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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''' 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.  


== BPS (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]].
The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243.  


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 }}]
 
[[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]]
 
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 ====
{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~3, ~9/7


Comma list: 55/54, 66/65, 77/75, 143/140
[[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 }}


Mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 11 }}]
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]], [[367edt|b367cdd]], [[462edt|b462cdd]]


POTE generator: ~9/7 = 443.506
[[Badness]] (Sintel): 0.0659


Vals: {{Val list| 19e, 27, 46ee }}
=== 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.
Badness: 0.023258
 
=== Hemisensi ===
Subgroup: 2.3.5.7.11


Comma list: 126/125, 243/242, 245/242
These temperaments are distributed into different family pages.
* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
* ''[[Hedgehog]]'' (+50/49) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple family]]
* ''[[Fourfives]]'' (+235298/234375) → [[Fifive family #Fourfives|Fifive family]]


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


POTE generator: ~25/22 = 221.605
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]]


Vals: {{Val list| 27e, 65, 157de, 222cde }}
For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].


Badness: 0.048714
Considered below are bohpier, pycnic, superenneadecal, superthird, magus and leapweek.


== 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]]:
* 5-odd-limit: ~27/25 = {{monzo| 0 0 1/19 }}
: Eigenmonzos (unchanged intervals): 2, 5/4


[[Tuning ranges]]:
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]].  
* 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 }}
[[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.


[[Badness]]: 0.860534
[[Subgroup]]: 2.3.5.7
 
=== 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 }}
: mapping generators: ~2, ~27/25


{{Multival|legend=1| 13 19 23 0 0 0 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9967{{c}}, ~27/25 = 146.4737{{c}}
[[POTE generator]]: ~27/25 = 146.474
: [[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 }}


[[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| 8d, …, 41, 131, 172, 213c }}


[[Badness]]: 0.068237
[[Badness]] (Sintel): 1.73


=== 11-limit ===
=== 11-limit ===
Line 260: Line 93:
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 tunings:  
* WE: ~2 = 1199.2309{{c}}, ~12/11 = 146.4507{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.5009{{c}}


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:
{{Optimal ET sequence|legend=0| 8d, …, 41, 90e, 131e }}
* 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 }}
Badness (Sintel): 1.12


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 tunings:  
* WE: ~2 = 1198.5478{{c}}, ~12/11 = 146.4252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.5230{{c}}


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:
* 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
 
; 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]
 
== 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 }}]
{{Optimal ET sequence|legend=0| 8d, …, 41, 90ef }}


{{Multival|legend=1| 9 -7 26 -32 16 80 }}
Badness (Sintel): 1.03


[[POTE generator]]: ~28/27 = 55.122
=== 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.  


{{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 }}
: mapping generators: ~2, ~77/75


POTE generator: ~28/27 = 55.126
Optimal tunings:  
* WE: ~2 = 1199.9966{{c}}, ~77/75 = 48.8281{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8282{{c}}


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


Badness: 0.035844
Badness (Sintel): 5.38


=== 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 tunings:  
* WE: ~2 = 1199.9962{{c}}, ~77/75 = 48.8219{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8219{{c}}


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


Badness: 0.031366
Badness (Sintel): 3.39


== Salsa ==
== Pycnic ==
{{see also| Schismatic family }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


Subgroup: 2.3.5.7
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]].


[[Comma list]]: 245/243, 32805/32768
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.


[[Mapping]]: [{{val| 1 1 7 -1 }}, {{val| 0 2 -16 13 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 2 -16 13 -30 15 75 }}
[[Comma list]]: 245/243, 525/512


[[POTE generator]]: ~128/105 = 351.049
{{Mapping|legend=1| 1 0 6 -3 | 0 3 -7 11 }}
: mapping generators: ~2, ~64/45


{{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
[[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 }}


[[Badness]]: 0.080152
{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}


=== 11-limit ===
[[Badness]] (Sintel): 1.87
Subgroup: 2.3.5.7.11


Comma list: 243/242, 245/242, 385/384
== Xenia ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''


Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]
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]].


