@@ Line 1: / Line 1: @@
-{{interwiki
-| de = Mitteltönig
-| en = Meantone family
-| es =
-| ja =
-}}
 {{Technical data page}}
 The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].
@@ Line 11: / Line 5: @@
 {{Main| Meantone }}
-Meantone is characterized by an [[2/1|octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].
+Meantone is characterized by an [[octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].
 [[Subgroup]]: 2.3.5
@@ Line 71: / Line 65: @@
 ==== 31edo as splitting the fifth into two, three and nine ====
-[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).
+[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out [[225/224]], which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).
 Temperaments discussed elsewhere include
@@ Line 77: / Line 71: @@
 * [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
 * [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
+* ''[[Mazzard]]'' (+12288/12005) → [[Buzzardsmic clan #Mazzard|Buzzardsmic clan]]
 * ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]
@@ Line 126: / Line 121: @@
 Comma list: 81/80, 99/98, 126/125
-Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 | 0 1 4 10 18 }}
 Optimal tunings:
@@ Line 157: / Line 152: @@
 Comma list: 81/80, 99/98, 126/125, 144/143
-Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}
 Optimal tunings:
@@ Line 182: / Line 177: @@
 Comma list: 81/80, 99/98, 120/119, 126/125, 144/143
-Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}
 Optimal tunings:
@@ Line 197: / Line 192: @@
 Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143
-Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}
 Optimal tunings:
@@ Line 216: / Line 211: @@
 Comma list: 66/65, 81/80, 99/98, 105/104
-Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}
 Optimal tunings:
@@ Line 235: / Line 230: @@
 Comma list: 66/65, 81/80, 99/98, 105/104, 120/119
-Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}
 Optimal tunings:
@@ Line 250: / Line 245: @@
 Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119
-Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}
 Optimal tunings:
@@ Line 267: / Line 262: @@
 Comma list: 78/77, 81/80, 99/98, 126/125
-Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}
 Optimal tunings:
@@ Line 286: / Line 281: @@
 Comma list: 78/77, 81/80, 99/98, 120/119, 126/125
-Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}
 Optimal tunings:
@@ Line 301: / Line 296: @@
 Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125
-Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}
 Optimal tunings:
@@ Line 316: / Line 311: @@
 Comma list: 81/80, 99/98, 126/125, 169/168
-Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}
 : mapping generators: ~2, ~26/15
@@ Line 333: / Line 328: @@
 Comma list: 81/80, 99/98, 126/125, 169/168, 221/220
-Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}
 Optimal tunings:
@@ Line 348: / Line 343: @@
 Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220
-Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
+{{Mapping|legend=0| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}
 Optimal tunings:
@@ Line 363: / Line 358: @@
 Comma list: 81/80, 99/98, 126/125, 847/845
-Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
+{{Mapping|legend=0| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}
 : mapping generators: ~55/39, ~3
@@ Line 380: / Line 375: @@
 Comma list: 81/80, 99/98, 126/125, 221/220, 289/288
-Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
+{{Mapping|legend=0| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}
 Optimal tunings:
@@ Line 395: / Line 390: @@
 Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220
-Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
+{{Mapping|legend=0| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}
 Optimal tunings:
@@ Line 406: / Line 401: @@
 === Meanpop ===
-{{See also| Meantone vs meanpop }}
+{{See also| Huygens vs meanpop }}
 Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.
