@@ 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''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the sensamagic comma, [[245/243]], a triprime [[comma]] with no factors of 2, {{val| 0 -5 1 2 }} to be exact.
-= Lambda =
+Tempering out 245/243 alone in the full 7-limit leads to a [[rank-3 temperament]], [[sensamagic]], for which [[283edo]] is the [[optimal patent val]].
-Subgroup: 3.5.7
+== BPS ==
+{{Main| BPS }}
-[[Comma list]]: 245/243
+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]].
-[[Sval]] [[mapping]]: [{{val| 1 1 2 }}, {{val| 0 -2 1 }}]
+[[Subgroup]]: 3.5.7
-Sval mapping generators: ~3, ~9/7
+[[Comma list]]: 245/243
-[[Gencom]] [[mapping]]: [{{val| 0 1 1 2 }}, {{val| 0 0 -2 1 }}]
+{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}
+: mapping generators: ~3, ~9/7
-[[POTE generator]]: ~9/7 = 440.4881
+[[Optimal tuning]]s:
+* [[WE]]: ~3 = 1903.7398{{c}}, ~9/7 = 440.9014{{c}}
+: [[error map]]: {{val| +1.785 -0.771 -2.248 }}
+* [[CWE]]: ~3 = 1901.9550{{c}}, ~9/7 = 440.6646{{c}}
+: error map: {{val| 0.000 -3.030 -5.580 }}
-[[Val]]s: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
+[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]], [[367edt|b367cdd]], [[462edt|b462cdd]]
-== Extensions ==
+[[Badness]] (Sintel): 0.0659
-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]], [[Augmented family #Hemiaug|hemiaug]], [[Magic family #magic|magic]], [[Gamelismic clan#Rodan|rodan]], [[Tetracot family #Octacot|octacot]], [[Diaschismic family #Shrutar|shrutar]], and [[Kleismic family #Clyde|clyde]] 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. Salsa tempers out [[32805/32768]], splitting the generator in fifteen. 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]]
+* ''[[Salsa]]'' (+32805/32768) → [[Schismatic family #Salsa|Schismatic 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, pycnic, superenneadecal, 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 tempers out 3125/3087 and may be described as the {{nowrap| 41 & 49 }} temperament. It is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].
-== 7-limit ==
+[[41edo]] itself makes for an excellent tuning, though [[90edo]] and [[131edo]] are interesting alternatives. Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 146.4741{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[WE]] and/or [[CWE]] (shown below) is largely unnoticeable.
-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 }}
+: mapping generators: ~2, ~27/25
-{{Multival|legend=1| 13 19 23 0 0 0 }}
-[[POTE generator]]: ~27/25 = 146.474
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9967{{c}}, ~27/25 = 146.4737{{c}}
+: [[error map]]: {{val| -0.003 +2.203 -3.314 +0.068 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 146.4739{{c}}
+: error map: {{val| 0.000 +2.205 -3.310 +0.073 }}
-{{Val list|legend=1| 41, 131 }}
+[[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
-[[Badness]]: 0.0682
+{{Optimal ET sequence|legend=1| 8d, …, 41, 131, 172, 213c }}
-== 11-limit ==
+[[Badness]] (Sintel): 1.73
+=== 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 tunings:
+* WE: ~2 = 1199.2309{{c}}, ~12/11 = 146.4507{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.5009{{c}}
-{{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| 8d, …, 41, 90e, 131e }}
-== 13-limit ==
+Badness (Sintel): 1.12
+==== 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 }}]
+Optimal tunings:
+* WE: ~2 = 1198.5478{{c}}, ~12/11 = 146.4252{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.5230{{c}}
-{{Val list|legend=1| 41, 90ef, 131ef, 221bdeff }}
+Minimax tuning:
+* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
+: unchanged-interval (eigenmonzo) basis: 2.5
-Badness: 0.0249
+{{Optimal ET sequence|legend=0| 8d, …, 41, 90ef }}
-== Music ==
+Badness (Sintel): 1.03
-by [[Chris Vaisvil]]:
+=== Triboh ===
-* [http://micro.soonlabel.com/bophier/bophier-1.mp3 bophier&#45;1.mp3]
+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.
