@@ Line 1: / Line 1: @@
+{{Technical data page}}
 The '''sensamagic clan''' tempers out the sensamagic comma, [[245/243]], a triprime [[comma]] with no factors of 2, {{val| 0 -5 1 2 }} to be exact. Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo]] is the [[optimal patent val]].
 == BPS ==
-The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called the ''lambda'' temperament, which was named after the [[4L 5s (tritave-equivalent)|lambda scale]].
+{{Main| BPS }}
+BPS, for ''Bohlen–Pierce–Stearns'', is the 3.5.7-subgroup temperament tempering out 245/243. This subgroup temperament was formerly called the ''lambda'' temperament, which was named after the [[4L 5s (tritave-equivalent)|lambda scale]].
 [[Subgroup]]: 3.5.7
@@ Line 8: / Line 11: @@
 [[Comma list]]: 245/243
-[[Sval]] [[mapping]]: [{{val| 1 1 2 }}, {{val| 0 -2 1 }}]
+{{Mapping|legend=2| 1 1 2 | 0 -2 1 }}
-Sval mapping generators: ~3, ~9/7
+: sval mapping generators: ~3, ~9/7
-[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881
+[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881
 [[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
 === Overview to extensions ===
-The full 7-limit extensions' relation to BPS is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are
+The full 7-limit extensions' relation to BPS is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are sensi, cohemiripple, hedgehog, and fourfives.
-* [[#Sensi]]
-* [[#Cohemiripple]]
-* ''[[Hedgehog]]'', {50/49, 245/243} → [[Porcupine family #Hedgehog|Porcupine family]]
-* ''[[Fourfives]]'', {245/243, 235298/234375} → [[Fifive family #Fourfives|Fifive family]]
-The others are weak extensions. See
-* ''[[Father]]'', {16/15, 28/27} → [[Father family #Father|Father family]]
-* ''[[Sidi]]'', {25/24, 245/243} → [[Dicot family #Sidi|Dicot family]]
-* [[Godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla|Meantone family]]
-* [[Superpyth]], {64/63, 245/243} → [[Archytas clan #Superpyth|Archytas clan]]
-* ''[[Hemiaug]]'', {128/125, 245/243} → [[Augmented family #Hemiaug|Augmented family]]
-* [[Magic]], {225/224, 245/243} → [[Magic family #Septimal magic|Magic family]]
-* [[Rodan]], {245/243, 1029/1024} → [[Gamelismic clan #Rodan|Gamelismic clan]]
-* ''[[Shrutar]]'', {245/243, 2048/2025} → [[Diaschismic family #Shrutar|Diaschismic family]]
-* ''[[Octacot]]'', {245/243, 2401/2400} → [[Tetracot family #Octacot|Tetracot family]]
-* ''[[Clyde]]'', {245/243, 3136/3125} → [[Kleismic family #Clyde|Kleismic family]]
-* ''[[Pental]]'', {245/243, 16807/16384} → [[Pental family #Septimal pental|Pental family]]
-* ''[[Bamity]]'', {245/243, 64827/64000} → [[Amity family #Bamity|Amity family]]
-* ''[[Escaped]]'', {245/243, 65625/65536} → [[Escapade family #Escaped|Escapade family]]
-as well as bohpier, salsa, pycnic, superthird, magus and leapweek discussed below.
-== Sensi ==
+These temperaments are distributed into different family pages.
-{{Main| Sensi }}
+* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
-{{See also| Sensipent family #Sensi }}
+* ''[[Hedgehog]]'' (+50/49) → [[Porcupine family #Hedgehog|Porcupine family]]
+* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple family]]
+* ''[[Fourfives]]'' (+235298/234375) → [[Fifive family #Fourfives|Fifive family]]
-Sensi tempers out [[126/125]], [[686/675]] and [[4375/4374]] in addition to [[245/243]], and can be described as the 19 &amp; 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of 8-, 11-, 19- and 27-tones are available. The name "sensi" is a play on the words "semi-" and "sixth."
