@@ Line 1: / Line 1: @@
-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.
+{{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]].
-For full 7-limit extensions, we have sensi, bohpier, escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as
+== BPS ==
-* ''[[Father]]'', {16/15, 28/27} → [[Father family #Father|Father family]]
+{{Main| BPS }}
-* ''[[Sidi]]'', {25/24, 245/243} → [[Dicot family #Sidi|Dicot family]]
-* [[Godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla|Meantone family]]
-* ''[[Hedgehog]]'', {50/49, 245/243} → [[Porcupine family #Hedgehog|Porcupine 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]]
-* ''[[Fourfives]]'', {245/243, 235298/234375} → [[Fifive family #Fourfives|Fifive family]]
-Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo|283EDO]] is the [[optimal patent val]].
-== 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]].
-The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called as ''lambda temperament'', which was named after [[4L 5s (tritave-equivalent)|lambda scale]].
-Subgroup: 3.5.7
+[[Subgroup]]: 3.5.7
 [[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
-[[POTE generator]]: ~9/7 = 440.4881
-[[Optimal GPV sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
-== Sensi ==
-{{main| Sensi }}
-{{see also| Sensipent family #Sensi }}
-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 of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."
-=== Septimal sensi ===
-Subgroup: 2.3.5.7
-[[Comma list]]: 126/125, 245/243
-[[Mapping]]: [{{val| 1 -1 -1 -2 }}, {{val| 0 7 9 13 }}]
-Mapping generators: ~2, ~9/7
-{{Multival|legend=1| 7 9 13 -2 1 5 }}
-[[POTE generator]]: ~9/7 = 443.383
-{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
-[[Badness]]: 0.025622
-==== Sensation ====
+: sval mapping generators: ~3, ~9/7
-Subgroup: 2.3.5.7.13
-Comma list: 91/90, 126/125, 169/168
+[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881
-Sval mapping: [{{val| 1 -1 -1 -2 0 }}, {{val| 0 7 9 13 10 }}]
+[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
-Gencom mapping: [{{val| 1 -1 -1 -2 0 0 }}, {{val| 0 7 9 13 0 10 }}]
+=== 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 sensi, cohemiripple, hedgehog, and fourfives.
-Gencom: [2 9/7; 91/90 126/125 169/168]
+These temperaments are distributed into different family pages.
+* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
+* ''[[Hedgehog]]'' (+50/49) → [[Porcupine family #Hedgehog|Porcupine family]]
+* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple family]]
+* ''[[Fourfives]]'' (+235298/234375) → [[Fifive family #Fourfives|Fifive family]]
-POTE generator: ~9/7 = 443.322
+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.
-Optimal GPV sequence: {{Val list| 19, 27, 46, 111de, 157de }}
+Discussed elsewhere are
+* [[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]]
-=== Sensor ===
+For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].
-Subgroup: 2.3.5.7.11
-Comma list: 126/125, 245/243, 385/384
+Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.
-Mapping: [{{val| 1 -1 -1 -2 9 }}, {{val| 0 7 9 13 -15 }}]
-POTE generator: ~9/7 = 443.294
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.321
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.365
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.962
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.945
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.626
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.559
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.551
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.518
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~9/7 = 443.506
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~25/22 = 221.605
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~25/22 = 221.556
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~11/10 = 156.692
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~11/10 = 156.725
-Optimal GPV sequence: {{Val list| 8d, …, 38df, 46 }}
-Badness: 0.026339
 == Bohpier ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Bohpier]].''
+{{Main| 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
+[[Subgroup]]: 2.3.5.7
 [[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 }}
-[[POTE generator]]: ~27/25 = 146.474
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 }}
-: Eigenmonzos (unchanged intervals): 2, 5/4
+: [[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 }}
-: Eigenmonzos (unchanged intervals): 2, 4/3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
-{{Val list|legend=1| 41, 131, 172, 213c }}
+{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}
 [[Badness]]: 0.068237
@@ Line 285: / 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 }}
-POTE generator: ~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 }}
-: Eigenmonzos (unchanged intervals): 2, 11/9
+: unchanged-interval (eigenmonzo) basis: 2.11/9
-Optimal GPV sequence: {{Val list| 41, 90e, 131e }}
+{{Optimal ET sequence|legend=0| 41, 90e, 131e }}
 Badness: 0.033949
@@ Line 302: / 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 }}
-POTE generator: ~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 }}
-: Eigenmonzos (unchanged intervals): 2, 5/4
+: Unchanged-interval (eigenmonzo) basis: 2.5
-Optimal GPV sequence: {{Val list| 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 326: / 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 }}
-POTE generator: ~77/75 = 48.828
+Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828
-Optimal GPV sequence: {{Val list| 49, 123ce, 172 }}
+{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}
 Badness: 0.162592
@@ Line 339: / 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 }}
-POTE generator: ~77/75 = 48.822
+Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822
-Optimal GPV sequence: {{Val list| 49f, 123ce, 172f, 295ce, 467bccef }}
+{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}
 Badness: 0.082158
-== Escaped ==
-{{see also| Escapade family #Escaped }}
-This temperament is also called as "sensa" because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with 19e&amp;27 temperament (sensi extension).''
