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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=2| 1 1 2 | 0 -2 1 }}


: sval mapping generators: ~3, ~9/7
: sval mapping generators: ~3, ~9/7
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== Bohpier ==
== Bohpier ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #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]].
Bohpier is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Multival|legend=1| 13 19 23 0 0 0 }}
{{Multival|legend=1| 13 19 23 0 0 0 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 146.474
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~27/25 = 146.474


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~27/25 = {{monzo| 0 0 1/19 }}
* [[7-odd-limit]]: ~27/25 = {{monzo| 0 0 1/19 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
* [[9-odd-limit]]: ~27/25 = {{monzo| 0 1/13 }}
* [[9-odd-limit]]: ~27/25 = {{monzo| 0 1/13 }}
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.3
: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3


{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}
{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}
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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 = 1\1, ~12/11 = 146.545
Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.545


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 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/9
: eigenmonzo (unchanged-interval) basis: 2.11/9


{{Optimal ET sequence|legend=1| 41, 90e, 131e }}
{{Optimal ET sequence|legend=0| 41, 90e, 131e }}


Badness: 0.033949
Badness: 0.033949
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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 = 1\1, ~12/11 = 146.603
Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.603


Minimax tuning:  
Minimax tuning:  
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: Eigenmonzo (unchanged-interval) basis: 2.5
: Eigenmonzo (unchanged-interval) basis: 2.5


{{Optimal ET sequence|legend=1| 41, 90ef, 131ef, 221bdeff }}
{{Optimal ET sequence|legend=0| 41, 90ef, 131ef, 221bdeff }}


Badness: 0.024864
Badness: 0.024864
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=== Triboh ===
=== Triboh ===
'''Triboh''' is named after "[[39edt|Triple Bohlen–Pierce scale]]", which divides each step of the [[13edt|equal-tempered]] [[Bohlen–Pierce]] scale into three equal parts.  
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.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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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 }}


Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.828
Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.828


{{Optimal ET sequence|legend=1| 49, 123ce, 172 }}
{{Optimal ET sequence|legend=0| 49, 123ce, 172 }}


Badness: 0.162592
Badness: 0.162592
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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 = 1\1, ~77/75 = 48.822
Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822


{{Optimal ET sequence|legend=1| 49f, 123ce, 172f, 295ce, 467bccef }}
{{Optimal ET sequence|legend=0| 49f, 123ce, 172f, 295ce, 467bccef }}


Badness: 0.082158
Badness: 0.082158
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{{Multival|legend=1| 2 -16 13 -30 15 75 }}
{{Multival|legend=1| 2 -16 13 -30 15 75 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 351.049
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~128/105 = 351.049


{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}
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Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}
Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014
Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.014


{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d }}
{{Optimal ET sequence|legend=0| 17, 24, 41, 106d, 147d }}


Badness: 0.039444
Badness: 0.039444
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Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}
Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.025
Optimal tuning (POTE): ~2 = 1200.000, ~11/9 = 351.025


{{Optimal ET sequence|legend=1| 17, 24, 41, 106df, 147df }}
{{Optimal ET sequence|legend=0| 17, 24, 41, 106df, 147df }}


Badness: 0.030793
Badness: 0.030793


== Pycnic ==
== Pycnic ==
{{See also| High badness temperaments #Stump }}
: ''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.
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.
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{{Multival|legend=1| 3 -7 11 -18 9 45 }}
{{Multival|legend=1| 3 -7 11 -18 9 45 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 567.720
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~45/32 = 567.720


{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}
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{{Multival|legend=1| 18 20 35 -10 5 25 }}
{{Multival|legend=1| 18 20 35 -10 5 25 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 439.076
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~9/7 = 439.076


{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 317bcc, 358bcc, 399bcc }}
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Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}
Mapping: {{mapping| 1 -5 -5 -10 2 | 0 18 20 35 4 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.152
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.152


