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]].

== BPS ==

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]].

[[Subgroup]]: 3.5.7

 

 

 

 

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

 

 

 

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.  

 

* [[Father]] (+16/15 or 28/27) → [[Father family #Father|Father family]]

 

 

 

: ''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]].

 

 

[[Comma list]]: 245/243, 3125/3087

 

 

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

 

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

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

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

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

 

 

 

Subgroup: 2.3.5.7.11

 

Comma list: 100/99, 245/243, 1344/1331

 

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

 

Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.545

 

Minimax tuning:  

* 11-odd-limit: ~12/11 = {{monzo| 1/7 1/7 0 0 -1/14 }}

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

 

 

 

==== 13-limit ====

 

 

 

Optimal tuning (POTE): ~2 = 1200.000, ~12/11 = 146.603

 

* 13- and 15-odd-limit: ~12/11 = {{monzo| 0 0 1/19 }}

 

 

 

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.

 

 

Comma list: 245/243, 1331/1323, 3125/3087

 

 

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

 

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

 

 

 

Comma list: 245/243, 275/273, 847/845, 1331/1323

 

 

Optimal tuning (POTE): ~2 = 1200.000, ~77/75 = 48.822

 

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

 

 

 

 

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

 

 

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

 

 

 

 

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

 

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

 

 

 

 

=== 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 }}

 

 

 

 

== 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.

 

 

 

{{Mapping|legend=1| 1 3 -1 8 | 0 -3 7 -11 }}

 

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

 

 

[[Badness]]: 0.073735

 

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''

 

 

[[Comma list]]: 245/243, 78125/76832

 

 

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

 

 

 

=== 11-limit ===

 

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

 

 

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

 

 

 

Subgroup: 2.3.5.7.11.13

 

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

 

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

 

 

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

 

Badness: 0.052835

 

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

 

 

[[Comma list]]: 245/243, 395136/390625

 

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

 

 

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

 

 

 

Comma list: 245/243, 2560/2541, 3773/3750

 

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

 

Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.667

 

 

Badness: 0.101496

 

Subgroup: 2.3.5.7.11.13

 

Comma list: 196/195, 245/243, 832/825, 1001/1000

 

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

 

Optimal tuning (POTE): ~33/32 = 63.158, ~3/2 = 705.801

 

 

 

: ''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 {{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]])³, 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)² 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]].

 

 

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

 

{{Mapping|legend=1| 1 -2 2 -6 | 0 11 1 27 }}

 

 

 

[[Badness]]: 0.108417

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

Mapping: {{mapping| 1 -2 2 -6 -6 | 0 11 1 27 29 }}

 

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

 

 

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

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

 

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

 

 

 

 

 

 

[[Comma list]]: 245/243, 2097152/2066715

 

{{Mapping|legend=1| 1 0 42 -21 | 0 1 -25 15 }}

 

: mapping generators: ~2, ~3

 

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

 

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

 

[[Badness]]: 0.140577

 

Subgroup: 2.3.5.7.11

 

Comma list: 245/243, 385/384, 1331/1323

 

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

 

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.554

 

 

 

=== 13-limit ===

 

Comma list: 169/168, 245/243, 352/351, 364/363

 

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

 

 

 

Badness: 0.032727

 

==== 17-limit ====

 

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 }}

 

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.540

 

 

Badness: 0.026243

 

Subgroup: 2.3.5.7.11.13.17

 

 

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

 

Optimal tuning (POTE): ~2 = 1200.000, ~3/2 = 704.537

 

 

 

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