shown here is [[311edo]], which is 140+171 so tuned between them.
+[[Subgroup]]: 2.5.7
-[[Subgroup]]: 2.5.7
+[[Comma list]]: 78125/76832
-[[Comma list]]: [[2100875/2097152]]
+{{Mapping|legend=2| 1 3 4 | 0 -4 -7 }}
+: mapping generators: ~2, ~28/25
-[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]
+[[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 }}
-[[Gencom]]: [2 256/245; 2100875/2097152]
+{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
-[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
+[[Badness]] (Sintel): 0.886
-Optimal tuning ([[POTE]]): ~256/245 = 77.205
+==== 2.5.7.11 subgroup ====
+Subgroup: 2.5.7.11
-{{Optimal ET sequence|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
+Comma list: 245/242, 625/616
-[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
+Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}
+: mapping generators: ~2, ~28/25
-=== French decimal ===
+Optimal tunings:
-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.
+* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~28/25 = 204.8199{{c}}
-Subgroup: 2.5.7
+{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
-Comma basis: {{monzo|372 -159 -1}}
+Badness (Sintel): 0.463
-Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]
+=== Mabilic ===
+{{Main| Mabilic and trismegistus }}
+{{See also| Chromatic pairs #Mabilic }}
-Optimal tuning (CTE): ~5/4 = 386.360
+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.
-{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
+[[Subgroup]]: 2.5.7
-[[Badness]] (Sintel): 148.6
+[[Comma list]]: 1071875/1048576
-==== 2.5.7.11 subgroup ====
+{{Mapping|legend=2| 1 1 5 | 0 3 -5 }}
-Subgroup: 2.5.7.11
-Comma basis: {{monzo|-49 8 17 -5}}, {{monzo|45 -27 10 -3}}
+{{Mapping|legend=3| 1 0 1 5 | 0 0 3 -5 }}
+: mapping generators: ~2, ~175/128
-Sval mapping:  [{{val| 1 2 54 -177}}, {{val|0 1 -159 -539}}]
+[[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 (CTE): ~5/4 = 386.361
+{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}
-{{Optimal ET sequence|legend=0|264, 733}}, ...
+[[Badness]] (Sintel): 1.70
-Badness (Sintel): 52.150
+=== Huntington ===
+{{See also| Chromatic pairs #Huntington }}
-==== 2.5.7.11.13 subgroup ====
+Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.
-Subgroup: 2.5.7.11.13
-Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
+[[Subgroup]]: 2.5.7
-Sval mapping: [{{val| 1 2 54 -177 52}}, {{val|0 1 -159 -539 173}}]
+[[Comma list]]: 40960000/40353607
-Optimal tuning (CTE): ~5/4 = 386.361
+{{Mapping|legend=2| 1 -4 0 | 0 9 4 }}
-{{Optimal ET sequence|legend=0|1525, 1789}}, ...
+{{Mapping|legend=3| 1 0 -4 0 | 0 0 9 4 }}
+: mapping generators: ~2, ~80/49
-Badness (Sintel): 10.518
+[[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 }}
-=== Bastille ===
+{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}
-{{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]].
+Badness (Sintel): 1.87
-Subgroup: 2.5.7
+==== 2.5.7.13 subgroup ====
+Subgroup: 2.5.7.13
-Comma list: {{Monzo|1426 -596 -15}}
+Comma list: 640/637, 10985/10976
-Sval mapping: [{{Val|1 -4 254}}, {{Val|0 -15 596}}]
+Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}
-Optimal tuning (CTE): ~{{Monzo|381 0 -159 -4}} = 694.243
+Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
+: mapping generators: ~2, ~13/8
-{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
+Optimal tunings:
+* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}
-[[Badness]] (Sintel): 7224.3
+{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}
-=== Augment ===
+Badness (Sintel): 0.319
-{{See also| Chromatic pairs #Augment }}
-Augment is related to [[augmented]].
+==== Silver ====
+{{See also| Chromatic pairs #Silver }}
-[[Subgroup]]: 2.5.7.11
+Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.
