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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]].


== BPS (Lambda) ==
== BPS ==
The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243.
{{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
 
[[Val]]s: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
 
=== Extensions ===
For full 7-limit extensions, we have sensi, bohpier, escaped, salsa, pycnic, cohemiripple, superthird, magus and leapweek discussed below, as well as
* ''[[father]]'', {16/15, 28/27} → [[Father family #Father]]
* ''[[sidi]]'', {25/24, 245/243} → [[Dicot family #Sidi]]
* [[godzilla]], {49/48, 81/80} → [[Meantone family #Godzilla]]
* ''[[hedgehog]]'', {50/49, 245/243} → [[Porcupine family #Hedgehog]]
* [[superpyth]], {64/63, 245/243} → [[Archytas clan #Superpyth]]
* ''[[hemiaug]]'', {128/125, 245/243} → [[Augmented family #Hemiaug]]
* [[magic]], {225/224, 245/243} → [[Magic family #Magic]]
* [[rodan]], {245/243, 1029/1024} → [[Gamelismic clan #Rodan]]
* ''[[shrutar]]'', {245/243, 2048/2025} → [[Diaschismic family #Shrutar]]
* ''[[octacot]]'', {245/243, 2401/2400} → [[Tetracot family #Octacot]]
* ''[[clyde]]'', {245/243, 3136/3125} → [[Kleismic family #Clyde]]
* ''[[bamity]]'', {245/243, 64827/64000} → [[Amity family #Bamity]]
* ''[[fourfives]]'', {245/243, 235298/234375} → [[Fifive family #Fourfives]]
 
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]].
 
== 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&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|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
 
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 1/13 0 0 7/13 }}, {{monzo| 5/13 0 0 9/13 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7
* [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 2/5 14/5 -7/5 0 }}, {{monzo| 4/5 18/5 -9/5 0 }}, {{monzo| 3/5 26/5 -13/5 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 9/5
 
[[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.
 
{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
 
[[Badness]]: 0.025622
 
==== Sensation ====
Subgroup: 2.3.5.7.13
 
Comma list: 91/90, 126/125, 169/168
 
Sval mapping: [{{val| 1 -1 -1 -2 0 }}, {{val| 0 7 9 13 10 }}]
 
Gencom mapping: [{{val| 1 -1 -1 -2 0 0 }}, {{val| 0 7 9 13 0 10 }}]
 
Gencom: [2 9/7; 91/90 126/125 169/168]
 
POTE generator: ~9/7 = 443.322
 
Vals: {{Val list| 19, 27, 46, 111de, 157de }}
 
=== Sensor ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 245/243, 385/384
 
Mapping: [{{val| 1 -1 -1 -2 9 }}, {{val| 0 7 9 13 -15 }}]
 
POTE generator: ~9/7 = 443.294
 
Vals: {{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
 
Vals: {{Val list| 19, 27, 46, 111df, 157df }}
 
Badness: 0.025575
 
=== 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
 
Vals: {{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
 
Vals: {{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
 
Vals: {{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
 
Vals: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
 
Badness: 0.020789
 
=== 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
 
Vals: {{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 }}]
: sval mapping generators: ~3, ~9/7


POTE generator: ~9/7 = 443.506
[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881


Vals: {{Val list| 19e, 27, 46ee }}
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]


Badness: 0.023258
=== 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.  


=== Hemisensi ===
These temperaments are distributed into different family pages.
Subgroup: 2.3.5.7.11
* [[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]]


Comma list: 126/125, 243/242, 245/242
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.


Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]
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]]


POTE generator: ~25/22 = 221.605
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].


Vals: {{Val list| 27e, 65, 157de, 222cde }}
Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.
 
Badness: 0.048714


== Bohpier ==
== 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]].
 
 
Subgroup: 2.3.5
 
[[Comma list]]: 1220703125/1162261467
 
[[Mapping]]: [{{val| 1 0 0 }}, {{val| 0 13 19 }}]
 
[[POTE generator]]: ~27/25 = 146.476


[[Minimax tuning]]:
Bohpier is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].
* 5-odd-limit: ~27/25 = {{monzo| 0 0 1/19 }}
: Eigenmonzos (unchanged intervals): 2, 5/4


[[Tuning ranges]]:
[[Subgroup]]: 2.3.5.7
* 5-odd-limit [[diamond monotone]]: ~27/25 = [144.000, 150.000] (3\25 to 1\8)
* 5-odd-limit [[diamond tradeoff]]: ~27/25 = [146.304, 147.393]
* 5-odd-limit diamond monotone and tradeoff: ~27/25 = [146.304, 147.393]
 
{{Val list|legend=1| 8, 41, 131, 172, 213c }}
 
[[Badness]]: 0.860534
 
=== 7-limit ===
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]]:  
[[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
 
