@@ 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]].
-= Lambda =
+== BPS ==
+{{Main| BPS }}
-Subgroup: 3.5.7
+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
 [[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
-[[Gencom]] [[mapping]]: [{{val| 0 1 1 2 }}, {{val| 0 0 -2 1 }}]
-[[POTE generator]]: ~9/7 = 440.4881
+: sval mapping generators: ~3, ~9/7
-[[Val]]s: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
+[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881
-== Extensions ==
+[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
-For full 7-limit extensions, we have bohpier, sensa/escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as [[Father family #father|father]], [[Dicot family #Sidi|sidi]], [[Meantone family #Godzilla|godzilla]], [[Porcupine family #Hedgehog|hedgehog]], [[Archytas clan #Superpyth|superpyth]], [[Sensipent family #Sensi|sensi]], [[Magic family #magic|magic]], [[Gamelismic clan#Rodan|rodan]], [[Tetracot family #Octacot|octacot]], [[Kleismic family #Clyde|clyde]], [[Diaschismic family #Shrutar|shrutar]] discussed elsewhere.
-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]].
+=== 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.
-= Bohpier =
+These temperaments are distributed into different family pages.
-== 5-limit ==
+* [[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]]
-Subgroup: 2.3.5
+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.
-[[Comma list]]: 1220703125/1162261467
+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]]
-[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 13 19 }}]
+For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].
-[[POTE generator]]: ~27/25 = 146.476
+Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.
-{{Val list|legend=1| 8, 41, 131, 172, 213c }}
+== Bohpier ==
+{{Main| Bohpier }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Bohpier]].''
-[[Badness]]: 0.8605
+Bohpier is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].
-== 7-limit ==
+[[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 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 146.474
-[[POTE generator]]: ~27/25 = 146.474
+[[Minimax tuning]]:
+* [[7-odd-limit]]: ~27/25 = {{monzo| 0 0 1/19 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
+* [[9-odd-limit]]: ~27/25 = {{monzo| 0 1/13 }}
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
-{{Val list|legend=1| 41, 49, 90, 131 }}
+{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}
-[[Badness]]: 0.0682
+[[Badness]]: 0.068237
-== 11-limit ==
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
 Comma list: 100/99, 245/243, 1344/1331
-POTE generator: ~12/11 = 146.545
+Mapping: {{mapping| 1 0 0 0 2 | 0 13 19 23 12 }}
-Mapping: [{{val| 1 0 0 0 2 }}, {{val| 0 13 19 23 12 }}]
+Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.545
-{{Val list|legend=1| 41, 90e, 131e }}
+Minimax tuning:
+* 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
+: unchanged-interval (eigenmonzo) basis: 2.11/9
-Badness: 0.0339
+{{Optimal ET sequence|legend=0| 41, 90e, 131e }}
-== 13-limit ==
+Badness: 0.033949
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 100/99, 144/143, 196/195, 275/273
-POTE generator: ~12/11 = 146.603
+Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}
-Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]
-{{Val list|legend=1| 41, 90ef, 131ef, 221bdeff }}
-Badness: 0.0249
-== 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]
-= Sensa aka escaped =
+Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.603
-{{see also| Escapade family #Escaped }}
-Subgroup: 2.3.5.7
+Minimax tuning:
+* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
+: Unchanged-interval (eigenmonzo) basis: 2.5
-[[Comma list]]: 245/243, 65625/65536
+{{Optimal ET sequence|legend=0| 41, 90ef, 131ef, 221bdeff }}
-[[Mapping]]: [{{val| 1 2 2 4 }}, {{val| 0 -9 7 -26 }}]
+Badness: 0.024864
-{{Multival|legend=1| 9 -7 26 -32 16 80 }}
+=== Triboh ===
+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.
-[[POTE generator]]: ~28/27 = 55.122
-{{Val list|legend=1| 22, 43d, 65, 87, 109, 196, 283 }}
-[[Badness]]: 0.0887
-== 11-limit ==
 Subgroup: 2.3.5.7.11
-Comma list: 245/243, 385/384, 4000/3993
+Comma list: 245/243, 1331/1323, 3125/3087
-Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]
+Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}
-POTE generator: ~28/27 = 55.126
+Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828
-{{Val list|legend=1| 22, 43d, 65, 87, 109, 196, 283 }}
+{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}
-Badness: 0.0358
+Badness: 0.162592
-== 13-limit ==
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 245/243, 352/351, 385/384, 625/624
+Comma list: 245/243, 275/273, 847/845, 1331/1323
-Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]
+Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}
-POTE generator: ~28/27 = 55.138
+Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822
-{{Val list|legend=1| 22, 65, 87, 109, 196, 283 }}
+{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}
-Badness: 0.0317
+Badness: 0.082158
-= Salsa =
+== 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
-{{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~128/105 = 351.049
-[[Badness]]: 0.08015
+{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
-== 11-limit ==
+[[Badness]]: 0.080152
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
 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
-{{Val list|legend=1| 17, 24, 41, 106d, 147d }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}
-Badness: 0.0394
+Badness: 0.039444
-== 13-limit ==
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
 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
-{{Val list|legend=1| 17, 24, 41, 106df, 147df }}
+{{Optimal ET sequence|legend=0| 17, 24, 41, 106df, 147df }}
-Badness: 0.0310
+Badness: 0.030793
-= Pycnic =
+== Pycnic ==
-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.