+The others are weak extensions. Father tempers out [[16/15]], splitting the generator in two. Godzilla tempers out [[49/48]] with a hemitwelfth period. Sidi tempers out [[25/24]], splitting the generator in two with a hemitwelfth period. Clyde tempers out [[3136/3125]] with a 1/6-twelfth period. Superpyth tempers out [[64/63]], splitting the generator in six. Magic tempers out [[225/224]] with a 1/5-twelfth period. Octacot tempers out [[2401/2400]], splitting the generator in five. Hemiaug tempers out [[128/125]]. Pentacloud tempers out [[16807/16384]]. These split the generator in seven. Bamity tempers out [[64827/64000]], splitting the generator in nine. Rodan tempers out [[1029/1024]], splitting the generator in ten. Shrutar tempers out [[2048/2025]], splitting the generator in eleven. Finally, escaped tempers out [[65625/65536]], splitting the generator in sixteen.
-=== Septimal sensi ===
+Discussed elsewhere are
-[[Subgroup]]: 2.3.5.7
+* [[Father]] (+16/15 or 28/27) → [[Father family #Father|Father family]]
+* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
+* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
+* ''[[Clyde]]'' (+3136/3125) → [[Kleismic family #Clyde|Kleismic family]]
+* [[Superpyth]] (+64/63) → [[Archytas clan #Superpyth|Archytas clan]]
+* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
+* ''[[Octacot]]'' (+2401/2400) → [[Tetracot family #Octacot|Tetracot family]]
+* ''[[Hemiaug]]'' (+128/125) → [[Augmented family #Hemiaug|Augmented family]]
+* ''[[Pentacloud]]'' (+16807/16384) → [[Quintile family #Pentacloud|Quintile family]]
+* ''[[Bamity]]'' (+64827/64000) → [[Amity family #Bamity|Amity family]]
+* [[Rodan]] (+1029/1024) → [[Gamelismic clan #Rodan|Gamelismic clan]]
+* ''[[Shrutar]]'' (+2048/2025) → [[Diaschismic family #Shrutar|Diaschismic family]]
+* ''[[Escaped]]'' (+65625/65536) → [[Escapade family #Escaped|Escapade family]]
-[[Comma list]]: 126/125, 245/243
+For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].
-[[Mapping]]: [{{val| 1 -1 -1 -2 }}, {{val| 0 7 9 13 }}]
+Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.
-Mapping generators: ~2, ~9/7
-{{Multival|legend=1| 7 9 13 -2 1 5 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 443.383
-{{Optimal ET sequence|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
-[[Badness]]: 0.025622
-==== Sensation ====
-Subgroup: 2.3.5.7.13
-Comma list: 91/90, 126/125, 169/168
-Sval mapping: [{{val| 1 -1 -1 -2 0 }}, {{val| 0 7 9 13 10 }}]
-Gencom mapping: [{{val| 1 -1 -1 -2 0 0 }}, {{val| 0 7 9 13 0 10 }}]
-Gencom: [2 9/7; 91/90 126/125 169/168]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.322
-{{Optimal ET sequence|legend=1| 19, 27, 46, 111de, 157de }}
-=== Sensor ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 245/243, 385/384
-Mapping: [{{val| 1 -1 -1 -2 9 }}, {{val| 0 7 9 13 -15 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.294
-{{Optimal ET sequence|legend=1| 19, 27, 46, 111d, 157d, 268cdd }}
-Badness: 0.037942
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 126/125, 169/168, 385/384
-Mapping: [{{val| 1 -1 -1 -2 9 0 }}, {{val| 0 7 9 13 -15 10 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.321
-{{Optimal ET sequence|legend=1| 19, 27, 46, 111df, 157df }}
-Badness: 0.025575
-==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
-Mapping: [{{val| 1 -1 -1 -2 9 0 10 }}, {{val| 0 7 9 13 -15 10 -16 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.365
-{{Optimal ET sequence|legend=1| 19, 27, 46, 157df, 203cdff, 249cddff }}
-Badness: 0.022908
-=== Sensis ===
-Subgroup: 2.3.5.7.11
-Comma list: 56/55, 100/99, 245/243
-Mapping: [{{val| 1 -1 -1 -2 2 }}, {{val| 0 7 9 13 4 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.962
-{{Optimal ET sequence|legend=1| 8d, 19, 27e, 73ee }}
-Badness: 0.028680
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 56/55, 78/77, 91/90, 100/99