-* [http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3 bophier&#45;12equal&#45;six&#45;octaves.mp3]
-= Sensa aka escaped =
+Subgroup: 2.3.5.7.11
-{{see also| Escapade family #Escaped }}
-Subgroup: 2.3.5.7
+Comma list: 245/243, 1331/1323, 3125/3087
-[[Comma list]]: 245/243, 65625/65536
+Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}
+: mapping generators: ~2, ~77/75
-[[Mapping]]: [{{val| 1 2 2 4 }}, {{val| 0 -9 7 -26 }}]
+Optimal tunings:
+* WE: ~2 = 1199.9966{{c}}, ~77/75 = 48.8281{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8282{{c}}
-{{Multival|legend=1| 9 -7 26 -32 16 80 }}
+{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}
-[[POTE generator]]: ~28/27 = 55.122
+Badness (Sintel): 5.38
-{{Val list|legend=1| 22, 65, 87, 196, 283 }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-[[Badness]]: 0.0887
+Comma list: 245/243, 275/273, 847/845, 1331/1323
-== 11-limit ==
+Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}
-Subgroup: 2.3.5.7.11
+Optimal tunings:
+* WE: ~2 = 1199.9962{{c}}, ~77/75 = 48.8219{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8219{{c}}
-Comma list: 245/243, 385/384, 4000/3993
+{{Optimal ET sequence|legend=0| 49f, 123ce, 172f }}
-Mapping: [{{val| 1 2 2 4 3 }}, {{val| 0 -9 7 -26 10 }}]
+Badness (Sintel): 3.39
-POTE generator: ~28/27 = 55.126
+== Pycnic ==
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
-{{Val list|legend=1| 22, 65, 87, 196, 283 }}
+Pycnic is related to [[triton]], but its mapping differs for the [[7/1|7th harmonic]]. It is also related to [[liese]], from which its mapping differs for the [[5/1|5th harmonic]].
-Badness: 0.0358
+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 two cents sharp of it in the CWE 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.
-== 13-limit ==
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13
+[[Comma list]]: 245/243, 525/512
-Comma list: 245/243, 352/351, 385/384, 625/624
+{{Mapping|legend=1| 1 0 6 -3 | 0 3 -7 11 }}
+: mapping generators: ~2, ~64/45
-Mapping: [{{val| 1 2 2 4 3 2 }}, {{val| 0 -9 7 -26 10 37 }}]
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1203.3437{{c}}, ~64/45 = 634.0416{{c}}
+: [[error map]]: {{val| +3.344 +0.170 -4.542 -4.400 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 632.3502{{c}}
+: error map: {{val| 0.000 -4.904 -12.765 -12.973 }}
-POTE generator: ~28/27 = 55.138
+{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
-{{Val list|legend=1| 22, 65, 87, 283 }}
+[[Badness]] (Sintel): 1.87
-Badness: 0.0317
+== Xenia ==
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''
-= Salsa =
+Xenia is related to [[Starling temperaments #Xenial|xenial]], but its mapping differs for the [[7/1|7th harmonic]]. It may be described as {{nowrap| 19 & 51c }} or {{nowrap| 19 & 70d }}, which tempers out the sensamagic and keega, [[1029/1000]].
-{{see also| Schismatic family }}
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
-[[Comma list]]: 245/243, 32805/32768
+[[Comma list]]: 245/243, 1029/1000
-[[Mapping]]: [{{val| 1 1 7 -1 }}, {{val| 0 2 -16 13 }}]
+{{Mapping|legend=1| 1 -6 -12 -9 | 0 9 17 14 }}
+: mapping generators: ~2, ~9/5
-{{Multival|legend=1| 2 -16 13 -30 15 75 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.0862{{c}}, ~9/5 = 1012.0503{{c}}
+: [[error map]]: {{val| +1.086 -0.020 +5.507 -9.898 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2199{{c}}
+: error map: {{val| 0.000 -0.976 +4.424 -11.748 }}
-[[POTE generator]]: ~128/105 = 351.049
+{{Optimal ET sequence|legend=1| 19, 70d, 89d }}
-{{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
+[[Badness]] (Sintel): 2.25
-[[Badness]]: 0.08015
+== Magus ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
-== 11-limit ==
+Magus temperament tempers out [[50331648/48828125]] in the 5-limit. This temperament can be described as {{nowrap| 46 & 49 }} temperament, which tempers out the sensamagic and [[28672/28125]]. 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: 2.3.5.7.11
+Magus has a generator of a sharp ~5/4, and ~[[25/16]] is twice as sharp so that it makes sense to equate with [[11/7]] by tempering out [[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]].