POTE generator: ~11/9 = 351.014
[[Subgroup]]: 2.3.5.7


Vals: {{Val list| 17, 24, 41, 106d, 147d }}
[[Comma list]]: 245/243, 1029/1000


Badness: 0.039444
{{Mapping|legend=1| 1 -6 -12 -9 | 0 9 17 14 }}
: mapping generators: ~2, ~9/5


=== 13-limit ===
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.11.13
* [[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 }}


Comma list: 105/104, 144/143, 243/242, 245/242
{{Optimal ET sequence|legend=1| 19, 70d, 89d }}


Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]
[[Badness]] (Sintel): 2.25


POTE generator: ~11/9 = 351.025
== Magus ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''


Vals: {{Val list| 17, 24, 41, 106df, 147df }}
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.


Badness: 0.030793
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]].


== Pycnic ==
[[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, 28672/28125


[[Comma list]]: 245/243, 525/512
{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
 
[[Mapping]]: [{{val| 1 3 -1 8 }}, {{val| 0 -3 7 -11 }}]
 
{{Multival|legend=1| 3 -7 11 -18 9 45 }}
 
[[POTE generator]]: ~45/32 = 567.720
 
{{Val list|legend=1| 17, 19, 55c, 74cd, 93cdd }}
 
[[Badness]]: 0.073735
 
== Cohemiripple ==
{{see also| Ripple family }}
 
Subgroup: 2.3.5.7
 
[[Comma list]]: 245/243, 1323/1250
 
[[Mapping]]: [{{val| 1 -3 -5 -5 }}, {{val| 0 10 16 17 }}]
 
{{Multival|legend=1| 10 16 17 2 -1 -5 }}


[[POTE generator]]: ~7/5 = 549.944
[[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 }}


{{Val list|legend=1| 11cd, 13cd, 24 }}
{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc }}


[[Badness]]: 0.190208
[[Badness]] (Sintel): 2.74


=== 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: 176/175, 245/243, 1331/1323


Mapping: [{{val| 1 -3 -5 -5 -8 }}, {{val| 0 10 16 17 25 }}]
Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}


POTE generator: ~7/5 = 549.945
Optimal tunings:  
* WE: ~2 = 1198.7144{{c}}, ~5/4 = 391.0836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.4506{{c}}


Vals: {{Val list| 11cdee, 13cdee, 24 }}
{{Optimal ET sequence|legend=0| 46, 95, 141bc }}


Badness: 0.082716
Badness (Sintel): 1.49


=== 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: 91/90, 176/175, 245/243, 1331/1323


Mapping: [{{val| 1 -3 -5 -5 -8 -5 }}, {{val| 0 -10 -16 -17 -25 -19 }}]
Mapping: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}


POTE generator: ~7/5 = 549.958
Optimal tunings:  
* WE: ~2 = 1199.7708{{c}}, ~5/4 = 391.2912{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.3597{{c}}


Vals: {{Val list| 11cdeef, 13cdeef, 24 }}
{{Optimal ET sequence|legend=0| 3de, 43de, 46 }}


Badness: 0.049933
Badness (Sintel): 1.78
 
== Superthird ==
{{see also| Shibboleth family }}


Subgroup: 2.3.5.7
== Superenneadecal ==
Superenneadecal is a cousin of [[enneadecal]] but a sharper fifth is used to temper out 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 }}
: mapping generators: ~392/375, ~3


[[POTE generator]]: ~9/7 = 439.076
[[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 }}


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


[[Badness]]: 0.139379
[[Badness]] (Sintel): 3.35


=== 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 tunings:  
* WE: ~33/32 = 63.0966{{c}}, ~3/2 = 704.9824{{c}}
* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.3096{{c}}


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


Badness: 0.070917
Badness (Sintel): 3.36


=== 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 tunings:  
* WE: ~33/32 = 63.0988{{c}}, ~3/2 = 705.1402{{c}}
* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.4315{{c}}


Vals: {{Val list| 11cdf, 30df, 41 }}
{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e, 209bcef }}


Badness: 0.052835
Badness (Sintel): 2.20


== Magus ==
== Superthird ==
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 #Shibboleth]].''


