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]]

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 ''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 }}

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

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

 

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

 

[[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

 

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

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

 

 

Badness: 0.033949

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

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

 

Mapping: {{mapping| 1 0 0 0 2 2 | 0 13 19 23 12 14 }}

 

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

 

Minimax tuning:  

* 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 }}

 

 

=== Triboh ===

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

 

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 }}

 

Badness: 0.162592

 

==== 13-limit ====

 

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

 

Mapping: {{mapping| 1 0 0 0 0 0 | 0 39 57 69 85 91 }}

 

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

 

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

 

 

== Salsa ==

 

 

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

 

 

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

 

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

 

[[Badness]]: 0.080152

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

 

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

 

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

 

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

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

 

 

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

 

 

 

== 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.

 

[[Subgroup]]: 2.3.5.7

 

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

 

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

 

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

 

 

 

== 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 }}

 

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

 

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

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

 

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

 

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

 

 

=== 13-limit ===

 

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

 

 

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

 

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

 

 

== Superenneadecal ==

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

 

[[Subgroup]]: 2.3.5.7

 

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

 

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

 

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

 

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

 

[[Badness]]: 0.132311

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

 

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

 

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

 

Badness: 0.101496

 

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

 

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

 

 

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

 

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

 

Badness: 0.053197

 

 

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]].