@@ Line 414: / Line 409: @@
 Comma list: 81/80, 126/125, 385/384
-Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}
+{{Mapping|legend=0| 1 0 -4 -13 24 | 0 1 4 10 -13 }}
 : mapping generator: ~2, ~3
@@ Line 446: / Line 441: @@
 Comma list: 81/80, 105/104, 126/125, 144/143
-Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
+{{Mapping|legend=0| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}
 Optimal tunings:
@@ Line 469: / Line 464: @@
 Comma list: 81/80, 105/104, 126/125, 144/143, 273/272
-Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
+{{Mapping|legend=0| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}
 Optimal tunings:
@@ Line 484: / Line 479: @@
 Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272
-Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
+{{Mapping|legend=0| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}
 Optimal tunings:
@@ Line 499: / Line 494: @@
 Comma list: 81/80, 105/104, 120/119, 126/125, 144/143
-Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
+{{Mapping|legend=0| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}
 Optimal tunings:
@@ Line 514: / Line 509: @@
 Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125
-Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
+{{Mapping|legend=0| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}
 Optimal tunings:
@@ Line 529: / Line 524: @@
 Comma list: 81/80, 126/125, 385/384, 847/845
-Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}
+{{Mapping|legend=0| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}
 : mapping generators: ~55/39, ~3
@@ Line 546: / Line 541: @@
 Comma list: 81/80, 126/125, 221/220, 273/272, 289/288
-Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}
+{{Mapping|legend=0| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}
 Optimal tunings:
@@ Line 561: / Line 556: @@
 Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272
-Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}
+{{Mapping|legend=0| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}
 Optimal tunings:
@@ Line 578: / Line 573: @@
 Comma list: 45/44, 56/55, 81/80
-Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
+{{Mapping|legend=0| 1 0 -4 -13 -6 | 0 1 4 10 6 }}
 Optimal tunings:
@@ Line 597: / Line 592: @@
 Comma list: 45/44, 56/55, 78/77, 81/80
-Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
+{{Mapping|legend=0| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}
 Optimal tunings:
@@ Line 612: / Line 607: @@
 Comma list: 45/44, 56/55, 78/77, 81/80, 120/119
-Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
+{{Mapping|legend=0| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}
 Optimal tunings:
@@ Line 627: / Line 622: @@
 Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119
-Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
+{{Mapping|legend=0| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}
 Optimal tunings:
@@ Line 642: / Line 637: @@
 Comma list: 45/44, 56/55, 65/64, 81/80
-Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
+{{Mapping|legend=0| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}
 Optimal tunings:
@@ Line 657: / Line 652: @@
 Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
-Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
+{{Mapping|legend=0| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}
 Optimal tunings:
@@ Line 672: / Line 667: @@
 Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
-Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
+{{Mapping|legend=0| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}
 Optimal tunings:
@@ Line 689: / Line 684: @@
 Comma list: 81/80, 126/125, 245/242
-Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
+{{Mapping|legend=0| 2 0 -8 -26 -31 | 0 1 4 10 12 }}
 : mapping generators: ~63/44, ~3
@@ Line 706: / Line 701: @@
 Comma list: 81/80, 105/104, 126/125, 245/242
-Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
+{{Mapping|legend=0| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}
 Optimal tunings:
@@ Line 721: / Line 716: @@
 Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
-Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
+{{Mapping|legend=0| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}
 Optimal tunings:
@@ Line 736: / Line 731: @@
 Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220
-Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
+{{Mapping|legend=0| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}
 Optimal tunings:
@@ Line 753: / Line 748: @@
 Comma list: 81/80, 126/125, 1344/1331
-Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
+{{Mapping|legend=0| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}
 : mapping generators: ~2, ~11/10
@@ Line 770: / Line 765: @@
 Comma list: 81/80, 126/125, 144/143, 364/363
-Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
+{{Mapping|legend=0| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}
 Optimal tunings:
@@ Line 785: / Line 780: @@
 Comma list: 81/80, 126/125, 144/143, 189/187, 221/220
-Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
+{{Mapping|legend=0| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}
 Optimal tunings:
@@ Line 794: / Line 789: @@
 Badness (Sintel): 1.28
+=== Migration ===
+See [[Rastmic clan #Migration|Rastmic clan]].