-Mapping: [{{val| 1 -1 -1 -2 2 0 }}, {{val| 0 7 9 13 4 10 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.945
-{{Optimal ET sequence|legend=1| 19, 27e, 46e, 73ee }}
-Badness: 0.020017
-=== Sensus ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 176/175, 245/243
-Mapping: [{{val| 1 -1 -1 -2 -8 }}, {{val| 0 7 9 13 31 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.626
-{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c, 165c }}
-Badness: 0.029486
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 126/125, 169/168, 352/351
-Mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.559
-{{Optimal ET sequence|legend=1| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
-Badness: 0.020789
-==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
-Mapping: [{{val| 1 -1 -1 -2 -8 0 -7 }}, {{val| 0 7 9 13 31 10 30 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.551
-{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
-Badness: 0.016238
-=== Sensa ===
-Subgroup: 2.3.5.7.11
-Comma list: 55/54, 77/75, 99/98
-Mapping: [{{val| 1 -1 -1 -2 -1 }}, {{val| 0 7 9 13 12 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.518
-{{Optimal ET sequence|legend=1| 19e, 27, 46ee }}
-Badness: 0.036835
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 55/54, 66/65, 77/75, 143/140
-Mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 11 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.506
-{{Optimal ET sequence|legend=1| 19e, 27, 46ee }}
-Badness: 0.023258
-=== Hemisensi ===
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 243/242, 245/242
-Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.605
-{{Optimal ET sequence|legend=1| 27e, 38d, 65, 157de, 222cde }}
-Badness: 0.048714
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 126/125, 169/168, 243/242
-Mapping: [{{val| 1 -1 -1 -2 -3 0 }}, {{val| 0 14 18 26 35 30 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~25/22 = 221.556
-{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
-Badness: 0.033016
-=== Bisensi ===
-Subgroup: 2.3.5.7.11
-Comma list: 121/120, 126/125, 245/243
-Mapping: [{{val| 2 5 7 9 9 }}, {{val| 0 -7 -9 -13 -8 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.692
-{{Optimal ET sequence|legend=1| 8d, …, 38d, 46, 176dde, 222cdde, 268cddee }}
-Badness: 0.041723
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 121/120, 126/125, 169/168
-Mapping: [{{val| 2 5 7 9 9 10 }}, {{val| 0 -7 -9 -13 -8 -10 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 156.725
-{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
-Badness: 0.026339
-== Cohemiripple ==
-{{See also| Ripple family }}
-[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 245/243, 1323/1250
-[[Mapping]]: [{{val| 1 -3 -5 -5 }}, {{val| 0 10 16 17 }}]
-{{Multival|legend=1| 10 16 17 2 -1 -5 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 549.944
-{{Optimal ET sequence|legend=1| 11cd, 13cd, 24 }}
-[[Badness]]: 0.190208
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 77/75, 243/242, 245/242
-Mapping: [{{val| 1 -3 -5 -5 -8 }}, {{val| 0 10 16 17 25 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.945
-{{Optimal ET sequence|legend=1| 11cdee, 13cdee, 24 }}
-Badness: 0.082716
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 77/75, 147/143, 243/242
-Mapping: [{{val| 1 -3 -5 -5 -8 -5 }}, {{val| 0 -10 -16 -17 -25 -19 }}]
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 549.958
-{{Optimal ET sequence|legend=1| 11cdeef, 13cdeef, 24 }}
-Badness: 0.049933
 == Bohpier ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Bohpier]].''
 {{Main| Bohpier }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Bohpier]].''
-'''[[Bohpier]]''' is named after its [[Relationship between Bohlen-Pierce and octave-ful temperaments|interesting relationship with the non-octave Bohlen-Pierce equal temperament]].
+Bohpier is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].