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~27/25 = [145.455, 150.000] (4\33 to 1\8)
* 9-odd-limit diamond monotone: ~27/25 = [145.455, 146.939] (4\33 to 6\49)
* 7-odd-limit [[diamond tradeoff]]: ~27/25 = [145.628, 147.393]
* 9-odd-limit diamond tradeoff: ~27/25 = [145.028, 147.393]
* 7-odd-limit diamond monotone and tradeoff: ~27/25 = [145.628, 147.393]
* 9-odd-limit diamond monotone and tradeoff: ~27/25 = [145.455, 146.939]


{{Val list|legend=1| 41, 131, 172, 213c }}
{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}


[[Badness]]: 0.068237
[[Badness]]: 0.068237
Line 262: Line 78:
Comma list: 100/99, 245/243, 1344/1331
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:  
Minimax tuning:  
* 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}
* 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
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~12/11 = [145.455, 146.939] (4\33 to 6\49)
* 11-odd-limit diamond tradeoff: ~12/11 = [145.028, 150.637]
* 11-odd-limit diamond monotone and tradeoff: ~12/11 = [145.455, 146.939]


Vals: {{Val list| 41, 90e, 131e }}
{{Optimal ET sequence|legend=0| 41, 90e, 131e }}


Badness: 0.033949
Badness: 0.033949


=== 13-limit ===
==== 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


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:  
Minimax tuning:  
* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}
: Eigenmonzos (unchanged intervals): 2, 5/4
: Unchanged-interval (eigenmonzo) basis: 2.5


Tuning ranges:
{{Optimal ET sequence|legend=0| 41, 90ef, 131ef, 221bdeff }}
* 13-odd-limit diamond monotone: ~12/11 = [145.455, 146.939] (4\33 to 6\49)
* 15-odd-limit diamond monotone: ~12/11 = [146.341, 146.939] (5\41 to 6\49)
* 13- and 15-odd-limit diamond tradeoff: ~12/11 = [138.573, 150.637]
* 13-odd-limit diamond monotone and tradeoff: ~12/11 = [145.455, 146.939]
* 15-odd-limit diamond monotone and tradeoff: ~12/11 = [146.341, 146.939]
 
Vals: {{Val list| 41, 90ef, 131ef, 221bdeff }}


Badness: 0.024864
Badness: 0.024864


; Music
=== Triboh ===
by [[Chris Vaisvil]]:
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-1.mp3 bophier&#45;1.mp3]
* [http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3 bophier&#45;12equal&#45;six&#45;octaves.mp3]
 
== 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
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


Vals: {{Val list| 22, 65, 87, 196, 283 }}
{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}


Badness: 0.035844
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


Vals: {{Val list| 22, 65, 87, 283 }}
{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}


Badness: 0.031366
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
[[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
[[Badness]]: 0.080152
Line 376: Line 155:
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


Vals: {{Val list| 17, 24, 41, 106d, 147d }}
{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}


Badness: 0.039444
Badness: 0.039444
Line 389: Line 168:
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


Vals: {{Val list| 17, 24, 41, 106df, 147df }}
{{Optimal ET sequence|legend=0| 17, 24, 41, 106df, 147df }}


Badness: 0.030793
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]].''


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
[[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
[[Badness]]: 0.073735


== Cohemiripple ==
== Superthird ==
{{see also| Ripple 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, 1323/1250
[[Comma list]]: 245/243, 78125/76832


[[Mapping]]: [{{val| 1 -3 -5 -5 }}, {{val| 0 10 16 17 }}]
{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}


{{Multival|legend=1| 10 16 17 2 -1 -5 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~9/7 = 439.076


[[POTE generator]]: ~7/5 = 549.944
{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}


{{Val list|legend=1| 11cd, 13cd, 24 }}
[[Badness]]: 0.139379
 
[[Badness]]: 0.190208


=== 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 -3 -5 -5 -8 }}, {{val| 0 10 16 17 25 }}]
Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}


POTE generator: ~7/5 = 549.945
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152


Vals: {{Val list| 11cdee, 13cdee, 24 }}
{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}


Badness: 0.082716
Badness: 0.070917


=== 13-limit ===
=== 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 -3 -5 -5 -8 -5 }}, {{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


Vals: {{Val list| 11cdeef, 13cdeef, 24 }}
{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}


Badness: 0.049933
Badness: 0.052835
 
== Superthird ==
{{see also| Shibboleth family }}


Subgroup: 2.3.5.7
== Superenneadecal ==
Superenneadecal is a cousin of [[enneadecal]] but sharper fifth is used to temper 245/243.


