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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
[[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]], [[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. 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
[[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, 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
Subgroup: 2.3.5.7.11


Comma list: 100/99, 245/243, 1344/1331
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
Subgroup: 2.3.5.7.11.13


Comma list: 100/99, 144/143, 196/195, 275/273
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-1.mp3]
* [http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3 bophier-12equal-six-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, 65, 87, 196, 283 }}
 
[[Badness]]: 0.0887
 
== 11-limit ==


Subgroup: 2.3.5.7.11
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, 65, 87, 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
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, 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
[[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
Subgroup: 2.3.5.7.11


Comma list: 243/242, 245/242, 385/384
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
Subgroup: 2.3.5.7.11.13


Comma list: 105/104, 144/143, 243/242, 245/242
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
[[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, 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
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
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| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
[[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| 11cd, 30d, 41, 153be, 194be, 235bcee }}
=== 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| 11cdf, 30df, 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| 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, 328bbcc }}
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, 233bcff, 279bccff }}
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, 29c, 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, 29c, 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, 29c, 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:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]

Latest revision as of 00:33, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The sensamagic clan tempers out the sensamagic comma, 245/243, a triprime comma with no factors of 2, ⟨0 -5 1 2] to be exact. 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.

BPS

Main article: 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 lambda scale.

Subgroup: 3.5.7

Comma list: 245/243

Sval mapping: [⟨1 1 2], ⟨0 -2 1]]

sval mapping generators: ~3, ~9/7

Optimal tuning (POTE): ~3 = 1901.9550, ~9/7 = 440.4881

Optimal ET sequence: b4, b9, b13, b56, b69, b82, 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 sensi, cohemiripple, hedgehog, and fourfives.

These temperaments are distributed into different family pages.

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.

Discussed elsewhere are

For no-twos extensions, see No-twos subgroup temperaments #BPS.

Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.

Bohpier

Main article: Bohpier
For the 5-limit version, see Miscellaneous 5-limit temperaments #Bohpier.

Bohpier is named after its interesting relationship with the non-octave Bohlen–Pierce equal temperament.

Subgroup: 2.3.5.7

Comma list: 245/243, 3125/3087

Mapping: [⟨1 0 0 0], ⟨0 13 19 23]]

Optimal tuning (POTE): ~2 = 1200.000, ~27/25 = 146.474

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5
unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 41, 131, 172, 213c

Badness: 0.068237

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 1344/1331

Mapping: [⟨1 0 0 0 2], ⟨0 13 19 23 12]]

Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.545

Minimax tuning:

  • 11-odd-limit: ~12/11 = [1/7 1/7 0 0 -1/14⟩
unchanged-interval (eigenmonzo) basis: 2.11/9

Optimal ET sequence: 41, 90e, 131e

Badness: 0.033949

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 275/273

Mapping: [⟨1 0 0 0 2 2], ⟨0 13 19 23 12 14]]

Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.603

Minimax tuning:

  • 13- and 15-odd-limit: ~12/11 = [0 0 1/19⟩
Unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 41, 90ef, 131ef, 221bdeff

Badness: 0.024864

Triboh

Triboh is named after the "Triple Bohlen–Pierce scale", which divides each step of the equal-tempered Bohlen–Pierce scale into three equal parts.

Subgroup: 2.3.5.7.11

Comma list: 245/243, 1331/1323, 3125/3087

Mapping: [⟨1 0 0 0 0], ⟨0 39 57 69 85]]

Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828

Optimal ET sequence: 49, 123ce, 172

Badness: 0.162592

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 245/243, 275/273, 847/845, 1331/1323

Mapping: [⟨1 0 0 0 0 0], ⟨0 39 57 69 85 91]]

Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822

Optimal ET sequence: 49f, 123ce, 172f, 295ce, 467bccef

Badness: 0.082158

Salsa

See also: Schismatic family

Subgroup: 2.3.5.7

Comma list: 245/243, 32805/32768

Mapping: [⟨1 1 7 -1], ⟨0 2 -16 13]]

Optimal tuning (POTE): ~2 = 1200.000, ~128/105 = 351.049

Optimal ET sequence: 17, 24, 41, 106d, 147d, 188cd, 335cd

Badness: 0.080152

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 245/242, 385/384

Mapping: [⟨1 1 7 -1 2], ⟨0 2 -16 13 5]]

Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.014

Optimal ET sequence: 17, 24, 41, 106d, 147d

Badness: 0.039444

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 245/242

Mapping: [⟨1 1 7 -1 2 4], ⟨0 2 -16 13 5 -1]]

Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.025

Optimal ET sequence: 17, 24, 41, 106df, 147df

Badness: 0.030793

Pycnic

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.