-[[Comma list]]: 56/55, 128/125
+Subgroup: 2.5.7.13.17
-{{Mapping|legend=2| 3 7 0 2 | 0 0 1 1 }}
+Comma list: 170/169, 640/637, 5525/5488
-{{Mapping|legend=3| 3 0 7 9 11| 0 0 0 -1 -1 }}
+Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}
-: [[gencom]]: [5/4 8/7; 56/55 128/125]
+Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
-[[Optimal tuning]] ([[POTE]]): ~5/4 = 1\3, ~8/7 = 228.275
+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=1| 3, 6, 9, 15, 21 }}
+{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}
-[[Tp tuning #T2 tuning|RMS error]]: 2.422 cents
+Badness (Sintel): 0.504
 === 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 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.
-Ostara can also refer to a collection of temperaments which temper out 16807/16796.
+Subgroup: 2.5.7.11
-[[Subgroup]]: 2.5.7.11
+Comma list: 8589934592/8544921875, 30691800524/30517578125
-[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
+Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}
+: mapping generators: ~2, ~5120/3773
-[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
+Optimal tunings:
+* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
-[[Optimal tuning]]s:
+* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}
-* [[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 }}
+{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}
-[[Badness]] (Sintel): 11.731
+Badness (Sintel): 11.7
 ==== 2.5.7.11.13 subgroup ====
@@ Line 195: / Line 231: @@
 Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
-Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
+Subgroup-val mapping: {{mapping| 1 1 20 -49 35 | 0 3 -39 119 -71 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
+* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
-* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
+* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}
 {{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
-Badness (Sintel): 3.415
+Badness (Sintel): 3.42
 ==== 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
-Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
+Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
+* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
-* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
+* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}
 {{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
-Badness (Sintel): 1.985
+Badness (Sintel): 1.99
 ==== 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
-Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
+Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
+* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
-* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢
+* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}
-{{Optimal ET sequence|legend=0| 93, 338, 431 }}
+{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
-Badness (Sintel): 1.285
+Badness (Sintel): 1.29
-=== Tricesimoprimal miracloid ===
+=== French decimal ===
-{{See also|Tricesimoprimal miracloid/Eliora's approach|l1=Eliora's approach to tricesimoprimal miracloid}}
+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.
-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
+[[Subgroup]]: 2.5.7
-Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
+[[Comma list]]: {{monzo| 372 -159 -1 }}
-Sval Mapping: [{{val| 1 419 48 177 157 624 625 }}, {{val| 0 -461 -50 -192 -169 -685 -686 }}]
+{{Mapping|legend=2| 1 0 372 | 0 1 -159 }}
+: mapping generators: ~2, ~5
-Optimal tuning (CTE): ~58/31 = 1084.628
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}
-{{Optimal ET sequence|legend=1| 52, 1737, 1789 }}, ...
+: [[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 }}
-=== Huntington ===
+{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}
-{{See also| Chromatic pairs #Huntington }}
-Huntington may be described as the 10 &amp; 27 temperament in the 2.5.7.13 subgroup.
+[[Badness]] (Sintel): 148
-[[Subgroup]]: 2.5.7.13
+==== 2.5.7.11 subgroup ====
+Subgroup: 2.5.7.11
-[[Comma list]]: [[640/637]], [[10985/10976]]
+Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}
-{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}
+Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}
-{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}
+Optimal tunings:
+* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
-: [[gencom]]: [2 16/13; 640/637 10985/10976]
+{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/13 = 357.002
+Badness (Sintel): 52.2
-{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}
+==== 2.5.7.11.13 subgroup ====
+Subgroup: 2.5.7.11.13
-[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents
+Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625
-==== Silver ====
+Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}
-{{See also| Chromatic pairs #Silver }}
-Silver can be described as the 10 &amp; 27 temperament in the 2.5.7.13.17 subgroup.
+Optimal tunings:
+* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}
-[[Subgroup]]: 2.5.7.13.17
+{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}
-[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]
+Badness (Sintel): 10.5
-{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}
+=== Bastille ===
+{{Main| Bastille }}
-{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
+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]].