-Comma list: 243/242, 245/242, 385/384
+[[Subgroup]]: 2.3.5.7
-Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]
+[[Comma list]]: 245/243, 28672/28125
-POTE generator: ~11/9 = 351.014
+{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
-{{Val list|legend=1| 17, 24, 41, 106d, 147d }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.7187{{c}}, ~5/4 = 391.0473{{c}}
+: [[error map]]: {{val| -1.281 +2.128 +2.171 -2.860 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.4129{{c}}
+: error map: {{val| 0.000 +3.587 +5.099 -0.678 }}
-Badness: 0.0394
+{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc }}
-== 13-limit ==
+[[Badness]] (Sintel): 2.74
-Subgroup: 2.3.5.7.11.13
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-Comma list: 105/104, 144/143, 243/242, 245/242
+Comma list: 176/175, 245/243, 1331/1323
-Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]
+Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}
-POTE generator: ~11/9 = 351.025
+Optimal tunings:
+* WE: ~2 = 1198.7144{{c}}, ~5/4 = 391.0836{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.4506{{c}}
-{{Val list|legend=1| 17, 24, 41, 106df, 147df }}
+{{Optimal ET sequence|legend=0| 46, 95, 141bc }}
-Badness: 0.0310
+Badness (Sintel): 1.49
-= Pycnic =
+=== 13-limit ===
-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: 2.3.5.7.11.13
-Subgroup: 2.3.5.7
+Comma list: 91/90, 176/175, 245/243, 1331/1323
-[[Comma list]]: 245/243, 525/512
-[[Mapping]]: [{{val| 1 3 -1 8 }}, {{val| 0 -3 7 -11 }}]
+Mapping: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}
-{{Multival|legend=1| 3 -7 11 -18 9 45 }}
+Optimal tunings:
+* WE: ~2 = 1199.7708{{c}}, ~5/4 = 391.2912{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.3597{{c}}
-[[POTE generator]]: ~45/32 = 567.720
+{{Optimal ET sequence|legend=0| 3de, 43de, 46 }}
-{{Val list|legend=1| 17, 19, 55c, 74cd, 93cdd }}
+Badness (Sintel): 1.78
-[[Badness]]: 0.0737
+== Superenneadecal ==
+Superenneadecal is a cousin of [[enneadecal]] but a sharper fifth is used to temper out 245/243.