Subgroup: 2.3.5.7

Comma list: 245/243, 525/512

Mapping: [⟨1 3 -1 8], ⟨0 -3 7 -11]]

Optimal tuning (POTE): ~2 = 1200.000, ~45/32 = 567.720

Optimal ET sequence: 17, 19, 55c, 74cd, 93cdd

Badness: 0.073735

Superthird

For the 5-limit version, see Miscellaneous 5-limit temperaments #Shibboleth.

Subgroup: 2.3.5.7

Comma list: 245/243, 78125/76832

Mapping: [⟨1 -5 -5 -10], ⟨0 18 20 35]]

Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.076

Optimal ET sequence: 11cd, 30d, 41, 317bcc, 358bcc, 399bcc

Badness: 0.139379

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 78125/76832

Mapping: [⟨1 -5 -5 -10 2], ⟨0 18 20 35 4]]

Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152

Optimal ET sequence: 11cd, 30d, 41, 153be, 194be, 235bcee

Badness: 0.070917

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 1375/1352

Mapping: [⟨1 -5 -5 -10 2 -8], ⟨0 18 20 35 4 32]]

Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119

Optimal ET sequence: 11cdf, 30df, 41

Badness: 0.052835

Superenneadecal

Superenneadecal is a cousin of enneadecal but sharper fifth is used to temper 245/243.

Subgroup: 2.3.5.7

Comma list: 245/243, 395136/390625

Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]

Optimal tuning (POTE): ~392/375 = 63.158, ~3/2 = 704.166

Optimal ET sequence: 19, 76bcd, 95, 114, 133, 247b, 380bcd

Badness: 0.132311

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 2560/2541, 3773/3750

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: 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

Mapping: [⟨19 0 14 -7 96 10], ⟨0 1 1 2 -1 2]]

Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801

Optimal ET sequence: 19, 76bcdf, 95, 114e

Badness: 0.053197

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 & 49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). The alternative extension amigo (43 & 46) tempers out the same 5-limit comma as the magus, but with the 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)3, that is, 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)2 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: [⟨1 -2 2 -6], ⟨0 11 1 27]]

Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.465

Optimal ET sequence: 46, 95, 141bc, 187bc, 328bbcc

Badness: 0.108417

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243, 1331/1323

Mapping: [⟨1 -2 2 -6 -6], ⟨0 11 1 27 29]]

Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.503

Optimal ET sequence: 46, 95, 141bc

Badness: 0.045108

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 176/175, 245/243, 1331/1323

Mapping: [⟨1 -2 2 -6 -6 5], ⟨0 11 1 27 29 -4]]

Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.366

Optimal ET sequence: 46, 233bcff, 279bccff

Badness: 0.043024

Leapweek

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: [⟨1 0 42 -21], ⟨0 1 -25 15]]

mapping generators: ~2, ~3

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.536

Optimal ET sequence: 17, 29c, 46, 109, 155, 264b, 419b

Badness: 0.140577

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 1331/1323

Mapping: [⟨1 0 42 -21 -14], ⟨0 1 -25 15 11]]

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.554

Optimal ET sequence: 17, 29c, 46, 109, 264b, 373b, 637bbe

Badness: 0.050679

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/243, 352/351, 364/363

Mapping: [⟨1 0 42 -21 -14 -9], ⟨0 1 -25 15 11 8]]

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.571

Optimal ET sequence: 17, 29c, 46, 63, 109

Badness: 0.032727

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 169/168, 245/243, 256/255, 273/272

Mapping: [⟨1 0 42 -21 -14 -9 -34], ⟨0 1 -25 15 11 8 24]]

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540

Optimal ET sequence: 17g, 29cg, 46, 109, 155f, 264bfg

Badness: 0.026243

Leapweeker

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 169/168, 221/220, 245/243, 364/363

Mapping: [⟨1 0 42 -21 -14 -9 39], ⟨0 1 -25 15 11 8 -22]]

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537

Optimal ET sequence: 17, 29c, 46, 109g, 155fg, 264bfgg

Badness: 0.026774

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