-: [[gencom]]: [2 13/8; 170/169 640/637 5525/5488]
+[[Subgroup]]: 2.5.7
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/8 = 842.711
+[[Comma list]]: {{monzo| 1426 -596 -15 }}
-{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}
+{{Mapping|legend=2| 1 -4 254 | 0 15 -596 }}
+: mapping generators: ~2, ~{{monzo| -380 159 4 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}
-=== Pakkanen ===
+: [[error map]]: {{val| -0.009 +0.020 +0.001 }}
-[[Subgroup]]: 2.5.7.11
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}
+: error map: {{val| 0.000 +0.041 +0.018 }}
-[[Comma list]]: 625/616
+{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}
-{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
+[[Badness]] (Sintel): 7.18 × 10<sup>3</sup>
-: mapping generators: ~2, ~5, ~11
+=== 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. }}
-[[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 }}
@@ Line 315: / Line 353: @@
 [[Comma list]]: 5021863/5000000
-{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
+{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
+: mapping generators: ~2, ~5, ~77/50
-: mapping generators: ~2, ~5, ~100/77
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1200.080¢, ~5 = 2786.820¢, ~100/77 = 454.618¢
+* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{c}}
-* [[CWE]]: ~2 = 1200.000¢, ~5 = 2786.740¢, ~100/77 = 454.590¢
+: [[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, 50, 58, 66, 87, 103, 124 }}
+{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}
-[[Badness]] (Sintel): 1.862
+[[Badness]] (Sintel): 1.86
 ==== 2.5.7.11.13 subgroup ====
@@ Line 332: / Line 371: @@
 Comma list: 847/845, 1001/1000
-Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}
+Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}
 Optimal tunings:
-* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
+* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
-* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}
-{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
+{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}
 Badness (Sintel): 0.179
@@ Line 347: / Line 386: @@
 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 }}
+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 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
+* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}
-* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}
-{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}
+{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}
 Badness (Sintel): 0.438
-== Higher 2.5 temperaments ==
+== Temperaments with a higher 2.5.''p'' gene ==
 Temperaments discussed elsewhere include:
 * Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]
-=== Movila ===
+=== Wizz ===
-This temperament has a structure very similar to [[mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].
+{{See also| Chromatic pairs #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]]: 1331/1280
+[[Comma list]]: 15625/15488
-[[Mapping]]: [{{val|1 1 3}}, {{val|0 3 1}}]
+{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
-[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[11/8]] = 529.846
+{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
+: mapping generators: ~125/88, ~5/4
-{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
+[[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 }}
-=== Wizz ===
+{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}
-{{See also| Chromatic pairs #Wizz }}
-Wizz, the 6 &amp; 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].
+[[Badness]] (Sintel): 0.266
+=== Insect ===
 [[Subgroup]]: 2.5.11
-[[Comma list]]: [[15625/15488]]
+[[Comma list]]: 33275/32768
-{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}
+{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}
+: mapping generators, ~2, ~55/32
-{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}
+[[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 }}
-: [[gencom]]: [125/88 5/4; 15625/15488]
+{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}
-[[Optimal tuning]] ([[POTE]]): ~125/88 = 1\2, ~5/4 = 383.768
+[[Badness]] (Sintel): 0.564
-{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222 }}
+=== 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]].
-[[Tp tuning #T2 tuning|RMS error]]: 0.3997
-=== Insect ===
 [[Subgroup]]: 2.5.11
-[[Comma list]]: 33275/32768
+[[Comma list]]: 1331/1280
-{{Mapping|legend=2|1 0 5|0 3 -2}}
+{{Mapping|legend=2| 1 1 3 | 0 3 1 }}
+: mapping generators: ~2, ~11/8
-: Mapping generators, ~2, ~[[55/32]]
+[[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]]): ~2 = 1\1, ~[[55/32]] = 928.032
+{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}
-{{Optimal ET sequence|legend=1|9, 13, 22, 97e, 119e, 141e, 163e, 304ceee}}
+[[Badness]] (Sintel): 0.718
 === Sephiroth ===
 {{See also| Chromatic pairs #Sephiroth }}
-Sephiroth is the no-7 restriction of [[muggles]].
+Sephiroth is the no-7 [[restriction]] of [[muggles]].
