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{{Technical data page}}
{{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]].
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.  
 
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 ==
== BPS ==
Line 11: Line 13:
[[Comma list]]: 245/243
[[Comma list]]: 245/243


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


: sval mapping generators: ~3, ~9/7
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]], [[367edt|b367cdd]], [[462edt|b462cdd]]


[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
[[Badness]] (Sintel): 0.0659


=== Overview to extensions ===
=== Overview to extensions ===
Line 28: Line 35:
* ''[[Fourfives]]'' (+235298/234375) → [[Fifive family #Fourfives|Fifive family]]
* ''[[Fourfives]]'' (+235298/234375) → [[Fifive family #Fourfives|Fifive family]]


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.  
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
Discussed elsewhere are
Line 43: Line 50:
* [[Rodan]] (+1029/1024) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* [[Rodan]] (+1029/1024) → [[Gamelismic clan #Rodan|Gamelismic clan]]
* ''[[Shrutar]]'' (+2048/2025) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Shrutar]]'' (+2048/2025) → [[Diaschismic family #Shrutar|Diaschismic family]]
* ''[[Salsa]]'' (+32805/32768) → [[Schismatic family #Salsa|Schismatic family]]
* ''[[Escaped]]'' (+65625/65536) → [[Escapade family #Escaped|Escapade family]]
* ''[[Escaped]]'' (+65625/65536) → [[Escapade family #Escaped|Escapade family]]


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


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


== Bohpier ==
== Bohpier ==
Line 53: Line 61:
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Bohpier]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Bohpier]].''


Bohpier is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].
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]].
 
[[41edo]] itself makes for an excellent tuning, though [[90edo]] and [[131edo]] are interesting alternatives. Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 146.4741{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[WE]] and/or [[CWE]] (shown below) is largely unnoticeable.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 60: Line 70:


{{Mapping|legend=1| 1 0 0 0 | 0 13 19 23 }}
{{Mapping|legend=1| 1 0 0 0 | 0 13 19 23 }}
: mapping generators: ~2, ~27/25


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 146.474
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9967{{c}}, ~27/25 = 146.4737{{c}}
: [[error map]]: {{val| -0.003 +2.203 -3.314 +0.068 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 146.4739{{c}}
: error map: {{val| 0.000 +2.205 -3.310 +0.073 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 69: Line 84:
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


{{Optimal ET sequence|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 80: Line 95:
Mapping: {{mapping| 1 0 0 0 2 | 0 13 19 23 12 }}
Mapping: {{mapping| 1 0 0 0 2 | 0 13 19 23 12 }}


Optimal tuning (POTE): ~2 = 1200.000, ~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:  
Line 86: Line 103:
: unchanged-interval (eigenmonzo) basis: 2.11/9
: unchanged-interval (eigenmonzo) basis: 2.11/9


{{Optimal ET sequence|legend=0| 41, 90e, 131e }}
{{Optimal ET sequence|legend=0| 8d, …, 41, 90e, 131e }}


Badness: 0.033949
Badness (Sintel): 1.12


==== 13-limit ====
==== 13-limit ====
Line 97: Line 114:
Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}
Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}


Optimal tuning (POTE): ~2 = 1200.000, ~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 }}
: Unchanged-interval (eigenmonzo) basis: 2.5
: unchanged-interval (eigenmonzo) basis: 2.5


{{Optimal ET sequence|legend=0| 41, 90ef, 131ef, 221bdeff }}
{{Optimal ET sequence|legend=0| 8d, …, 41, 90ef }}


Badness: 0.024864
Badness (Sintel): 1.03


=== Triboh ===
=== Triboh ===
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Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}
Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}
: mapping generators: ~2, ~77/75


Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828
Optimal tunings:
* WE: ~2 = 1199.9966{{c}}, ~77/75 = 48.8281{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8282{{c}}


{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}
{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}


Badness: 0.162592
Badness (Sintel): 5.38


==== 13-limit ====
==== 13-limit ====
Line 129: Line 151:
Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}
Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}


Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822
Optimal tunings:
* WE: ~2 = 1199.9962{{c}}, ~77/75 = 48.8219{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8219{{c}}


{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}
{{Optimal ET sequence|legend=0| 49f, 123ce, 172f }}


Badness: 0.082158
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|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
[[Subgroup]]: 2.3.5.7


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~128/105 = 351.049
[[Comma list]]: 245/243, 525/512


{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
{{Mapping|legend=1| 1 0 6 -3 | 0 3 -7 11 }}
: mapping generators: ~2, ~64/45


[[Badness]]: 0.080152
[[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 }}


=== 11-limit ===
{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
Subgroup: 2.3.5.7.11


Comma list: 243/242, 245/242, 385/384
[[Badness]] (Sintel): 1.87


Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
== Xenia ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''