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Stump]].''
-Subgroup: 2.3.5.7
+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
 [[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 = 1200.000, ~45/32 = 567.720
-[[POTE generator]]: ~45/32 = 567.720
+{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
-{{Val list|legend=1| 17, 19, 36c, 55c, 74cd, 93cdd }}
+[[Badness]]: 0.073735
-[[Badness]]: 0.0737
+== Superthird ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
-= Cohemiripple =
+[[Subgroup]]: 2.3.5.7
-{{see also| Ripple family }}
-Subgroup: 2.3.5.7
+[[Comma list]]: 245/243, 78125/76832
-[[Comma list]]: 245/243, 1323/1250
+{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
-[[Mapping]]: [{{val| 1 7 11 12 }}, {{val| 0 -10 -16 -17 }}]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~9/7 = 439.076
-{{Multival|legend=1| 10 16 17 2 -1 -5 }}
+{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
-[[POTE generator]]: ~7/5 = 549.944
+[[Badness]]: 0.139379
-{{Val list|legend=1| 11cd, 13cd, 24 }}
-[[Badness]]: 0.1902
-== 11-limit ==
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 77/75, 243/242, 245/242
+Comma list: 100/99, 245/243, 78125/76832
-Mapping: [{{val| 1 7 11 12 17 }}, {{val| 0 -10 -16 -17 -25 }}]
-POTE generator: ~7/5 = 549.945
+Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}
-{{Val list|legend=1| 11cdee, 13cdee, 24 }}
+Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152
-Badness: 0.0827
+{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}
-== 13-limit ==
+Badness: 0.070917
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 77/75, 147/143, 243/242
+Comma list: 100/99, 144/143, 196/195, 1375/1352
-Mapping: [{{val| 1 7 11 12 17 14 }}, {{val| 0 -10 -16 -17 -25 -19 }}]
+Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}
-POTE generator: ~7/5 = 549.958
+Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119
-{{Val list|legend=1| 11cdeef, 13cdeef, 24 }}
+{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}
-Badness: 0.0499
+Badness: 0.052835
-= Superthird =
+== Superenneadecal ==
+Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 245/243, 78125/76832
+[[Comma list]]: 245/243, 395136/390625
-[[Mapping]]: [{{val| 1 13 15 25 }}, {{val| 0 -18 -20 -35 }}]
+{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
-{{Multival|legend=1| 18 20 35 -10 5 25 }}
+[[Optimal tuning]] ([[POTE]]): ~392/375 = 63.158, ~3/2 = 704.166
-[[POTE generator]]: ~9/7 = 439.076
+{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
-{{Val list|legend=1| 41, 317bc, 358bc, 399bc }}
+[[Badness]]: 0.132311
-[[Badness]]: 0.1394
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 11-limit ==
+Comma list: 245/243, 2560/2541, 3773/3750
-Subgroup: 2.3.5.7.11
+Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
-Comma list: 100/99, 245/243, 78125/76832
+Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667
-Mapping: [{{val| 1 13 15 25 6 }}, {{val| 0 -18 -20 -35 -4 }}]
+{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}
-POTE generator: ~9/7 = 439.152
+Badness: 0.101496
-{{Val list|legend=1| 41, 153be, 194be, 235bce }}
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Badness: 0.0709
+Comma list: 196/195, 245/243, 832/825, 1001/1000
-== 13-limit ==
+Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
-Subgroup: 2.3.5.7.11.13
+Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801
-Comma list: 100/99, 144/143, 196/195, 1375/1352
+{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e }}
-Mapping: [{{val| 1 13 15 25 6 24 }}, {{val| 0 -18 -20 -35 -4 -32 }}]
+Badness: 0.053197
-POTE generator: ~9/7 = 439.119
+== Magus ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
-{{Val list|legend=1| 41 }}
+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.
-Badness: 0.0528
+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]].