 == Flattone ==
@@ Line 838: / Line 836: @@
 Comma list: 45/44, 81/80, 385/384
-Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
+{{Mapping|legend=0| 1 0 -4 17 -6 | 0 1 4 -9 6 }}
 Optimal tuning:
@@ Line 852: / Line 850: @@
 Badness (Sintel): 1.12
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 45/44, 65/64, 78/77, 81/80
-Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
+{{Mapping|legend=0| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}
 Optimal tunings:
@@ Line 870: / Line 868: @@
 Badness (Sintel): 0.920
+=== Ptolemy ===
+See [[Rastmic clan #Ptolemy|Rastmic clan]].
 == Dominant ==
@@ Line 905: / Line 906: @@
 Comma list: 36/35, 56/55, 64/63
-Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
+{{Mapping|legend=0| 1 0 -4 6 13 | 0 1 4 -2 -6 }}
 Tuning ranges:
@@ Line 924: / Line 925: @@
 Comma list: 36/35, 56/55, 64/63, 66/65
-Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
+{{Mapping|legend=0| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}
 Optimal tunings:
@@ Line 943: / Line 944: @@
 Comma list: 26/25, 36/35, 56/55, 64/63
-Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
+{{Mapping|legend=0| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}
 Optimal tunings:
@@ Line 958: / Line 959: @@
 Comma list: 36/35, 64/63, 77/75
-Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
+{{Mapping|legend=0| 1 0 -4 6 -14 | 0 1 4 -2 11 }}
 Optimal tunings:
@@ Line 973: / Line 974: @@
 Comma list: 26/25, 36/35, 64/63, 66/65
-Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
+{{Mapping|legend=0| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}
 Optimal tunings:
@@ Line 988: / Line 989: @@
 Comma list: 36/35, 45/44, 64/63
-Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
+{{Mapping|legend=0| 1 0 -4 6 -6 | 0 1 4 -2 6 }}
 Optimal tunings:
@@ Line 1,003: / Line 1,004: @@
 Comma list: 22/21, 33/32, 36/35
-Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
+{{Mapping|legend=0| 1 0 -4 6 5 | 0 1 4 -2 -1 }}
 Optimal tunings:
@@ Line 1,012: / Line 1,013: @@
 Badness (Sintel): 0.864
+=== Neutrominant ===
+See [[Rastmic clan #Neutrominant|Rastmic clan]].
 == Flattertone ==
@@ Line 1,041: / Line 1,045: @@
 Comma list: 45/44, 81/80, 1375/1344
-Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}
+{{Mapping|legend=0| 1 0 -4 -24 -6 | 0 1 4 17 6 }}
 Optimal tunings:
@@ Line 1,082: / Line 1,086: @@
 Comma list: 21/20, 28/27, 33/32
-Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}
+{{Mapping|legend=0| 1 0 -4 -2 5 | 0 1 4 3 -1 }}
 Optimal tunings:
@@ Line 1,112: / Line 1,116: @@
 [[Badness]] (Sintel): 2.67
+=== 11-limit ===
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 81/80, 176/175, 7056/6875
+{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}
+{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}
+[[Badness]] (Sintel): 2.15
+=== 13-limit ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 81/80, 176/175, 196/195, 832/825
+{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
+[[Badness]] (Sintel): 2.04
+=== 17-limit ===
+[[Subgroup]]: 2.3.5.7.11.13.17
+[[Comma list]]: 81/80, 176/175, 189/187, 196/195, 832/825
+{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
+[[Badness]] (Sintel): 1.98
+=== 19-limit ===
+[[Subgroup]]: 2.3.5.7.11.13.17.19
+[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825
+{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}
+{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}
+[[Badness]] (Sintel): 1.95
+{{Todo|unify precision|review}}
 == Supermean ==
@@ Line 1,137: / Line 1,207: @@
 Comma list: 56/55, 81/80, 132/125
-Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}
+{{Mapping|legend=0| 1 0 -4 -21 -14 | 0 1 4 15 11 }}
 Optimal tunings:
@@ Line 1,152: / Line 1,222: @@
 Comma list: 26/25, 56/55, 66/65, 81/80
-Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}
+{{Mapping|legend=0| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}
 Optimal tunings:
@@ Line 1,165: / Line 1,235: @@
 {{Main| Mohajira }}
-Mohajira can be viewed as derived from mohaha which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31.
+Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,204: / Line 1,274: @@
 Comma list: 81/80, 121/120, 176/175
-Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}
+{{Mapping|legend=0| 1 1 0 6 2 | 0 2 8 -11 5 }}
 Optimal tunings:
@@ Line 1,230: / Line 1,300: @@
 Comma list: 66/65, 81/80, 105/104, 121/120
-Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}
+{{Mapping|legend=0| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}
 Optimal tunings:
@@ Line 1,247: / Line 1,317: @@
 Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
-Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}
+{{Mapping|legend=0| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}
 Optimal tunings:
@@ Line 1,264: / Line 1,334: @@
 Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
-Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}
+{{Mapping|legend=0| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}
 Optimal tunings:
@@ Line 1,304: / Line 1,374: @@
 Comma list: 56/55, 77/75, 243/242
-Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}
+{{Mapping|legend=0| 1 1 0 -1 2 | 0 2 8 13 5 }}
 Optimal tunings:
@@ Line 1,321: / Line 1,391: @@
 Comma list: 56/55, 66/65, 77/75, 243/242
-Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}
+{{Mapping|legend=0| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}
 Optimal tunings:
@@ Line 1,368: / Line 1,438: @@
 Comma list: 56/55, 81/80, 540/539
-Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}
+{{Mapping|legend=0| 1 0 -4 -3 4 | 0 3 12 11 -1 }}
 Optimal tunings:
@@ Line 1,383: / Line 1,453: @@
 Comma list: 56/55, 78/77, 81/80, 91/90
-Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}
+{{Mapping|legend=0| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}
 Optimal tunings:
@@ Line 1,398: / Line 1,468: @@
 Comma list: 77/75, 81/80, 99/98
-Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}
+{{Mapping|legend=0| 1 0 -4 -3 -5 | 0 3 12 11 16 }}
 Optimal tunings:
@@ Line 1,413: / Line 1,483: @@
 Comma list: 66/65, 77/75, 81/80, 99/98
-Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}
+{{Mapping|legend=0| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}
 Optimal tunings:
@@ Line 1,428: / Line 1,498: @@
 Comma list: 45/44, 81/80, 343/330
-Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}
+{{Mapping|legend=0| 1 0 -4 -3 -6 | 0 3 12 11 18 }}
 Optimal tunings:
@@ Line 1,443: / Line 1,513: @@
 Comma list: 45/44, 81/80, 91/88, 147/143
-Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}
+{{Mapping|legend=0| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}
 Optimal tunings:
@@ Line 1,479: / Line 1,549: @@
 Comma list: 81/80, 176/175, 864/847
-Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}
+{{Mapping|legend=0| 1 2 4 1 5 | 0 -3 -12 13 -11 }}
 Optimal tunings:
@@ Line 1,494: / Line 1,564: @@
 Comma list: 78/77, 81/80, 144/143, 176/175
-Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
+{{Mapping|legend=0| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}
 Optimal tunings:
@@ Line 1,562: / Line 1,632: @@
 Comma list: 81/80, 99/98, 121/120
-Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}
+{{Mapping|legend=0| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}
 Optimal tunings:
@@ Line 1,577: / Line 1,647: @@
 Comma list: 66/65, 81/80, 99/98, 121/120
-Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}
+{{Mapping|legend=0| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}
 Optimal tunings:
@@ Line 1,592: / Line 1,662: @@
 Comma list: 78/77, 81/80, 91/90, 99/98
-Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}
+{{Mapping|legend=0| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}
 Optimal tunings:
@@ Line 1,607: / Line 1,677: @@
 Comma list: 81/80, 99/98, 105/104, 121/120
-Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}
+{{Mapping|legend=0| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}
 Optimal tunings:
@@ Line 1,622: / Line 1,692: @@
 Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
-Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}
+{{Mapping|legend=0| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}
 Optimal tunings:
@@ Line 1,637: / Line 1,707: @@
 Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
-Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}
+{{Mapping|legend=0| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}
 Optimal tunings:
@@ Line 1,652: / Line 1,722: @@
 Comma list: 81/80, 385/384, 1375/1372
-Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}
+{{Mapping|legend=0| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}
 Optimal tunings:
@@ Line 1,663: / Line 1,733: @@
 == Jerome ==
-Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.
+Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 5<sup>1/20</sup>, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. Related temperaments include [[progression]] and [[secund]], which only differ by the mapping of 5. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.
 [[Subgroup]]: 2.3.5.7
@@ Line 1,688: / Line 1,758: @@
 Comma list: 81/80, 99/98, 864/847
-Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}
+{{Mapping|legend=0| 1 1 0 2 3 | 0 5 20 7 4 }}
 Optimal tunings:
@@ Line 1,703: / Line 1,773: @@
 Comma list: 78/77, 81/80, 99/98, 144/143
-Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}
+{{Mapping|legend=0| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}
 Optimal tunings:
@@ Line 1,718: / Line 1,788: @@
 Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
-Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}
+{{Mapping|legend=0| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}
 Optimal tunings:
@@ Line 1,733: / Line 1,803: @@
 Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
-Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}
+{{Mapping|legend=0| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}
 Optimal tunings:
@@ Line 1,769: / Line 1,839: @@
 Comma list: 81/80, 99/98, 2541/2500
-Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}
+{{Mapping|legend=0| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}
 Optimal tunings:
@@ Line 1,815: / Line 1,885: @@
 Comma list: 45/44, 50/49, 81/80
-Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}
+{{Mapping|legend=0| 2 0 -8 -7 -12 | 0 1 4 4 6 }}
 Optimal tunings:
@@ Line 1,834: / Line 1,904: @@
 Comma list: 45/44, 50/49, 78/77, 81/80
-Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}
+{{Mapping|legend=0| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}
 Optimal tunings:
@@ Line 1,853: / Line 1,923: @@
 Comma list: 45/44, 50/49, 78/77, 81/80, 85/84
-Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}
+{{Mapping|legend=0| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}
 Optimal tunings:
@@ Line 1,868: / Line 1,938: @@
 Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84
-Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}
+{{Mapping|legend=0| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}
 Optimal tunings:
@@ Line 1,883: / Line 1,953: @@
 Comma list: 27/26, 40/39, 45/44, 50/49
-Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}
+{{Mapping|legend=0| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}
 Optimal tunings:
@@ Line 1,898: / Line 1,968: @@
 Comma list: 33/32, 50/49, 55/54
-Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}
+{{Mapping|legend=0| 2 0 -8 -7 10 | 0 1 4 4 -1 }}
 Optimal tunings:
@@ Line 1,913: / Line 1,983: @@
 Comma list: 50/49, 56/55, 81/77
-Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}
+{{Mapping|legend=0| 2 0 -8 -7 7 | 0 1 4 4 0 }}
 Optimal tunings:
@@ Line 1,932: / Line 2,002: @@
 Comma list: 50/49, 81/80, 864/847
-Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}
+{{Mapping|legend=0| 2 1 -4 -3 8 | 0 2 8 8 -1 }}
 : mapping generators: ~7/5, ~16/11
@@ Line 1,949: / Line 2,019: @@
 Comma list: 50/49, 78/77, 81/80, 144/143
-Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}
+{{Mapping|legend=0| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}
 Optimal tunings:
@@ Line 1,964: / Line 2,034: @@
 Comma list: 50/49, 78/77, 81/80, 85/84, 144/143
-Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}
+{{Mapping|legend=0| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}
 Optimal tunings:
@@ Line 1,979: / Line 2,049: @@
 Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143
-Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}
+{{Mapping|legend=0| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}