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50331648/48828125
[[Comma list]]: 245/243, 78125/76832
 
[[Mapping]]: [{{val| 1 -2 2 }}, {{val| 0 11 1 }}]
 
[[POTE generator]]: ~5/4 = 391.225
 
{{Val list|legend=1| 46, 181c, 227c, 273c, 319c }}
 
[[Badness]]: 0.360162
 
=== 7-limit ===
Subgroup: 2.3.5.7


[[Comma list]]: 245/243, 28672/28125
{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
: mapping generators: ~2, ~9/7


[[Mapping]]: [{{val| 1 -2 2 -6 }}, {{val| 0 11 1 27 }}]
[[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 }}


{{Multival|legend=1| 11 1 27 -24 12 60 }}
{{Optimal ET sequence|legend=1| 11cd, 30d, 41 }}


[[POTE generator]]: ~5/4 = 391.465
[[Badness]] (Sintel): 3.53
 
{{Val list|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
 
[[Badness]]: 0.108417


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


Comma list: 176/175, 245/243, 1331/1323
Comma list: 100/99, 245/243, 78125/76832


Mapping: [{{val| 1 -2 2 -6 -6 }}, {{val| 0 11 1 27 29 }}]
Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}


POTE generator: ~5/4 = 391.503
Optimal tunings:  
* WE: ~2 = 1199.5116{c}}, ~9/7 = 438.9734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.1362{{c}}


Vals: {{Val list| 46, 95, 141bc }}
{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be }}


Badness: 0.045108
Badness (Sintel): 2.34


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


Comma list: 91/90, 176/175, 245/243, 1331/1323
Comma list: 100/99, 144/143, 196/195, 1375/1352


Mapping: [{{val| 1 -2 2 -6 -6 5 }}, {{val| 0 11 1 27 29 -4 }}]
Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}


POTE generator: ~5/4 = 391.366
Optimal tunings:  
* WE: ~2 = 1199.2631{c}}, ~9/7 = 438.8494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.0943{{c}}


Vals: {{Val list| 46, 233bcff, 279bccff }}
{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}


Badness: 0.043024
Badness (Sintel): 2.18


== Leapweek ==
== Leapweek ==
Subgroup: 2.3.5.7
: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
 
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.
 
[[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 }}
: mapping generators: ~2, ~3


[[POTE generator]]: ~3/2 = 704.536
[[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 }}


{{Val list|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
{{Optimal ET sequence|legend=1| 17, 46, 109, 155, 264b }}


[[Badness]]: 0.140577
[[Badness]] (Sintel): 3.56


=== 11-limit ===
=== 11-limit ===
Line 571: Line 385:
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 tunings:  
* WE: ~2 = 1199.7910{{c}}, ~3/2 = 704.4312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5542{{c}}


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


Badness: 0.050679
Badness (Sintel): 1.68


=== 13-limit ===
=== 13-limit ===
Line 584: Line 400:
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
 
Vals: {{Val list| 17, 29c, 46, 63, 109, 218f, 373bf }}
 
Badness: 0.032727
 
== Semiwolf ==
[[Subgroup]]: 3/2.7/4.5/2


[[Comma]] list: 245/243
Optimal tunings:  
* WE: ~2 = 1200.0070{{c}}, ~3/2 = 704.5751{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5709{{c}}


[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
{{Optimal ET sequence|legend=0| 17, 46, 63, 109 }}


[[POL2]] generator: ~7/6 = 262.1728
Badness (Sintel): 1.35


[[Vals]]: [[3edf]], [[5edf]], [[8edf]]
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=== 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 tunings:  
* WE: ~2 = 1199.8670{{c}}, ~3/2 = 704.4620{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5395{{c}}


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


[[Vals]]: [[8edf]], [[13edf]]
Badness (Sintel): 1.34


=== 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 tunings:  
* WE: ~2 = 1200.1737{{c}}, ~3/2 = 704.6390{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5364{{c}}


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


Badness (Sintel): 1.36


[[Category:Regular temperament theory]]
[[Category:Temperament clan]]
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Temperament clans]]
[[Category:Listen]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 07:41, 15 August 2026

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 of temperaments 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

Subgroup-val mapping: [⟨1 1 2], ⟨0 2 -1]]

mapping generators: ~3, ~9/7

Optimal tunings:

  • WE: ~3 = 1903.7398 ¢, ~9/7 = 440.9014 ¢
error map: ⟨+1.785 -0.771 -2.248]
  • CWE: ~3 = 1901.9550 ¢, ~9/7 = 440.6646 ¢
error map: ⟨0.000 -3.030 -5.580]

Optimal ET sequence: b4, b9, b13, b56, b69, b82, b95, b367cdd, b462cdd

Badness (Sintel): 0.0659

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. Salsa tempers out 32805/32768, splitting the generator in fifteen. 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, pycnic, superenneadecal, superthird, magus and leapweek.