-= Magus =
+[[Subgroup]]: 2.3.5.7
-== 5-limit ==
-Subgroup: 2.3.5
+[[Comma list]]: 245/243, 28672/28125
-[[Comma list]]: 50331648/48828125
+{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
-[[Mapping]]: [{{val| 1 9 3 }}, {{val| 0 -11 -1 }}]
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 391.465
-[[POTE generator]]: ~5/4 = 391.225
+{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
-{{Val list|legend=1| 40, 43, 46, 181c, 227c, 273c, 319c }}
+[[Badness]]: 0.108417
-[[Badness]]: 0.3602
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-== 7-limit ==
+Comma list: 176/175, 245/243, 1331/1323
-Subgroup: 2.3.5.7
+Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}
-[[Comma list]]: 245/243, 28672/28125
+Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.503
-[[Mapping]]: [{{val| 1 9 3 21 }}, {{val| 0 -11 -1 -27 }}]
+{{Optimal ET sequence|legend=0| 46, 95, 141bc }}
-{{Multival|legend=1| 11 1 27 -24 12 60 }}
+Badness: 0.045108
-[[POTE generator]]: ~5/4 = 391.465
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-{{Val list|legend=1| 46, 95, 141bc, 187bc, 328bc }}
+Comma list: 91/90, 176/175, 245/243, 1331/1323
-[[Badness]]: 0.1084
+Mapping: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}
-== 11-limit ==
+Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.366
-Subgroup: 2.3.5.7.11
+{{Optimal ET sequence|legend=0| 46, 233bcff, 279bccff }}
-Comma list: 176/175, 245/243, 1331/1323
+Badness: 0.043024
-Mapping: [{{val| 1 9 3 21 23 }}, {{val| 0 -11 -1 -27 -29 }}]
+== Leapweek ==
+: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
-POTE generator: ~5/4 = 391.503
+[[Subgroup]]: 2.3.5.7
-{{Val list|legend=1| 46, 95, 141bc }}
+[[Comma list]]: 245/243, 2097152/2066715
-Badness: 0.0451
+{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}
-== 13-limit ==
+: mapping generators: ~2, ~3
-Subgroup: 2.3.5.7.11.13
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 704.536
-Comma list: 91/90, 176/175, 245/243, 1331/1323
+{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
-Mapping: [{{val| 1 9 3 21 23 1 }}, {{val| 0 -11 -1 -27 -29 4 }}]
+[[Badness]]: 0.140577
-POTE generator: ~5/4 = 391.366
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-{{Val list|legend=1| 46, 233bcf, 279bcf }}
+Comma list: 245/243, 385/384, 1331/1323
-Badness: 0.0430
+Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}
-= Leapweek =
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.554
-Subgroup: 2.3.5.7
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
-[[Comma list]]: 245/243, 2097152/2066715
+Badness: 0.050679
-[[Mapping]]: [{{val| 1 1 17 -6 }}, {{val| 0 1 -25 15 }}]
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-[[POTE generator]]: ~3/2 = 704.536
+Comma list: 169/168, 245/243, 352/351, 364/363
-{{Val list|legend=1| 17, 46, 109, 155, 264b, 419b }}
+Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}
-[[Badness]]: 0.14058
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.571
-== 11-limit ==
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 63, 109 }}
-Subgroup: 2.3.5.7.11
+Badness: 0.032727
-Comma list: 245/243, 385/384, 1331/1323
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-Mapping: [{{val| 1 1 17 -6 -3 }}, {{val| 0 1 -25 15 11 }}]
+Comma list: 154/153, 169/168, 245/243, 256/255, 273/272
-POTE generator: ~3/2 = 704.554
+Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}
-{{Val list|legend=1| 17, 46, 109, 264b, 373b, 637be }}
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540
-Badness: 0.0507
+{{Optimal ET sequence|legend=0| 17g, 29cg, 46, 109, 155f, 264bfg }}
-== 13-limit ==
+Badness: 0.026243
-Subgroup: 2.3.5.7.11.13
+==== Leapweeker ====
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 245/243, 352/351, 364/363
+Comma list: 136/135, 169/168, 221/220, 245/243, 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 39 | 0 1 -25 15 11 8 -22 }}
-POTE generator: ~3/2 = 704.571
+Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537
-{{Val list|legend=1| 17, 46, 63, 109, 218f, 373bf }}
+{{Optimal ET sequence|legend=0| 17, 29c, 46, 109g, 155fg, 264bfgg }}
-Badness: 0.0327
+Badness: 0.026774
-[[Category:Regular temperament theory]]
+[[Category:Temperament clans]]
-[[Category:Temperament clan]]
+[[Category:Pages with mostly numerical content]]
-[[Category:Sensamagic]]
+[[Category:Sensamagic clan| ]] <!-- main article -->
 [[Category:Rank 2]]
 [[Category:Listen]]

Sensamagic clan: Difference between revisions