{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 153be, 194be, 235bcee }}
{{Optimal ET sequence|legend=0| 11cd, 30d, 41, 153be, 194be, 235bcee }}


Badness: 0.070917
Badness: 0.070917
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Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}
Mapping: {{mapping| 1 -5 -5 -10 2 -8 | 0 18 20 35 4 32 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.119
Optimal tuning (POTE): ~2 = 1200.000, ~9/7 = 439.119


{{Optimal ET sequence|legend=1| 11cdf, 30df, 41 }}
{{Optimal ET sequence|legend=0| 11cdf, 30df, 41 }}


Badness: 0.052835
Badness: 0.052835
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{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}
{{Mapping|legend=1| 19 0 14 -7 | 0 1 1 2 }}


[[Optimal tuning]] ([[POTE]]): ~392/375 = 1\19, ~3/2 = 704.166
[[Optimal tuning]] ([[POTE]]): ~392/375 = 63.158, ~3/2 = 704.166


{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114, 133, 247b, 380bcd }}
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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 = 1\19, ~3/2 = 705.667
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667


{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114e }}
{{Optimal ET sequence|legend=0| 19, 76bcd, 95, 114e }}


Badness: 0.101496
Badness: 0.101496
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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 = 1\19, ~3/2 = 705.801
Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801


{{Optimal ET sequence|legend=1| 19, 76bcdf, 95, 114e }}
{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e }}


Badness: 0.053197
Badness: 0.053197


== Magus ==
== Magus ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Magus]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''


Magus temperament tempers out [[50331648/48828125]] (salegu) in the 5-limit. This temperament can be described as 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 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]].
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
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{{Multival|legend=1| 11 1 27 -24 12 60 }}
{{Multival|legend=1| 11 1 27 -24 12 60 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.465
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~5/4 = 391.465


{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}
Line 315: Line 315:
Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}
Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.503
Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.503


{{Optimal ET sequence|legend=1| 46, 95, 141bc }}
{{Optimal ET sequence|legend=0| 46, 95, 141bc }}


Badness: 0.045108
Badness: 0.045108
Line 328: Line 328:
Mapping: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}
Mapping: {{mapping| 1 -2 2 -6 -6 5 | 0 11 1 27 29 -4 }}


Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.366
Optimal tuning (POTE): ~2 = 1200.000, ~5/4 = 391.366


{{Optimal ET sequence|legend=1| 46, 233bcff, 279bccff }}
{{Optimal ET sequence|legend=0| 46, 233bcff, 279bccff }}


Badness: 0.043024
Badness: 0.043024


== 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]].''


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.536
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~3/2 = 704.536


{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}
Line 358: Line 358:
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 = 1\1, ~3/2 = 704.554
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.554


{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 264b, 373b, 637bbe }}
{{Optimal ET sequence|legend=0| 17, 29c, 46, 109, 264b, 373b, 637bbe }}


Badness: 0.050679
Badness: 0.050679
Line 371: Line 371:
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 = 1\1, ~3/2 = 704.571
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.571


{{Optimal ET sequence|legend=1| 17, 29c, 46, 63, 109 }}
{{Optimal ET sequence|legend=0| 17, 29c, 46, 63, 109 }}


Badness: 0.032727
Badness: 0.032727
Line 384: Line 384:
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 = 1\1, ~3/2 = 704.540
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540


{{Optimal ET sequence|legend=1| 17g, 29cg, 46, 109, 155f, 264bfg }}
{{Optimal ET sequence|legend=0| 17g, 29cg, 46, 109, 155f, 264bfg }}


Badness: 0.026243
Badness: 0.026243
Line 397: Line 397:
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 = 1\1, ~3/2 = 704.537
Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537


{{Optimal ET sequence|legend=1| 17, 29c, 46, 109g, 155fg, 264bfgg }}
{{Optimal ET sequence|legend=0| 17, 29c, 46, 109g, 155fg, 264bfgg }}


Badness: 0.026774
Badness: 0.026774
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