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


{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}
[[Subgroup]]: 2.3.5.7


Badness: 0.039444
[[Comma list]]: 245/243, 1029/1000


=== 13-limit ===
{{Mapping|legend=1| 1 -6 -12 -9 | 0 9 17 14 }}
Subgroup: 2.3.5.7.11.13
: mapping generators: ~2, ~9/5


Comma list: 105/104, 144/143, 243/242, 245/242
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.0862{{c}}, ~9/5 = 1012.0503{{c}}
: [[error map]]: {{val| +1.086 -0.020 +5.507 -9.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2199{{c}}
: error map: {{val| 0.000 -0.976 +4.424 -11.748 }}


Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
{{Optimal ET sequence|legend=1| 19, 70d, 89d }}


Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.025
[[Badness]] (Sintel): 2.25


{{Optimal ET sequence|legend=0| 17, 24, 41, 106df, 147df }}
== Magus ==
 
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
Badness: 0.030793


== Pycnic ==
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.
: ''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.
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]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 525/512
[[Comma list]]: 245/243, 28672/28125


{{Mapping|legend=1| 1 3 -1 8 | 0 -3 7 -11 }}
{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~45/32 = 567.720
 
{{Optimal ET sequence|legend=1| 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|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}


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


{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc }}


[[Badness]]: 0.139379
[[Badness]] (Sintel): 2.74


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


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


Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152
Optimal tunings:
* WE: ~2 = 1198.7144{{c}}, ~5/4 = 391.0836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.4506{{c}}


{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}
{{Optimal ET sequence|legend=0| 46, 95, 141bc }}


Badness: 0.070917
Badness (Sintel): 1.49


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


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


Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119
Optimal tunings:
* WE: ~2 = 1199.7708{{c}}, ~5/4 = 391.2912{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.3597{{c}}


{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}
{{Optimal ET sequence|legend=0| 3de, 43de, 46 }}


Badness: 0.052835
Badness (Sintel): 1.78


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 242: Line 266:


{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
: mapping generators: ~392/375, ~3


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


{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b }}


[[Badness]]: 0.132311
[[Badness]] (Sintel): 3.35


=== 11-limit ===
=== 11-limit ===
Line 256: Line 285:
Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}


Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667
Optimal tunings:
* WE: ~33/32 = 63.0966{{c}}, ~3/2 = 704.9824{{c}}
* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.3096{{c}}


{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}
{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}


Badness: 0.101496
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Line 269: Line 300:
Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
Mapping: {{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 tunings:
* WE: ~33/32 = 63.0988{{c}}, ~3/2 = 705.1402{{c}}
* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.4315{{c}}


{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e }}
{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e, 209bcef }}


Badness: 0.053197
Badness (Sintel): 2.20


== Magus ==
== Superthird ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
 
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.
 
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]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 28672/28125
[[Comma list]]: 245/243, 78125/76832


{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
: mapping generators: ~2, ~9/7


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


{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
{{Optimal ET sequence|legend=1| 11cd, 30d, 41 }}


[[Badness]]: 0.108417
[[Badness]] (Sintel): 3.53


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


Optimal tuning (POTE): ~2 = 1200.000, ~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}}


{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}
Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}


Optimal tuning (POTE): ~2 = 1200.000, ~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}}


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


Badness: 0.043024
Badness (Sintel): 2.18


== Leapweek ==
== Leapweek ==
: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''
: ''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
[[Subgroup]]: 2.3.5.7
Line 328: Line 368:


{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}
{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 }}


{{Optimal ET sequence|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 344: Line 387:
Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}
Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}


Optimal tuning (POTE): ~2 = 1200.000, ~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}}


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


Badness: 0.050679
Badness (Sintel): 1.68


=== 13-limit ===
=== 13-limit ===
Line 357: Line 402:
Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}
Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}


Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.571
Optimal tunings:
* WE: ~2 = 1200.0070{{c}}, ~3/2 = 704.5751{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5709{{c}}


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


Badness: 0.032727
Badness (Sintel): 1.35


==== 17-limit ====
==== 17-limit ====
Line 370: Line 417:
Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}
Mapping: {{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 tunings:
* WE: ~2 = 1199.8670{{c}}, ~3/2 = 704.4620{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5395{{c}}


{{Optimal ET sequence|legend=0| 17g, 29cg, 46, 109, 155f, 264bfg }}
{{Optimal ET sequence|legend=0| 17g, 46, 109 }}


Badness: 0.026243
Badness (Sintel): 1.34


==== Leapweeker ====
==== Leapweeker ====
Line 383: Line 432:
Mapping: {{mapping| 1 0 42 -21 -14 -9 39 | 0 1 -25 15 11 8 -22 }}
Mapping: {{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 tunings:
* WE: ~2 = 1200.1737{{c}}, ~3/2 = 704.6390{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5364{{c}}


{{Optimal ET sequence|legend=0| 17, 29c, 46, 109g, 155fg, 264bfgg }}
{{Optimal ET sequence|legend=0| 17, 46, 109g, 155fg }}


Badness: 0.026774
Badness (Sintel): 1.36


[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Listen]]
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