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Sensamagic clan: Difference between revisions

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[[Comma list]]: 245/243
[[Comma list]]: 245/243


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


[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550{{c}}, ~9/7 = 440.4881{{c}}
[[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 ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]], [[367edt|b367cdd]], [[462edt|b462cdd]]
 
[[Badness]] (Sintel): 0.0659


=== Overview to extensions ===
=== 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.  
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.
These temperaments are distributed into different family and clan pages.
* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
* ''[[Hedgehog]]'' (+50/49) → [[Porcupine family #Hedgehog|Porcupine family]]
* ''[[Hedgehog]]'' (+50/49) → [[Stearnsmic clan #Hedgehog|Stearnsmic clan]]
* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple family]]
* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple family]]
* ''[[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 44: 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 54: 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 61: 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{{c}}, ~27/25 = 146.474{{c}}
[[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 70: 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]] (Smith): 0.068237
[[Badness]] (Sintel): 1.73


=== 11-limit ===
=== 11-limit ===
Line 79: Line 93:
Comma list: 100/99, 245/243, 1344/1331
Comma list: 100/99, 245/243, 1344/1331


Mapping: {{mapping| 1 0 0 0 2 | 0 13 19 23 12 }}
{{Mapping|legend=0| 1 0 0 0 2 | 0 13 19 23 12 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 146.545{{c}}
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:  
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: 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 (Smith): 0.033949
Badness (Sintel): 1.12


==== 13-limit ====
==== 13-limit ====
Line 96: Line 112:
Comma list: 100/99, 144/143, 196/195, 275/273
Comma list: 100/99, 144/143, 196/195, 275/273


Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}
{{Mapping|legend=0| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~12/11 = 146.603{{c}}
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 (Smith): 0.024864
Badness (Sintel): 1.03


=== Triboh ===
=== Triboh ===
Line 115: Line 133:
Comma list: 245/243, 1331/1323, 3125/3087
Comma list: 245/243, 1331/1323, 3125/3087


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


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/75 = 48.828{{c}}
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 (Smith): 0.162592
Badness (Sintel): 5.38


==== 13-limit ====
==== 13-limit ====
Line 128: Line 149:
Comma list: 245/243, 275/273, 847/845, 1331/1323
Comma list: 245/243, 275/273, 847/845, 1331/1323


Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}
{{Mapping|legend=0| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/75 = 48.822{{c}}
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 (Smith): 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{{c}}, ~128/105 = 351.049{{c}}
[[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]] (Smith): 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{{c}}, ~11/9 = 351.014{{c}}
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 (Smith): 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{{c}}, ~11/9 = 351.025{{c}}
[[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 (Smith): 0.030793
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.


== Pycnic ==
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]].
: ''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, 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{{c}}, ~45/32 = 567.720{{c}}
 
{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}


[[Badness]] (Smith): 0.073735
[[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 }}


== Superthird ==
{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''


[[Subgroup]]: 2.3.5.7
[[Badness]] (Sintel): 2.74
 
[[Comma list]]: 245/243, 78125/76832
 
{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 439.076{{c}}
 
{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
 
[[Badness]] (Smith): 0.139379


=== 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|legend=0| 1 -2 2 -6 -6 | 0 11 1 27 29 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 439.152{{c}}
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 (Smith): 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|legend=0| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 439.119{{c}}
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 (Smith): 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 243: 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{{c}}, ~3/2 = 704.166{{c}}
[[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]] (Smith): 0.132311
[[Badness]] (Sintel): 3.35


=== 11-limit ===
=== 11-limit ===
Line 255: Line 283:
Comma list: 245/243, 2560/2541, 3773/3750
Comma list: 245/243, 2560/2541, 3773/3750


Mapping: {{mapping| 19 0 14 -7 96 | 0 1 1 2 -1 }}
{{Mapping|legend=0| 19 0 14 -7 96 | 0 1 1 2 -1 }}


Optimal tuning (POTE): ~33/32 = 63.158{{c}}, ~3/2 = 705.667{{c}}
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 (Smith): 0.101496
Badness (Sintel): 3.36


=== 13-limit ===
=== 13-limit ===
Line 268: Line 298:
Comma list: 196/195, 245/243, 832/825, 1001/1000
Comma list: 196/195, 245/243, 832/825, 1001/1000


Mapping: {{mapping| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}
{{Mapping|legend=0| 19 0 14 -7 96 10 | 0 1 1 2 -1 2 }}


Optimal tuning (POTE): ~33/32 = 63.158{{c}}, ~3/2 = 705.801{{c}}
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 (Smith): 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{{c}}, ~5/4 = 391.465{{c}}
[[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]] (Smith): 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|legend=0| 1 -5 -5 -10 2 | 0 18 20 35 4 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.503{{c}}
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 (Smith): 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|legend=0| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~5/4 = 391.366{{c}}
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 (Smith): 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 331: Line 370:
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3/2 = 704.536{{c}}
[[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]] (Smith): 0.140577
[[Badness]] (Sintel): 3.56


=== 11-limit ===
=== 11-limit ===
Line 342: Line 385:
Comma list: 245/243, 385/384, 1331/1323
Comma list: 245/243, 385/384, 1331/1323


Mapping: {{mapping| 1 0 42 -21 -14 | 0 1 -25 15 11 }}
{{Mapping|legend=0| 1 0 42 -21 -14 | 0 1 -25 15 11 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.554{{c}}
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 (Smith): 0.050679
Badness (Sintel): 1.68


=== 13-limit ===
=== 13-limit ===
Line 355: Line 400:
Comma list: 169/168, 245/243, 352/351, 364/363
Comma list: 169/168, 245/243, 352/351, 364/363


Mapping: {{mapping| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}
{{Mapping|legend=0| 1 0 42 -21 -14 -9 | 0 1 -25 15 11 8 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.571{{c}}
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 (Smith): 0.032727
Badness (Sintel): 1.35


==== 17-limit ====
==== 17-limit ====
Line 368: Line 415:
Comma list: 154/153, 169/168, 245/243, 256/255, 273/272
Comma list: 154/153, 169/168, 245/243, 256/255, 273/272


Mapping: {{mapping| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}
{{Mapping|legend=0| 1 0 42 -21 -14 -9 -34 | 0 1 -25 15 11 8 24 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.540{{c}}
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 (Smith): 0.026243
Badness (Sintel): 1.34


==== Leapweeker ====
==== Leapweeker ====
Line 381: Line 430:
Comma list: 136/135, 169/168, 221/220, 245/243, 364/363
Comma list: 136/135, 169/168, 221/220, 245/243, 364/363


Mapping: {{mapping| 1 0 42 -21 -14 -9 39 | 0 1 -25 15 11 8 -22 }}
{{Mapping|legend=0| 1 0 42 -21 -14 -9 39 | 0 1 -25 15 11 8 -22 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~3/2 = 704.537{{c}}
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 (Smith): 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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