-= Cohemiripple =
+[[Subgroup]]: 2.3.5.7
-{{see also| Ripple family }}
-Subgroup: 2.3.5.7
+[[Comma list]]: 245/243, 395136/390625
-[[Comma list]]: 245/243, 1323/1250
+{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
+: mapping generators: ~392/375, ~3
-[[Mapping]]: [{{val| 1 7 11 12 }}, {{val| 0 -10 -16 -17 }}]
+[[Optimal tuning]]s:
+* [[WE]]: ~392/375 = 63.1399{{c}}, ~3/2 = 703.9652{{c}}
+: [[error map]]: {{val| -0.343 +1.668 +1.267 -3.560 }}
+* [[CWE]]: ~392/375 = 63.1579{{c}}, ~3/2 = 703.9028{{c}}
+: error map: {{val| 0.000 +1.948 +1.800 -3.126 }}
-{{Multival|legend=1| 10 16 17 2 -1 -5 }}
+{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b }}
-[[POTE generator]]: ~7/5 = 549.944
+[[Badness]] (Sintel): 3.35
-{{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: 245/243, 2560/2541, 3773/3750
-Mapping: [{{val| 1 7 11 12 17 }}, {{val| 0 -10 -16 -17 -25 }}]
+Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
-POTE generator: ~7/5 = 549.945
+Optimal tunings:
+* WE: ~33/32 = 63.0966{{c}}, ~3/2 = 704.9824{{c}}
+* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.3096{{c}}
-{{Val list|legend=1| 11cdee, 13cdee, 24 }}
+{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}
-Badness: 0.0827
+Badness (Sintel): 3.36
-== 13-limit ==
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 66/65, 77/75, 147/143, 243/242
+Comma list: 196/195, 245/243, 832/825, 1001/1000
-Mapping: [{{val| 1 7 11 12 17 14 }}, {{val| 0 -10 -16 -17 -25 -19 }}]
+Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
-POTE generator: ~7/5 = 549.958
+Optimal tunings:
+* WE: ~33/32 = 63.0988{{c}}, ~3/2 = 705.1402{{c}}
+* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.4315{{c}}
-{{Val list|legend=1| 11cdeef, 13cdeef, 24 }}
+{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e, 209bcef }}
-Badness: 0.0499
+Badness (Sintel): 2.20
-= Superthird =
+== Superthird ==
+: ''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 13 15 25 }}, {{val| 0 -18 -20 -35 }}]
+{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
+: mapping generators: ~2, ~9/7
-{{Multival|legend=1| 18 20 35 -10 5 25 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3935{{c}}, ~9/7 = 439.2199{{c}}
+: [[error map]]: {{val| +0.394 +2.035 -3.884 -0.066 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 439.0931{{c}}
+: error map: {{val| 0.000 +1.721 -4.452 -0.568 }}
-[[POTE generator]]: ~9/7 = 439.076
+{{Optimal ET sequence|legend=1| 11cd, 30d, 41 }}
-{{Val list|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
+[[Badness]] (Sintel): 3.53
-[[Badness]]: 0.1394
-== 11-limit ==
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
 Comma list: 100/99, 245/243, 78125/76832
-Mapping: [{{val| 1 13 15 25 6 }}, {{val| 0 -18 -20 -35 -4 }}]
+Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}
-POTE generator: ~9/7 = 439.152
-{{Val list|legend=1| 11cd, 30d, 41, 153be, 194be, 235bcee }}
+Optimal tunings:
+* WE: ~2 = 1199.5116{c}}, ~9/7 = 438.9734{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.1362{{c}}
-Badness: 0.0709
+{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be }}
-== 13-limit ==
+Badness (Sintel): 2.34
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
 Comma list: 100/99, 144/143, 196/195, 1375/1352
-Mapping: [{{val| 1 13 15 25 6 24 }}, {{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 tunings:
+* WE: ~2 = 1199.2631{c}}, ~9/7 = 438.8494{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.0943{{c}}
-{{Val list|legend=1| 11cdf, 30df, 41 }}
+{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}
-Badness: 0.0528
+Badness (Sintel): 2.18
-= Magus =
+== Leapweek ==
-== 5-limit ==
+: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
-Subgroup: 2.3.5
+Leapweek may be described as the {{nowrap| 46 & 63 }} temperament, generated by a perfect fifth and being a strong extension of [[leapfrog]]. [[109edo]] makes for an excellent tuning.