-Subgroup: 2.3.5.7
-[[Comma list]]: 245/243, 65625/65536
-[[Mapping]]: [{{val| 1 2 2 4 }}, {{val| 0 -9 7 -26 }}]
-{{Multival|legend=1| 9 -7 26 -32 16 80 }}
-[[POTE generator]]: ~28/27 = 55.122
-{{Val list|legend=1| 22, 65, 87, 196, 283 }}
-[[Badness]]: 0.088746
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 245/243, 385/384, 4000/3993
-Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]
-POTE generator: ~28/27 = 55.126
-Optimal GPV sequence: {{Val list| 22, 65, 87, 196, 283 }}
-Badness: 0.035844
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 245/243, 352/351, 385/384, 625/624
-Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]
-POTE generator: ~28/27 = 55.138
-Optimal GPV sequence: {{Val list| 22, 65, 87, 283 }}
-Badness: 0.031366
 == Salsa ==
-{{see also| Schismatic family }}
+{{See also| Schismatic family }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}
-[[POTE generator]]: ~128/105 = 351.049
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~128/105 = 351.049
-{{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
+{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
 [[Badness]]: 0.080152
@@ Line 414: / 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 }}
-POTE generator: ~11/9 = 351.014
+Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.014
-Optimal GPV sequence: {{Val list| 17, 24, 41, 106d, 147d }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}
 Badness: 0.039444
@@ Line 427: / 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 }}
-POTE generator: ~11/9 = 351.025
+Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.025
-Optimal GPV sequence: {{Val list| 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.
+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.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}
-[[POTE generator]]: ~45/32 = 567.720
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~45/32 = 567.720
-{{Val list|legend=1| 17, 19, 55c, 74cd, 93cdd }}
+{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
 [[Badness]]: 0.073735
-== 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 }}
-[[POTE generator]]: ~7/5 = 549.944
-{{Val list|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 }}]
-POTE generator: ~7/5 = 549.945
-Optimal GPV sequence: {{Val list| 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 }}]
-POTE generator: ~7/5 = 549.958
-Optimal GPV sequence: {{Val list| 11cdeef, 13cdeef, 24 }}
-Badness: 0.049933
 == Superthird ==
-{{see also| Shibboleth family }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}
-[[POTE generator]]: ~9/7 = 439.076
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~9/7 = 439.076
-{{Val list|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
+{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
 [[Badness]]: 0.139379
@@ Line 519: / 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 }}
-POTE generator: ~9/7 = 439.152
+Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152
-Optimal GPV sequence: {{Val list| 11cd, 30d, 41, 153be, 194be, 235bcee }}
+{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}
 Badness: 0.070917
@@ Line 532: / 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 }}
-POTE generator: ~9/7 = 439.119
+Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119
-Optimal GPV sequence: {{Val list| 11cdf, 30df, 41 }}
+{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}
 Badness: 0.052835
 == Superenneadecal ==
+Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.
-Superenneadeal is another cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 245/243, 395136/390625
-Subgroup: 2.3.5.7
+{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
-[[Comma list]]: 245/243, 395136/390625
+[[Optimal tuning]] ([[POTE]]): ~392/375 = 63.158, ~3/2 = 704.166
-Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]
+{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
-{{Val list|legend=1| 19, 76bcd, 95, 114 }}
+[[Badness]]: 0.132311
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-[[Comma list]]: 245/243, 2560/2541, 3773/3750
+Comma list: 245/243, 2560/2541, 3773/3750
+Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
+Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667
-Mapping: [⟨19 0 14 -7 96], ⟨0 1 1 2 -1]]
+{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}
-{{Val list|legend=1| 19, 76bcd, 95, 114e }}
+Badness: 0.101496
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-[[Comma list]]: 196/195, 245/243, 832/825, 1001/1000
+Comma list: 196/195, 245/243, 832/825, 1001/1000
-Mapping: [⟨19 0 14 -7 96 10], ⟨0 1 1 2 -1 2]]
+Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
-{{Val list|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). 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.
