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* ''[[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
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* [[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 ==
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: ''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
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Badness (Sintel): 3.39
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
 
[[Comma list]]: 245/243, 32805/32768
 
{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }}
: mapping generators: ~2, ~128/105
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}}
: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}}
: error map: {{val| 0.000 +0.139 -3.068 -5.213 }}
 
{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }}
 
[[Badness]] (Sintel): 2.03
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 245/242, 385/384
 
Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}}


{{Optimal ET sequence|legend=0| 17, 24, 41, 106d }}
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]].


Badness (Sintel): 1.30
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.
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 105/104, 144/143, 243/242, 245/242
 
Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}}
 
{{Optimal ET sequence|legend=0| 17, 24, 41 }}
 
Badness (Sintel): 1.27
 
== Pycnic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Stump]].''
 
The fifth of pycnic in size is a meantone fifth, but four of them are not used to reach 5. This has the effect of making the Pythagorean major third, nominally 81/64, very close to 5/4 in tuning, being a cent sharp of it in the POTE tuning for instance. Pycnic has [[mos]] of size 9, 11, 13, 15, 17… which contain these alternative thirds, leading to two kinds of major triads, an official one and a nominally Pythagorean one which is actually in better tune.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Badness]] (Sintel): 1.87
[[Badness]] (Sintel): 1.87


== Superthird ==
== Xenia ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''
 
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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 1029/1000
 
{{Mapping|legend=1| 1 -6 -12 -9 | 0 9 17 14 }}
: mapping generators: ~2, ~9/5
 
[[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 }}
 
{{Optimal ET sequence|legend=1| 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 {{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.
 
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, 78125/76832
[[Comma list]]: 245/243, 28672/28125


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3935{{c}}, ~9/7 = 439.2199{{c}}
* [[WE]]: ~2 = 1198.7187{{c}}, ~5/4 = 391.0473{{c}}
: [[error map]]: {{val| +0.394 +2.035 -3.884 -0.066 }}
: [[error map]]: {{val| -1.281 +2.128 +2.171 -2.860 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 439.0931{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.4129{{c}}
: error map: {{val| 0.000 +1.721 -4.452 -0.568 }}
: error map: {{val| 0.000 +3.587 +5.099 -0.678 }}


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


[[Badness]] (Sintel): 3.53
[[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 tunings:  
Optimal tunings:  
* WE: ~2 = 1199.5116{c}}, ~9/7 = 438.9734{{c}}
* WE: ~2 = 1198.7144{{c}}, ~5/4 = 391.0836{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.1362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.4506{{c}}


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


Badness (Sintel): 2.34
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 tunings:  
Optimal tunings:  
* WE: ~2 = 1199.2631{c}}, ~9/7 = 438.8494{{c}}
* WE: ~2 = 1199.7708{{c}}, ~5/4 = 391.2912{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.0943{{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 (Sintel): 2.18
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
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Badness (Sintel): 2.20
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]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.7187{{c}}, ~5/4 = 391.0473{{c}}
* [[WE]]: ~2 = 1200.3935{{c}}, ~9/7 = 439.2199{{c}}
: [[error map]]: {{val| -1.281 +2.128 +2.171 -2.860 }}
: [[error map]]: {{val| +0.394 +2.035 -3.884 -0.066 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.4129{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 439.0931{{c}}
: error map: {{val| 0.000 +3.587 +5.099 -0.678 }}
: error map: {{val| 0.000 +1.721 -4.452 -0.568 }}


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


[[Badness]] (Sintel): 2.74
[[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 tunings:  
Optimal tunings:  
* WE: ~2 = 1198.7144{{c}}, ~5/4 = 391.0836{{c}}
* WE: ~2 = 1199.5116{c}}, ~9/7 = 438.9734{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.4506{{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 (Sintel): 1.49
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 tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7708{{c}}, ~5/4 = 391.2912{{c}}
* WE: ~2 = 1199.2631{c}}, ~9/7 = 438.8494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.3597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.0943{{c}}


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


Badness (Sintel): 1.78
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
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Sensamagic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/Sensamagic_clan"