-[[Comma list]]: 50331648/48828125
+[[Subgroup]]: 2.3.5.7
-[[Mapping]]: [{{val| 1 9 3 }}, {{val| 0 -11 -1 }}]
+[[Comma list]]: 245/243, 2097152/2066715
-[[POTE generator]]: ~5/4 = 391.225
+{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}
+: mapping generators: ~2, ~3
-{{Val list|legend=1| 46, 181c, 227c, 273c, 319c }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.6301{{c}}, ~3/2 = 704.3191{{c}}
+: [[error map]]: {{val| -0.370 +1.994 -0.578 -1.821 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5387{{c}}
+: error map: {{val| 0.000 +2.584 +0.218 -0.745 }}
-[[Badness]]: 0.3602
+{{Optimal ET sequence|legend=1| 17, 46, 109, 155, 264b }}
-== 7-limit ==
+[[Badness]] (Sintel): 3.56
-Subgroup: 2.3.5.7
-[[Comma list]]: 245/243, 28672/28125
-[[Mapping]]: [{{val| 1 9 3 21 }}, {{val| 0 -11 -1 -27 }}]
-{{Multival|legend=1| 11 1 27 -24 12 60 }}
-[[POTE generator]]: ~5/4 = 391.465
-{{Val list|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
-[[Badness]]: 0.1084
-== 11-limit ==
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 176/175, 245/243, 1331/1323
+Comma list: 245/243, 385/384, 1331/1323
-Mapping: [{{val| 1 9 3 21 23 }}, {{val| 0 -11 -1 -27 -29 }}]
-POTE generator: ~5/4 = 391.503
+Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}
-{{Val list|legend=1| 46, 95, 141bc }}
+Optimal tunings:
+* WE: ~2 = 1199.7910{{c}}, ~3/2 = 704.4312{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5542{{c}}
-Badness: 0.0451
+{{Optimal ET sequence|legend=0| 17, 46, 109, 264b }}
-== 13-limit ==
+Badness (Sintel): 1.68
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 176/175, 245/243, 1331/1323
+Comma list: 169/168, 245/243, 352/351, 364/363
-Mapping: [{{val| 1 9 3 21 23 1 }}, {{val| 0 -11 -1 -27 -29 4 }}]
-POTE generator: ~5/4 = 391.366
-{{Val list|legend=1| 46, 233bcff, 279bccff }}
+Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}
-Badness: 0.0430
+Optimal tunings:
+* WE: ~2 = 1200.0070{{c}}, ~3/2 = 704.5751{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5709{{c}}
-= Leapweek =
+{{Optimal ET sequence|legend=0| 17, 46, 63, 109 }}
-Subgroup: 2.3.5.7
+Badness (Sintel): 1.35
-[[Comma list]]: 245/243, 2097152/2066715
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-[[Mapping]]: [{{val| 1 1 17 -6 }}, {{val| 0 1 -25 15 }}]
+Comma list: 154/153, 169/168, 245/243, 256/255, 273/272
-[[POTE generator]]: ~3/2 = 704.536
+Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}
-{{Val list|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
+Optimal tunings:
+* WE: ~2 = 1199.8670{{c}}, ~3/2 = 704.4620{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5395{{c}}
-[[Badness]]: 0.14058
+{{Optimal ET sequence|legend=0| 17g, 46, 109 }}
-== 11-limit ==
+Badness (Sintel): 1.34
-Subgroup: 2.3.5.7.11
+==== Leapweeker ====
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 245/243, 385/384, 1331/1323
+Comma list: 136/135, 169/168, 221/220, 245/243, 364/363
-Mapping: [{{val| 1 1 17 -6 -3 }}, {{val| 0 1 -25 15 11 }}]
-POTE generator: ~3/2 = 704.554
-{{Val list|legend=1| 17, 29c, 46, 109, 264b, 373b, 637be }}
-Badness: 0.0507
-== 13-limit ==
-Subgroup: 2.3.5.7.11.13
-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 39 | 0 1 -25 15 11 8 -22 }}
-POTE generator: ~3/2 = 704.571
+Optimal tunings:
+* WE: ~2 = 1200.1737{{c}}, ~3/2 = 704.6390{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5364{{c}}
-{{Val list|legend=1| 17, 29c, 46, 63, 109, 218f, 373bf }}
+{{Optimal ET sequence|legend=0| 17, 46, 109g, 155fg }}
-Badness: 0.0327
+Badness (Sintel): 1.36
-[[Category:Regular temperament theory]]
+[[Category:Sensamagic clan| ]] <!-- main article -->
-[[Category:Temperament clan]]
+[[Category:Temperament clans]]
-[[Category:Sensamagic]]
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]
-[[Category:Listen]]

Sensamagic clan: Difference between revisions