 [[Subgroup]]: 2.3.5.7
@@ Line 314: / Line 59: @@
 [[Comma list]]: 245/243, 3125/3087
-[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 13 19 23 }}]
+{{Mapping|legend=1| 1 0 0 0 | 0 13 19 23 }}
-{{Multival|legend=1| 13 19 23 0 0 0 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 146.474
-Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 146.474
 [[Minimax tuning]]:
-* 7-odd-limit: ~27/25 = {{monzo| 0 0 1/19 }}
+* [[7-odd-limit]]: ~27/25 = {{monzo| 0 0 1/19 }}
-: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-* 9-odd-limit: ~27/25 = {{monzo| 0 1/13 }}
+* [[9-odd-limit]]: ~27/25 = {{monzo| 0 1/13 }}
-: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 {{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}
@@ Line 335: / Line 78: @@
 Comma list: 100/99, 245/243, 1344/1331
-Mapping: [{{val| 1 0 0 0 2 }}, {{val| 0 13 19 23 12 }}]
+Mapping: {{mapping| 1 0 0 0 2 | 0 13 19 23 12 }}
-Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.545
+Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.545
 Minimax tuning:
 * 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
-: Eigenmonzo basis (unchanged-interval basis): 2.11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
-{{Optimal ET sequence|legend=1| 41, 90e, 131e }}
+{{Optimal ET sequence|legend=0| 41, 90e, 131e }}
 Badness: 0.033949
@@ Line 352: / Line 95: @@
 Comma list: 100/99, 144/143, 196/195, 275/273
-Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]
+Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}
-Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.603
+Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.603
 Minimax tuning:
 * 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
-: Eigenmonzo basis (unchanged-interval basis): 2.5
+: Unchanged-interval (eigenmonzo) basis: 2.5
-{{Optimal ET sequence|legend=1| 41, 90ef, 131ef, 221bdeff }}
+{{Optimal ET sequence|legend=0| 41, 90ef, 131ef, 221bdeff }}
 Badness: 0.024864
-; Music
-by [[Chris Vaisvil]]:
-* [http://micro.soonlabel.com/bophier/bophier-1.mp3 bophier&#45;1.mp3]
-* [http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3 bophier&#45;12equal&#45;six&#45;octaves.mp3]
 === Triboh ===
-'''Triboh''' is named after "[[39edt|Triple Bohlen-Pierce scale]]", which divides each step of the [[13edt|equal-tempered]] [[Bohlen-Pierce]] scale into three equal parts.
+Triboh is named after the "[[39edt|Triple Bohlen–Pierce scale]]", which divides each step of the [[13edt|equal-tempered]] [[Bohlen–Pierce]] scale into three equal parts.
 Subgroup: 2.3.5.7.11
@@ Line 376: / Line 114: @@
 Comma list: 245/243, 1331/1323, 3125/3087
-Mapping: [{{val| 1 0 0 0 0 }}, {{val| 0 39 57 69 85 }}]
+Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}
-Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.828
+Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828
-{{Optimal ET sequence|legend=1| 49, 123ce, 172 }}
+{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}
 Badness: 0.162592
@@ Line 389: / Line 127: @@
 Comma list: 245/243, 275/273, 847/845, 1331/1323
-Mapping: [{{val| 1 0 0 0 0 0 }}, {{val| 0 39 57 69 85 91 }}]
+Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}
-Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.822
+Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822
-{{Optimal ET sequence|legend=1| 49f, 123ce, 172f, 295ce, 467bccef }}
+{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}
 Badness: 0.082158
@@ Line 404: / Line 142: @@
 [[Comma list]]: 245/243, 32805/32768
-[[Mapping]]: [{{val| 1 1 7 -1 }}, {{val| 0 2 -16 13 }}]
+{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
-{{Multival|legend=1| 2 -16 13 -30 15 75 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 351.049
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~128/105 = 351.049
 {{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
@@ Line 419: / Line 155: @@
 Comma list: 243/242, 245/242, 385/384
-Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]
+Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014
+Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.014
-{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}
 Badness: 0.039444
@@ Line 432: / Line 168: @@
 Comma list: 105/104, 144/143, 243/242, 245/242
-Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]
+Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.025
+Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.025
-{{Optimal ET sequence|legend=1| 17, 24, 41, 106df, 147df }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106df, 147df }}
 Badness: 0.030793
 == Pycnic ==
-{{See also| High badness temperaments #Stump }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Stump]].''