+[[Subgroup]]: 2.5.11
+[[Comma list]]: 34375/32768
+{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}
+{{Mapping|legend=3| 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
+Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}
+Gencom mapping: {{mapping| 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 }}
-[[Subgroup]]: 2.5.11.13.17
+Badness (Sintel): 0.410
-[[Comma list]]: 65/64, 170/169, 221/220
+==== 2.5.11.13.17 subgroup ====
+Subgroup: 2.5.11.13.17
-{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
+Comma list: 65/64, 170/169, 221/220
-{{Mapping|legend=3| 1 0 2 0 5 4 5 | 0 0 1 0 -5 -1 -3 }}
+Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}
-: [[gencom]]: [2 5/4; 65/64 170/169 221/220]
+Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 372.236
+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| 10, 13, 16, 29 }}
+{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}
-[[Tp tuning #T2 tuning|RMS error]]: 1.774 cents
+Badness (Sintel): 0.299
 === Trader ===
 [[Subgroup]]: 2.5.13
-[[Comma list]]: [[26/25]]
+[[Comma list]]: 26/25
-{{Mapping|legend=2|1 2 3|0 1 2}}
+{{Mapping|legend=2| 1 2 3 | 0 1 2 }}
+: mapping generators, ~2, ~5/4
-: 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 tuning]] ([[CTE]]): ~2 = 1\1, ~[[5/4]] = 407.079
+{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}
-{{Optimal ET sequence|legend=1|3, 20c, 23c, 26c}}
+[[Badness]] (Sintel): 0.138
 === Superquintal ===
@@ Line 448: / Line 543: @@
 [[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
-== No-threes-or-fives subgroup temperaments ==
+== No-threes no-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]]
@@ Line 468: / Line 567: @@
 {{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]].
+Amaranthine is the high-accuracy 2.7.11-subgroup strong [[restriction]] of [[Gamelismic clan #11-limit 3|undecimal mothra]].
 [[Subgroup]]: 2.7.11
@@ Line 474: / Line 573: @@
 [[Comma list]]: 5767168/5764801
-{{Mapping|legend=2| 1 2 -3 | 0 1 8 }}
+{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}
+: mapping generators: ~2, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~7/4 = 968.913
+[[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 }}
+{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}
-Badness (Sintel): 0.031
+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
@@ Line 491: / Line 617: @@
 {{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|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}
+: mapping generators: ~2, ~11/8
-: [[gencom]]: [2 11/8; 343/338 847/832]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 540.099
+{{Optimal ET sequence|legend=1| 9, 11, 20 }}
-{{Optimal ET sequence|legend=1| 5, 7, 9, 11, 20 }}
+[[Badness]] (Sintel): 0.368
-[[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.
+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 }}
+{{Mapping|legend=3| 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 }}
-[[Subgroup]]: 2.7.11.13
+[[Badness]] (Sintel): 1.73
-[[Comma list]]: [[1573/1568]], [[15488/15379]]
+==== 2.7.11.13 subgroup ====
+Subgroup: 2.7.11.13
-{{Mapping|legend=2| 1 0 1 3 | 0 8 7 2 }}
+Comma list: 1573/1568, 15488/15379
-{{Mapping|legend=3| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
+Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}
-: [[gencom]]: [2 14/11; 1573/1568 15488/15379]
+Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/11 = 421.309
+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=1| 17, 20, 37, 57, 94, 225, 319cd, 413bcd }}
+{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.4043 cents
+Badness (Sintel): 0.307
 === Voltage ===
-Voltage is the 3 &amp; 7 temperament in the 2.7.13 subgroup.
+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]]
+[[Comma list]]: 28672/28561
-{{Mapping|legend=2| 1 4 4 | 0 -4 -1 }}
+{{Mapping|legend=2| 1 0 3 | 0 4 1 }}
-{{Mapping|legend=3| 1 0 0 4 0 4 | 0 0 0 -4 0 -1 }}
+{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}
+: mapping generators: ~2, ~13
-: [[gencom]]: [2, 16/13; 28672/28561]
+[[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 tuning]]:
+{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}
-* [[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 }}
+[[Badness]] (Sintel): 0.115
-[[Tp tuning #T2 tuning|RMS error]]: 0.1423 cents
 === Ultrakleismic ===
@@ Line 548: / 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 561: / 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
-: 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 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 }}
-{{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
-[[Gencom]]: [2 64/53; 1048576/1042139]
+{{Mapping|legend=2| 1 0 6 | 0 3 -1 }}]
+: mapping generators, ~2, ~64/53
-[[Mapping]]: [{{val|1 0 6}}, {{val|0 3 -1}}]]
+[[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 }}
-[[POTE generator]]: ~64/53 = 323.034
+{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
-{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}
+[[Badness]] (Sintel): 0.224
 === 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.
