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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, salsa, superenneadecal, superthird, magus and leapweek.


== Bohpier ==
== Bohpier ==
Line 157: Line 157:


Badness (Sintel): 3.39
Badness (Sintel): 3.39
== Pycnic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Stump]].''
Pycnic is related to [[triton]], but its mapping differs for the [[5/1|5th harmonic]]. It is also related to [[liese]], from which its mapping differs for the [[7/1|7th 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 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, 525/512
{{Mapping|legend=1| 1 0 6 -3 | 0 3 -7 11 }}
: mapping generators: ~2, ~64/45
[[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 }}
{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
[[Badness]] (Sintel): 1.87


== Salsa ==
== Salsa ==
Line 208: Line 232:
Badness (Sintel): 1.27
Badness (Sintel): 1.27


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


Pycnic is related to [[triton]], but its mapping differs for the [[5/1|5th harmonic]]. It is also related to [[liese]], from which its mapping differs for the [[7/1|7th harmonic]].
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.


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 0 6 -3 | 0 3 -7 11 }}
{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}
: mapping generators: ~2, ~64/45


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.3437{{c}}, ~64/45 = 634.0416{{c}}
* [[WE]]: ~2 = 1198.7187{{c}}, ~5/4 = 391.0473{{c}}
: [[error map]]: {{val| +3.344 +0.170 -4.542 -4.400 }}
: [[error map]]: {{val| -1.281 +2.128 +2.171 -2.860 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 632.3502{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.4129{{c}}
: error map: {{val| 0.000 -4.904 -12.765 -12.973 }}
: error map: {{val| 0.000 +3.587 +5.099 -0.678 }}


{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc }}


[[Badness]] (Sintel): 1.87
[[Badness]] (Sintel): 2.74
 
== 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 }}
: mapping generators: ~2, ~9/7
 
[[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| 11cd, 30d, 41 }}
 
[[Badness]] (Sintel): 3.53


=== 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 ==
Line 332: Line 335:
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]] 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, 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 ==
Retrieved from "https://en.xen.wiki/w/Sensamagic_clan"