 The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has [[mos]] of size 9, 11, 13, 15, 17… which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.
@@ Line 449: / Line 185: @@
 [[Comma list]]: 245/243, 525/512
-[[Mapping]]: [{{val| 1 3 -1 8 }}, {{val| 0 -3 7 -11 }}]
+{{Mapping|legend=1| 1 3 -1 8 | 0 -3 7 -11 }}
-{{Multival|legend=1| 3 -7 11 -18 9 45 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 567.720
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~45/32 = 567.720
 {{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
@@ Line 460: / Line 194: @@
 == Superthird ==
-{{See also| Shibboleth family }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 466: / Line 200: @@
 [[Comma list]]: 245/243, 78125/76832
-[[Mapping]]: [{{val| 1 -5 -5 -10 }}, {{val| 0 18 20 35 }}]
+{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
-{{Multival|legend=1| 18 20 35 -10 5 25 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~9/7 = 439.076
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 439.076
 {{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
@@ Line 481: / Line 213: @@
 Comma list: 100/99, 245/243, 78125/76832
-Mapping: [{{val| 1 -5 -5 -10 2 }}, {{val| 0 18 20 35 4 }}]
+Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.152
+Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152
-{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 153be, 194be, 235bcee }}
+{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}
 Badness: 0.070917
@@ Line 494: / Line 226: @@
 Comma list: 100/99, 144/143, 196/195, 1375/1352
-Mapping: [{{val| 1 -5 -5 -10 2 -8 }}, {{val| 0 18 20 35 4 32 }}]
+Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}
-Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.119
+Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119
-{{Optimal ET sequence|legend=1| 11cdf, 30df, 41 }}
+{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}
 Badness: 0.052835
@@ Line 509: / Line 241: @@
 [[Comma list]]: 245/243, 395136/390625
-[[Mapping]]: [{{val| 19 0 14 -7 }}, {{val| 0 1 1 2 }}]
+{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
+[[Optimal tuning]] ([[POTE]]): ~392/375 = 63.158, ~3/2 = 704.166
 {{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
+[[Badness]]: 0.132311
 === 11-limit ===
@@ Line 518: / Line 254: @@
 Comma list: 245/243, 2560/2541, 3773/3750
-Mapping: [{{val| 19 0 14 -7 96 }}, {{val| 0 1 1 2 -1 }}]
+Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
+Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667
+{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}
-{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114e }}
+Badness: 0.101496
 === 13-limit ===
@@ Line 527: / Line 267: @@
 Comma list: 196/195, 245/243, 832/825, 1001/1000
-Mapping: [{{val| 19 0 14 -7 96 10 }}, {{val| 0 1 1 2 -1 2 }}]
+Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
-{{Optimal ET sequence|legend=1| 19, 76bcdf, 95, 114e }}
+Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801
+{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e }}
+Badness: 0.053197
 == Magus ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Magus]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
-Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46 &amp; 49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). The alternative extension [[Starling temperaments #Amigo|amigo]] (43 &amp; 46) tempers out the same 5-limit comma as the magus, but with the [[126/125|starling comma]] (126/125) rather than the sensamagic tempered out.
+Magus temperament tempers out [[50331648/48828125]] (salegu) in the 5-limit. This temperament can be described as {{nowrap| 46 & 49 }} temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). The alternative extension [[starling temperaments #Amigo|amigo]] ({{nowrap|43 & 46}}) tempers out the same 5-limit comma as the magus, but with the [[126/125|starling comma]] (126/125) rather than the sensamagic tempered out.
