Sensamagic clan: Difference between revisions

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 ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called the ''lambda'' temperament, which was named after the [[4L 5s (tritave-equivalent)|lambda scale]].

[[Sval]] [[mapping]]: [{{val| 1 1 2 }}, {{val| 0 -2 1 }}]

Sval mapping generators: ~3, ~9/7

[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881

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

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 tempers out [[126/125]], [[686/675]] and [[4375/4374]] in addition to [[245/243]], and can be described as the 19 & 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of 8-, 11-, 19- and 27-tones are available.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 136/135, 154/153, 169/168

Mapping: [{{val| 1 -1 -1 -2 -8 0 -7 }}, {{val| 0 7 9 13 31 10 30 }}]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.551

{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}

 

 

 

 

 

 

 

 

 

 

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 443.506

 

{{Optimal ET sequence|legend=1| 19e, 27, 46ee }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{{Optimal ET sequence|legend=1| 11cdeef, 13cdeef, 24 }}

 

Badness: 0.049933

'''[[Bohpier]]''' is named after its [[Relationship between Bohlen-Pierce and octave-ful temperaments|interesting relationship with the non-octave Bohlen-Pierce equal temperament]].

[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 13 19 23 }}]

{{Multival|legend=1| 13 19 23 0 0 0 }}

 

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

* 7-odd-limit: ~27/25 = {{monzo| 0 0 1/19 }}

: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5

* 9-odd-limit: ~27/25 = {{monzo| 0 1/13 }}

: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.3

{{Optimal ET sequence|legend=1| 41, 131, 172, 213c }}

[[Badness]]: 0.068237

Mapping: [{{val| 1 0 0 0 2 }}, {{val| 0 13 19 23 12 }}]

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.545

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

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

Badness: 0.033949

Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]

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

: Eigenmonzo basis (unchanged-interval basis): 2.5

 

Badness: 0.024864

by [[Chris Vaisvil]]:

* [http://micro.soonlabel.com/bophier/bophier-1.mp3 bophier-1.mp3]

'''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.  

Mapping: [{{val| 1 0 0 0 0 }}, {{val| 0 39 57 69 85 }}]

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

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

Badness: 0.162592

Mapping: [{{val| 1 0 0 0 0 0 }}, {{val| 0 39 57 69 85 91 }}]

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

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

Badness: 0.082158

== Salsa ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 32805/32768

[[Mapping]]: [{{val| 1 1 7 -1 }}, {{val| 0 2 -16 13 }}]

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

{{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}

Subgroup: 2.3.5.7.11

Comma list: 243/242, 245/242, 385/384

Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]

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

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

Badness: 0.039444

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 243/242, 245/242

Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]

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

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

 

 

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.

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

 

 

 

{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}

[[Badness]]: 0.073735

{{See also| Shibboleth family }}

[[Subgroup]]: 2.3.5.7

 

 

 

 

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

 

 

[[Badness]]: 0.139379

Comma list: 100/99, 245/243, 78125/76832

Mapping: [{{val| 1 -5 -5 -10 2 }}, {{val| 0 18 20 35 4 }}]

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

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

Badness: 0.070917

Comma list: 100/99, 144/143, 196/195, 1375/1352

Mapping: [{{val| 1 -5 -5 -10 2 -8 }}, {{val| 0 18 20 35 4 32 }}]

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

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

Badness: 0.052835

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

[[Mapping]]: [{{val| 19 0 14 -7 }}, {{val| 0 1 1 2 }}]

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

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

[[Badness]]: 0.132311

Mapping: [{{val| 19 0 14 -7 96 }}, {{val| 0 1 1 2 -1 }}]

Optimal tuning (POTE): ~33/32 = 1\19, ~3/2 = 705.667

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

Badness: 0.101496

Mapping: [{{val| 19 0 14 -7 96 10 }}, {{val| 0 1 1 2 -1 2 }}]

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

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

Badness: 0.053197

== Magus ==

: ''For the 5-limit version of this temperament, see [[High badness 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]] (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.

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

 

{{Multival|legend=1| 11 1 27 -24 12 60 }}

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

{{Optimal ET sequence|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}

[[Badness]]: 0.108417

Comma list: 176/175, 245/243, 1331/1323

Mapping: [{{val| 1 -2 2 -6 -6 }}, {{val| 0 11 1 27 29 }}]

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

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

Badness: 0.045108

Comma list: 91/90, 176/175, 245/243, 1331/1323

Mapping: [{{val| 1 -2 2 -6 -6 5 }}, {{val| 0 11 1 27 29 -4 }}]

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

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

Badness: 0.043024

:''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''

[[Mapping]]: [{{val| 1 0 42 -21 }}, {{val| 0 1 -25 15 }}]

 

Mapping generators: ~2, ~3

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

{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}

[[Badness]]: 0.140577

Mapping: [{{val| 1 0 42 -21 -14 }}, {{val| 0 1 -25 15 11 }}]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.554

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

Badness: 0.050679

Mapping: [{{val| 1 0 42 -21 -14 -9 }}, {{val| 0 1 -25 15 11 8 }}]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.571

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

Badness: 0.032727

Mapping: [{{val| 1 0 42 -21 -14 -9 -34 }}, {{val| 0 1 -25 15 11 8 24 }}]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.540

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

Badness: 0.026243

Mapping: [{{val| 1 0 42 -21 -14 -9 39 }}, {{val| 0 1 -25 15 11 8 -22 }}]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.537

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

Badness: 0.026774

[[Category:Sensamagic clan| ]] <!-- main article -->

[[Category:Rank 2]]

[[Category:Listen]]