-Subgroup: 2.3.5.7
+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
 [[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 = 1200.000, ~5/4 = 391.465
-[[POTE generator]]: ~5/4 = 391.465
+{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
-{{Val list|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
 [[Badness]]: 0.108417
@@ Line 594: / 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 }}
-POTE generator: ~5/4 = 391.503
+Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.503
-Optimal GPV sequence: {{Val list| 46, 95, 141bc }}
+{{Optimal ET sequence|legend=0| 46, 95, 141bc }}
 Badness: 0.045108
@@ Line 607: / 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 }}
-POTE generator: ~5/4 = 391.366
+Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.366
-Optimal GPV sequence: {{Val list| 46, 233bcff, 279bccff }}
+{{Optimal ET sequence|legend=0| 46, 233bcff, 279bccff }}
 Badness: 0.043024
 == Leapweek ==
-Subgroup: 2.3.5.7
+: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243, 2097152/2066715
-[[Mapping]]: [{{val| 1 1 17 -6 }}, {{val| 0 1 -25 15 }}]
+{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}
-[[POTE generator]]: ~3/2 = 704.536
+: mapping generators: ~2, ~3
-{{Val list|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 704.536
+{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
 [[Badness]]: 0.140577
@@ Line 633: / Line 342: @@
 Comma list: 245/243, 385/384, 1331/1323
-Mapping: [{{val| 1 1 17 -6 -3 }}, {{val| 0 1 -25 15 11 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}
-POTE generator: ~3/2 = 704.554
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.554
-Optimal GPV sequence: {{Val list| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
 Badness: 0.050679
@@ Line 646: / Line 355: @@
 Comma list: 169/168, 245/243, 352/351, 364/363
-Mapping: [{{val| 1 1 17 -6 -3 -1 }}, {{val| 0 1 -25 15 11 8 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}
-POTE generator: ~3/2 = 704.571
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.571
-Optimal GPV sequence: {{Val list| 17, 29c, 46, 63, 109 }}
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 63, 109 }}
 Badness: 0.032727
@@ Line 659: / Line 368: @@
 Comma list: 154/153, 169/168, 245/243, 256/255, 273/272
-Mapping: [{{val| 1 1 17 -6 -3 -1 -10 }}, {{val| 0 1 -25 15 11 8 24 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}
-POTE generator: ~3/2 = 704.540
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540
-Optimal GPV sequence: {{Val list| 17g, 29cg, 46, 109, 155f, 264bfg }}
+{{Optimal ET sequence|legend=0| 17g, 29cg, 46, 109, 155f, 264bfg }}
 Badness: 0.026243
@@ Line 672: / Line 381: @@
 Comma list: 136/135, 169/168, 221/220, 245/243, 364/363
-Mapping: [{{val| 1 1 17 -6 -3 -1 17 }}, {{val| 0 1 -25 15 11 8 -22 }}]
+Mapping: {{mapping| 1 0 42 -21 -14 -9 39 | 0 1 -25 15 11 8 -22 }}
-POTE generator: ~3/2 = 704.537
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537
-Optimal GPV sequence: {{Val list| 17, 29c, 46, 109g, 155fg, 264bfgg }}
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 109g, 155fg, 264bfgg }}
 Badness: 0.026774
-== Semiwolf ==
-[[Subgroup]]: 3/2.7/4.5/2
-[[Comma list]]: 245/243
-[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
-[[POL2]] generator: ~7/6 = 262.1728
-[[Optimal GPV sequence]]: [[3edf]], [[5edf]], [[8edf]]
-=== Semilupine ===
-[[Subgroup]]: 3/2.7/4.5/2.11/4
-[[Comma list]]: 100/99, 245/243
-[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]
-[[POL2]] generator: ~7/6 = 264.3771
-[[Optimal GPV sequence]]: [[8edf]], [[13edf]]
-=== Hemilycan ===
-[[Subgroup]]: 3/2.7/4.5/2.11/4
-[[Comma list]]: 245/243, 441/440
-[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]
-[[POL2]] generator: ~7/6 = 261.5939
-[[Optimal GPV sequence]]: [[8edf]], [[11edf]]
 [[Category:Temperament clans]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Sensamagic clan| ]] <!-- main article -->
 [[Category:Rank 2]]
 [[Category:Listen]]

Sensamagic clan: Difference between revisions