+Magus has a generator of a sharp ~5/4 (so that ~[[25/16]] is twice as sharp so that it makes sense to equate with [[11/7]] by tempering [[176/175]]), so that three reaches [[128/125]] short of the octave (where 128/125 is tuned narrow); this is significant because magus reaches [[3/2]] as ([[25/16]])/([[128/125]])<sup>3</sup>, that is, {{nowrap|2 + 3 × 3 {{=}} 11}} generators. Therefore, it implies that [[25/24]] is split into three [[128/125]]'s. Therefore, in the 5-limit, magus can be thought of as a higher-complexity and sharper analogue of [[würschmidt]] (which reaches [[3/2]] as (25/16)/(128/125)<sup>2</sup> implying 25/24 is split into two 128/125's thus having a guaranteed neutral third), which itself is a higher-complexity and sharper analogue of [[magic]] (which equates 25/24 with 128/125 by flattening 5). For more details on these connections see [[Würschmidt comma]].
 [[Subgroup]]: 2.3.5.7
@@ Line 540: / Line 286: @@
 [[Comma list]]: 245/243, 28672/28125
-[[Mapping]]: [{{val| 1 -2 2 -6 }}, {{val| 0 11 1 27 }}]
+{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
-{{Multival|legend=1| 11 1 27 -24 12 60 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.465
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 391.465
 {{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
@@ Line 555: / Line 299: @@
 Comma list: 176/175, 245/243, 1331/1323
-Mapping: [{{val| 1 -2 2 -6 -6 }}, {{val| 0 11 1 27 29 }}]
+Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}
-Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.503
+Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.503
-{{Optimal ET sequence|legend=1| 46, 95, 141bc }}
+{{Optimal ET sequence|legend=0| 46, 95, 141bc }}
 Badness: 0.045108
@@ Line 568: / Line 312: @@
 Comma list: 91/90, 176/175, 245/243, 1331/1323
-Mapping: [{{val| 1 -2 2 -6 -6 5 }}, {{val| 0 11 1 27 29 -4 }}]
+Mapping: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}
-Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.366
+Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.366
-{{Optimal ET sequence|legend=1| 46, 233bcff, 279bccff }}
+{{Optimal ET sequence|legend=0| 46, 233bcff, 279bccff }}
 Badness: 0.043024
 == Leapweek ==
-:''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
+: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 583: / Line 327: @@
 [[Comma list]]: 245/243, 2097152/2066715
-[[Mapping]]: [{{val| 1 0 42 -21 }}, {{val| 0 1 -25 15 }}]
+{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}
-Mapping generators: ~2, ~3
+: mapping generators: ~2, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.536
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 704.536
 {{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
@@ Line 598: / Line 342: @@
 Comma list: 245/243, 385/384, 1331/1323
-Mapping: [{{val| 1 0 42 -21 -14 }}, {{val| 0 1 -25 15 11 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.554
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.554
-{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
 Badness: 0.050679
@@ Line 611: / Line 355: @@
 Comma list: 169/168, 245/243, 352/351, 364/363
-Mapping: [{{val| 1 0 42 -21 -14 -9 }}, {{val| 0 1 -25 15 11 8 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.571
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.571
-{{Optimal ET sequence|legend=1| 17, 29c, 46, 63, 109 }}
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 63, 109 }}
 Badness: 0.032727
@@ Line 624: / Line 368: @@
 Comma list: 154/153, 169/168, 245/243, 256/255, 273/272
-Mapping: [{{val| 1 0 42 -21 -14 -9 -34 }}, {{val| 0 1 -25 15 11 8 24 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.540
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540
-{{Optimal ET sequence|legend=1| 17g, 29cg, 46, 109, 155f, 264bfg }}
+{{Optimal ET sequence|legend=0| 17g, 29cg, 46, 109, 155f, 264bfg }}
 Badness: 0.026243
@@ Line 637: / Line 381: @@
 Comma list: 136/135, 169/168, 221/220, 245/243, 364/363
-Mapping: [{{val| 1 0 42 -21 -14 -9 39 }}, {{val| 0 1 -25 15 11 8 -22 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 -9 39 | 0 1 -25 15 11 8 -22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.537
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537
-{{Optimal ET sequence|legend=1| 17, 29c, 46, 109g, 155fg, 264bfgg }}
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 109g, 155fg, 264bfgg }}
 Badness: 0.026774
 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Sensamagic clan| ]] <!-- main article -->
 [[Category:Rank 2]]
 [[Category:Listen]]

Sensamagic clan: Difference between revisions