Bohpier

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

Bohpier tempers out 3125/3087 and may be described as the 41 & 49 temperament. It is named after its interesting relationship with the non-octave Bohlen–Pierce equal temperament.

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 ¢ 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

Comma list: 245/243, 3125/3087

Mapping: [⟨1 0 0 0], ⟨0 13 19 23]]

mapping generators: ~2, ~27/25

Optimal tunings:

  • WE: ~2 = 1199.9967 ¢, ~27/25 = 146.4737 ¢
error map: ⟨-0.003 +2.203 -3.314 +0.068]
  • CWE: ~2 = 1200.0000 ¢, ~27/25 = 146.4739 ¢
error map: ⟨0.000 +2.205 -3.310 +0.073]

Minimax tuning:

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

Optimal ET sequence: 8d, …, 41, 131, 172, 213c

Badness (Sintel): 1.73

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 tunings:

  • WE: ~2 = 1199.2309 ¢, ~12/11 = 146.4507 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 146.5009 ¢

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: 8d, …, 41, 90e, 131e

Badness (Sintel): 1.12

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 tunings:

  • WE: ~2 = 1198.5478 ¢, ~12/11 = 146.4252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~12/11 = 146.5230 ¢

Minimax tuning:

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

Optimal ET sequence: 8d, …, 41, 90ef

Badness (Sintel): 1.03

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]]

mapping generators: ~2, ~77/75

Optimal tunings:

  • WE: ~2 = 1199.9966 ¢, ~77/75 = 48.8281 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/75 = 48.8282 ¢

Optimal ET sequence: 49, 123ce, 172

Badness (Sintel): 5.38

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 tunings:

  • WE: ~2 = 1199.9962 ¢, ~77/75 = 48.8219 ¢
  • CWE: ~2 = 1200.0000 ¢, ~77/75 = 48.8219 ¢

Optimal ET sequence: 49f, 123ce, 172f

Badness (Sintel): 3.39

Pycnic

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Stump.

Pycnic is related to triton, but its mapping differs for the 7th harmonic. It is also related to liese, from which its mapping differs for the 5th harmonic.

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.

Subgroup: 2.3.5.7

Comma list: 245/243, 525/512

Mapping: [⟨1 0 6 -3], ⟨0 3 -7 11]]

mapping generators: ~2, ~64/45

Optimal tunings:

  • WE: ~2 = 1203.3437 ¢, ~64/45 = 634.0416 ¢
error map: ⟨+3.344 +0.170 -4.542 -4.400]
  • CWE: ~2 = 1200.0000 ¢, ~64/45 = 632.3502 ¢
error map: ⟨0.000 -4.904 -12.765 -12.973]

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

Badness (Sintel): 1.87

Xenia

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Xenial.

Xenia is related to xenial, but its mapping differs for the 7th harmonic. It may be described as 19 & 51c or 19 & 70d, which tempers out the sensamagic and keega, 1029/1000.

Subgroup: 2.3.5.7

Comma list: 245/243, 1029/1000

Mapping: [⟨1 -6 -12 -9], ⟨0 9 17 14]]

mapping generators: ~2, ~9/5

Optimal tunings:

  • WE: ~2 = 1201.0862 ¢, ~9/5 = 1012.0503 ¢
error map: ⟨+1.086 -0.020 +5.507 -9.898]
  • CWE: ~2 = 1200.0000 ¢, ~9/5 = 1011.2199 ¢
error map: ⟨0.000 -0.976 +4.424 -11.748]

Optimal ET sequence: 19, 70d, 89d

Badness (Sintel): 2.25

Magus

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

Magus temperament tempers out 50331648/48828125 in the 5-limit. This temperament can be described as 46 & 49 temperament, which tempers out the sensamagic and 28672/28125. 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, 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)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 tunings:

  • WE: ~2 = 1198.7187 ¢, ~5/4 = 391.0473 ¢
error map: ⟨-1.281 +2.128 +2.171 -2.860]
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.4129 ¢
error map: ⟨0.000 +3.587 +5.099 -0.678]

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

Badness (Sintel): 2.74

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 tunings:

  • WE: ~2 = 1198.7144 ¢, ~5/4 = 391.0836 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.4506 ¢

Optimal ET sequence: 46, 95, 141bc

Badness (Sintel): 1.49

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 tunings:

  • WE: ~2 = 1199.7708 ¢, ~5/4 = 391.2912 ¢
  • CWE: ~2 = 1200.0000 ¢, ~5/4 = 391.3597 ¢

Optimal ET sequence: 3de, 43de, 46

Badness (Sintel): 1.78

Superenneadecal

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

Subgroup: 2.3.5.7

Comma list: 245/243, 395136/390625

Mapping: [⟨19 0 14 -7], ⟨0 1 1 2]]

mapping generators: ~392/375, ~3

Optimal tunings:

  • WE: ~392/375 = 63.1399 ¢, ~3/2 = 703.9652 ¢
error map: ⟨-0.343 +1.668 +1.267 -3.560]
  • CWE: ~392/375 = 63.1579 ¢, ~3/2 = 703.9028 ¢
error map: ⟨0.000 +1.948 +1.800 -3.126]

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

Badness (Sintel): 3.35

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 tunings:

  • WE: ~33/32 = 63.0966 ¢, ~3/2 = 704.9824 ¢
  • CWE: ~33/32 = 63.1579 ¢, ~3/2 = 705.3096 ¢

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

Badness (Sintel): 3.36

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 tunings:

  • WE: ~33/32 = 63.0988 ¢, ~3/2 = 705.1402 ¢
  • CWE: ~33/32 = 63.1579 ¢, ~3/2 = 705.4315 ¢

Optimal ET sequence: 19, 76bcdf, 95, 114e, 209bcef

Badness (Sintel): 2.20

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]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • WE: ~2 = 1200.3935 ¢, ~9/7 = 439.2199 ¢
error map: ⟨+0.394 +2.035 -3.884 -0.066]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 439.0931 ¢
error map: ⟨0.000 +1.721 -4.452 -0.568]

Optimal ET sequence: 11cd, 30d, 41

Badness (Sintel): 3.53

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 tunings:

  • WE: ~2 = 1199.5116{c}}, ~9/7 = 438.9734 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 439.1362 ¢

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

Badness (Sintel): 2.34

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 tunings:

  • WE: ~2 = 1199.2631{c}}, ~9/7 = 438.8494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 439.0943 ¢

Optimal ET sequence: 11cdf, 30df, 41

Badness (Sintel): 2.18

Leapweek

Not to be confused with scales produced by leap week calendars such as Symmetry454.

Leapweek may be described as the 46 & 63 temperament, generated by a perfect fifth and being a strong extension of leapfrog. 109edo makes for an excellent tuning.

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 tunings:

  • WE: ~2 = 1199.6301 ¢, ~3/2 = 704.3191 ¢
error map: ⟨-0.370 +1.994 -0.578 -1.821]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5387 ¢
error map: ⟨0.000 +2.584 +0.218 -0.745]

Optimal ET sequence: 17, 46, 109, 155, 264b

Badness (Sintel): 3.56

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 tunings:

  • WE: ~2 = 1199.7910 ¢, ~3/2 = 704.4312 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5542 ¢

Optimal ET sequence: 17, 46, 109, 264b

Badness (Sintel): 1.68

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 tunings:

  • WE: ~2 = 1200.0070 ¢, ~3/2 = 704.5751 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5709 ¢

Optimal ET sequence: 17, 46, 63, 109

Badness (Sintel): 1.35

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 tunings:

  • WE: ~2 = 1199.8670 ¢, ~3/2 = 704.4620 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5395 ¢

Optimal ET sequence: 17g, 46, 109

Badness (Sintel): 1.34

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 tunings:

  • WE: ~2 = 1200.1737 ¢, ~3/2 = 704.6390 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5364 ¢

Optimal ET sequence: 17, 46, 109g, 155fg

Badness (Sintel): 1.36

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