+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]]
+[[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
-: [[gencom]]: [2 13/11; 1352/1331]
+{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~13/11 = 279.318
+{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
+: mapping generators, ~2, ~143/128
-{{Optimal ET sequence|legend=1| 13, 30, 43, 73, 116 }}
+[[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 }}
-[[Tp tuning #T2 tuning|RMS error]]: 0.8449 cents
+{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}
+[[Badness]] (Sintel): 0.451
 === Blackbirds ===
@@ Line 611: / Line 799: @@
 [[Subgroup]]: 2.11.13
-[[Comma list]]: [[29282/28561]]
+[[Comma list]]: 29282/28561
 {{Mapping|legend=2| 4 0 1 | 0 1 1 }}
 {{Mapping|legend=3| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}
+: mapping generators, ~13/11, ~11
-: [[gencom]]: [13/11 11/8; 29282/28561]
+[[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]] ([[POTE]]): ~13/11 = 1\4, ~11/8 = 546.660
+{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}
-{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 44, 68, 112c, 180bc }}
+[[Badness]] (Sintel): 0.668
-[[Tp tuning #T2 tuning|RMS error]]: 0.8685 cents
+=== 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]].
-=== Bluebirds ===
+[[Subgroup]]: 2.11.13.17.19
-{{Distinguish| Bluebird }}
-{{See also| Chromatic pairs #Bluebirds }}
-[[Subgroup]]: 2.11.13
+[[Comma list]]: 209/208, 2057/2048, 83521/83486
-[[Comma list]]: [[265837/262144]]
+{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}
+: mapping generators: ~2, ~26/17
-{{Mapping|legend=2| 1 0 6 | 0 3 -2 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}
-{{Mapping|legend=3| 1 0 0 0 3 4 | 0 0 0 0 3 -2 }}
+{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}
-: [[gencom]]: [2 143/128; 265837/262144]
+[[Badness]] (Sintel): 0.386
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~143/128 = 182.368
+=== 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.
-{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc }}
+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.
-[[Tp tuning #T2 tuning|RMS error]]: 0.4444 cents
+[[Subgroup]]: 2.11.37
-=== Yamablu ===
+[[Comma list]]: 1874161/1874048
-Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].
-[[Subgroup]]: 2.11.13.17.19
+{{Mapping|legend=2| 4 0 7 | 0 1 1 }}
+: mapping generators: ~44/37, ~11
-[[Comma list]]: 209/208, 2057/2048, 83521/83486
+[[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 }}
-[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]
+{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}
-Optimal tuning ([[POTE]]): ~17/13 = 462.9606
+[[Badness]] (Sintel): 0.00188
-{{Optimal ET sequence|legend=1| 13, 44, 57, 70}}
-[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
 === Mavericks ===
 [[Subgroup]]: 2.13.19
 [[Comma list]]: 47525504/47045881
-[[Mapping]]: [{{val|1 1 2}}, {{val|0 6 5}}]
+{{Mapping|legend=2| 1 1 2 | 0 6 5 }}
+: mapping generators: ~2, ~26/19
+[[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 tuning]] ([[CTE]]): ~2 = 1\1, ~26/19 = 539.886
+{{Optimal ET sequence|legend=1| 9, 11, 20 }}
-{{Optimal ET sequence|legend=1| 7fh, 9, 11, 20 }}
+[[Badness]] (Sintel): 0.559
 === Yer (rank 3) ===
@@ Line 677: / Line 879: @@
 [[Comma list]]: 209/208, 2057/2048
-[[Sval]] [[mapping]]: {{mapping| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}
+{{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]] ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
+[[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 }}
+[[Badness]] (Sintel): 0.106
 [[Category:Temperament collections]]
 [[Category:Subgroup temperaments]]

No-threes subgroup temperaments: Difference between revisions