[[Comma list]]: 245/243, 78125/76832
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 1 -5 -5 -10 }}, {{val| 0 18 20 35 }}]
[[Comma list]]: 245/243, 395136/390625


{{Multival|legend=1| 18 20 35 -10 5 25 }}
{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}


[[POTE generator]]: ~9/7 = 439.076
[[Optimal tuning]] ([[POTE]]): ~392/375 = 63.158, ~3/2 = 704.166


{{Val list|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}


[[Badness]]: 0.139379
[[Badness]]: 0.132311


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 100/99, 245/243, 78125/76832
Comma list: 245/243, 2560/2541, 3773/3750


Mapping: [{{val| 1 -5 -5 -10 2 }}, {{val| 0 18 20 35 4 }}]
Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}


POTE generator: ~9/7 = 439.152
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667


Vals: {{Val list| 11cd, 30d, 41, 153be, 194be, 235bcee }}
{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}


Badness: 0.070917
Badness: 0.101496


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 100/99, 144/143, 196/195, 1375/1352
Comma list: 196/195, 245/243, 832/825, 1001/1000


Mapping: [{{val| 1 -5 -5 -10 2 -8 }}, {{val| 0 18 20 35 4 32 }}]
Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}


POTE generator: ~9/7 = 439.119
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801


Vals: {{Val list| 11cdf, 30df, 41 }}
{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e }}


Badness: 0.052835
Badness: 0.053197


== Magus ==
== 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.
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
 
 
Subgroup: 2.3.5
 
[[Comma list]]: 50331648/48828125
 
[[Mapping]]: [{{val| 1 -2 2 }}, {{val| 0 11 1 }}]


[[POTE generator]]: ~5/4 = 391.225
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.


{{Val list|legend=1| 46, 181c, 227c, 273c, 319c }}
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]].


[[Badness]]: 0.360162
[[Subgroup]]: 2.3.5.7
 
=== 7-limit ===
Subgroup: 2.3.5.7


[[Comma list]]: 245/243, 28672/28125
[[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
[[Badness]]: 0.108417
Line 537: Line 299:
Comma list: 176/175, 245/243, 1331/1323
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


Vals: {{Val list| 46, 95, 141bc }}
{{Optimal ET sequence|legend=0| 46, 95, 141bc }}


Badness: 0.045108
Badness: 0.045108
Line 550: Line 312:
Comma list: 91/90, 176/175, 245/243, 1331/1323
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


Vals: {{Val list| 46, 233bcff, 279bccff }}
{{Optimal ET sequence|legend=0| 46, 233bcff, 279bccff }}


Badness: 0.043024
Badness: 0.043024


== Leapweek ==
== 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
[[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
[[Badness]]: 0.140577
Line 576: Line 342:
Comma list: 245/243, 385/384, 1331/1323
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


Vals: {{Val list| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
{{Optimal ET sequence|legend=0| 17, 29c, 46, 109, 264b, 373b, 637bbe }}


Badness: 0.050679
Badness: 0.050679
Line 589: Line 355:
Comma list: 169/168, 245/243, 352/351, 364/363
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


Vals: {{Val list| 17, 29c, 46, 63, 109 }}
{{Optimal ET sequence|legend=0| 17, 29c, 46, 63, 109 }}


Badness: 0.032727
Badness: 0.032727


== Semiwolf ==
==== 17-limit ====
[[Subgroup]]: 3/2.7/4.5/2
Subgroup: 2.3.5.7.11.13.17
 
[[Comma]] list: 245/243
 
[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
 
[[POL2]] generator: ~7/6 = 262.1728
 
[[Vals]]: [[3edf]], [[5edf]], [[8edf]]


=== Semilupine ===
Comma list: 154/153, 169/168, 245/243, 256/255, 273/272
[[Subgroup]]: 3/2.7/4.5/2.11/4


[[Comma]] list: 100/99, 245/243
Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}


[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540


[[POL2]] generator: ~7/6 = 264.3771
{{Optimal ET sequence|legend=0| 17g, 29cg, 46, 109, 155f, 264bfg }}


[[Vals]]: [[8edf]], [[13edf]]
Badness: 0.026243


=== Hemilycan ===
==== Leapweeker ====
[[Subgroup]]: 3/2.7/4.5/2.11/4
Subgroup: 2.3.5.7.11.13.17


[[Comma]] list: 245/243, 441/440
Comma list: 136/135, 169/168, 221/220, 245/243, 364/363


[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]
Mapping: {{mapping| 1 0 42 -21 -14 -9 39 | 0 1 -25 15 11 8 -22 }}


[[POL2]] generator: ~7/6 = 261.5939
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537


[[Vals]]: [[8edf]], [[11edf]]
{{Optimal ET sequence|legend=0| 17, 29c, 46, 109g, 155fg, 264bfgg }}


Badness: 0.026774


[[Category:Regular temperament theory]]
[[Category:Temperament clans]]
[[Category:Temperament clan]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Sensamagic_clan"