 Optimal tunings:
@@ Line 2,015: / Line 2,085: @@
 Comma list: 81/80, 245/242, 385/384
-Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}
+{{Mapping|legend=0| 2 1 -4 11 8 | 0 2 8 -5 -1 }}
 Optimal tunings:
@@ Line 2,030: / Line 2,100: @@
 Comma list: 81/80, 105/104, 144/143, 245/242
-Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}
+{{Mapping|legend=0| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}
 Optimal tunings:
@@ Line 2,045: / Line 2,115: @@
 Comma list: 81/80, 105/104, 144/143, 245/242, 273/272
-Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}
+{{Mapping|legend=0| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}
 Optimal tunings:
@@ Line 2,060: / Line 2,130: @@
 Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209
-Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}
+{{Mapping|legend=0| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}
 Optimal tunings:
@@ Line 2,096: / Line 2,166: @@
 Comma list: 81/80, 99/98, 73728/73205
-Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}
+{{Mapping|legend=0| 2 1 -4 4 8 | 0 4 16 3 -2 }}
 Optimal tunings:
@@ Line 2,111: / Line 2,181: @@
 Comma list: 81/80, 99/98, 144/143, 2200/2197
-Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}
+{{Mapping|legend=0| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}
 Optimal tunings:
@@ Line 2,147: / Line 2,217: @@
 Comma list: 81/80, 385/384, 2401/2376
-Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}
+{{Mapping|legend=0| 5 0 -20 14 41 | 0 1 4 0 -3 }}
 Optimal tunings:
@@ Line 2,162: / Line 2,232: @@
 Comma list: 81/80, 105/104, 144/143, 2401/2376
-Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}
+{{Mapping|legend=0| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}
 Optimal tunings:
@@ Line 2,178: / Line 2,248: @@
 [[Comma list]]: 81/80, 96/95
-{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}
+{{Mapping|legend=3| 1 0 -4 9 | 0 1 4 -3 }}
-{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}
+{{Mapping|legend=5| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}
 : mapping generators: ~2, ~3
@@ Line 2,201: / Line 2,271: @@
 [[Comma list]]: 45/44, 81/80
-{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}
+{{Mapping|legend=3| 1 0 -4 -6 | 0 1 4 6 }}
-{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}
+{{Mapping|legend=5| 1 0 -4 0 -6 | 0 1 4 0 6 }}
 : mapping generators: ~2, ~3
@@ Line 2,222: / Line 2,292: @@
 Comma list: 45/44, 65/64, 81/80
-Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}
+{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 4 6 -4 }}
-Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}
+{{Mapping|legend=4| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}
 Optimal tunings:
@@ Line 2,239: / Line 2,309: @@
 [[Comma list]]: 33/32, 55/54
-{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}
+{{Mapping|legend=3| 1 0 -4 5 | 0 1 4 -1 }}
-{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}
+{{Mapping|legend=5| 1 0 -4 0 5 | 0 1 4 0 -1 }}
 : mapping generators: ~2, ~3
@@ Line 2,260: / Line 2,330: @@
 Comma list: 33/32, 55/54, 975/968
-Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
+{{Mapping|legend=2| 1 0 -4 5 21 | 0 1 4 -1 -11 }}
-Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}
+{{Mapping|legend=4| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}
 Optimal tunings:
@@ Line 2,275: / Line 2,345: @@
 <references/>
-[[Category:Temperament families]]
 [[Category:Meantone family| ]] <!-- main article -->
 [[Category:Meantone| ]] <!-- key article -->
-[[Category:Rank 2]]
+[[Category:Temperament families]]
+[[Category:Catalogs of rank-2 temperaments]]
 [[Category:Listen]]

Meantone family: Difference between revisions