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

[[Comma list]]: 245/243

{{Mapping|legend=2| 1 1 2 | 0 -2 1 }}

: sval mapping generators: ~3, ~9/7

[[Optimal tuning]] ([[POTE]]): ~3 = 1901.9550, ~9/7 = 440.4881

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

=== Overview to extensions ===

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, cohemiripple, hedgehog, and fourfives.  

* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]

* ''[[Hedgehog]]'' (+50/49) → [[Porcupine family #Hedgehog|Porcupine family]]

* ''[[Cohemiripple]]'' (+1323/1250) → [[Ripple family #Cohemiripple|Ripple 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.  

 

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

* ''[[Clyde]]'' (+3136/3125) → [[Kleismic family #Clyde|Kleismic family]]

* [[Superpyth]] (+64/63) → [[Archytas clan #Superpyth|Archytas clan]]

* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]

* ''[[Octacot]]'' (+2401/2400) → [[Tetracot family #Octacot|Tetracot family]]

* ''[[Hemiaug]]'' (+128/125) → [[Augmented family #Hemiaug|Augmented family]]

* ''[[Pentacloud]]'' (+16807/16384) → [[Quintile family #Pentacloud|Quintile family]]

* ''[[Bamity]]'' (+64827/64000) → [[Amity family #Bamity|Amity family]]

* [[Rodan]] (+1029/1024) → [[Gamelismic clan #Rodan|Gamelismic clan]]

* ''[[Shrutar]]'' (+2048/2025) → [[Diaschismic family #Shrutar|Diaschismic family]]

* ''[[Escaped]]'' (+65625/65536) → [[Escapade family #Escaped|Escapade family]]

 

For ''no-twos'' extensions, see [[No-twos subgroup temperaments #BPS]].

 

Considered below are bohpier, salsa, pycnic, superthird, magus and leapweek.

 

== 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]].

 

[[Subgroup]]: 2.3.5.7

 

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

 

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

 

[[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

 

 

[[Badness]]: 0.068237

 

=== 11-limit ===

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

 

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

 

Badness: 0.033949

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

Comma list: 100/99, 144/143, 196/195, 275/273

 

 

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

 

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

: Unchanged-interval (eigenmonzo) basis: 2.5

 

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

 

Badness: 0.024864

 

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

 

Mapping: {{mapping| 1 0 0 0 0 | 0 39 57 69 85 }}

 

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

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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]].