Sensamagic clan: Difference between revisions

The '''sensamagic clan''' of [[regular temperament|temperaments]] [[tempering out|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 [[rank-3 temperament]], [[sensamagic]], for which [[283edo]] is the [[optimal patent val]].

{{Main| 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]].

 

 

 

 

 

[[Badness]] (Sintel): 0.0659

 

 

* ''[[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. Salsa tempers out [[32805/32768]], splitting the generator in fifteen. Finally, escaped tempers out [[65625/65536]], splitting the generator in sixteen.

 

* ''[[Salsa]]'' (+32805/32768) → [[Schismatic family #Salsa|Schismatic family]]

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

 

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

 

 

 

Bohpier tempers out 3125/3087 and may be described as the {{nowrap| 41 & 49 }} temperament. It is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].

 

[[41edo]] itself makes for an excellent tuning, though [[90edo]] and [[131edo]] are interesting alternatives. Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 146.4741{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[WE]] and/or [[CWE]] (shown below) is largely unnoticeable.

 

[[Subgroup]]: 2.3.5.7

 

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

 

: mapping generators: ~2, ~27/25

 

* [[WE]]: ~2 = 1199.9967{{c}}, ~27/25 = 146.4737{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 146.4739{{c}}

: error map: {{val| 0.000 +2.205 -3.310 +0.073 }}

 

[[Minimax tuning]]:  

: [[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]] (Sintel): 1.73

 

=== 11-limit ===

 

 

 

* WE: ~2 = 1199.2309{{c}}, ~12/11 = 146.4507{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.5009{{c}}

 

Minimax tuning:  

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

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

 

 

Badness (Sintel): 1.12

 

==== 13-limit ====

 

 

 

* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.5230{{c}}

 

* 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

 

 

* WE: ~2 = 1199.9966{{c}}, ~77/75 = 48.8281{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8282{{c}}

 

 

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

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

 

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

 

* CWE: ~2 = 1200.0000{{c}}, ~77/75 = 48.8219{{c}}

 

 

 

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #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 two cents sharp of it in the CWE 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 0 6 -3 | 0 3 -7 11 }}

 

* [[WE]]: ~2 = 1203.3437{{c}}, ~64/45 = 634.0416{{c}}

: [[error map]]: {{val| +3.344 +0.170 -4.542 -4.400 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 632.3502{{c}}

: error map: {{val| 0.000 -4.904 -12.765 -12.973 }}

 

 

 

 

Xenia is related to [[Starling temperaments #Xenial|xenial]], but its mapping differs for the [[7/1|7th harmonic]]. It may be described as {{nowrap| 19 & 51c }} or {{nowrap| 19 & 70d }}, which tempers out the sensamagic and keega, [[1029/1000]].

 

 

[[Comma list]]: 245/243, 1029/1000

 

: mapping generators: ~2, ~9/5

 

* [[WE]]: ~2 = 1201.0862{{c}}, ~9/5 = 1012.0503{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2199{{c}}

 

 

 

== Magus ==

 

Magus temperament tempers out [[50331648/48828125]] in the 5-limit. This temperament can be described as {{nowrap| 46 & 49 }} temperament, which tempers out the sensamagic and [[28672/28125]]. 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, and ~[[25/16]] is twice as sharp so that it makes sense to equate with [[11/7]] by tempering out [[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]].

 

[[Subgroup]]: 2.3.5.7

 

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

 

 

* [[WE]]: ~2 = 1198.7187{{c}}, ~5/4 = 391.0473{{c}}

: [[error map]]: {{val| -1.281 +2.128 +2.171 -2.860 }}

: error map: {{val| 0.000 +3.587 +5.099 -0.678 }}

 

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

 

[[Badness]] (Sintel): 2.74

 

 

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

 

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

 

 

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

 

 

 

 

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

 

* WE: ~2 = 1199.7708{{c}}, ~5/4 = 391.2912{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.3597{{c}}

 

 

 

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

 

 

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

 

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

: mapping generators: ~392/375, ~3

 

[[Optimal tuning]]s:  

* [[WE]]: ~392/375 = 63.1399{{c}}, ~3/2 = 703.9652{{c}}

: [[error map]]: {{val| -0.343 +1.668 +1.267 -3.560 }}

* [[CWE]]: ~392/375 = 63.1579{{c}}, ~3/2 = 703.9028{{c}}

: error map: {{val| 0.000 +1.948 +1.800 -3.126 }}

 

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

 

[[Badness]] (Sintel): 3.35

 

=== 11-limit ===

 

 

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

 

* WE: ~33/32 = 63.0966{{c}}, ~3/2 = 704.9824{{c}}

* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.3096{{c}}

 

 

 

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 }}

 

* WE: ~33/32 = 63.0988{{c}}, ~3/2 = 705.1402{{c}}

* CWE: ~33/32 = 63.1579{{c}}, ~3/2 = 705.4315{{c}}

 

 

 

 

[[Subgroup]]: 2.3.5.7

 

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

 

{{Mapping|legend=1| 1 -5 -5 -10 | 0 18 20 35 }}

 

* [[WE]]: ~2 = 1200.3935{{c}}, ~9/7 = 439.2199{{c}}

: [[error map]]: {{val| +0.394 +2.035 -3.884 -0.066 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 439.0931{{c}}

: error map: {{val| 0.000 +1.721 -4.452 -0.568 }}

 

 

 

=== 11-limit ===

 

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

 

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

 

 

 

 

=== 13-limit ===

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 tunings:  

* WE: ~2 = 1199.2631{c}}, ~9/7 = 438.8494{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.0943{{c}}

 

 

 

 

 

[[Subgroup]]: 2.3.5.7

 

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

 

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

 

: [[error map]]: {{val| -0.370 +1.994 -0.578 -1.821 }}

: error map: {{val| 0.000 +2.584 +0.218 -0.745 }}

 

 

 

=== 11-limit ===

 

 

 

* WE: ~2 = 1199.7910{{c}}, ~3/2 = 704.4312{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5542{{c}}

 

 

 

Subgroup: 2.3.5.7.11.13

 

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

 

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

 

* WE: ~2 = 1200.0070{{c}}, ~3/2 = 704.5751{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5709{{c}}

 

 

 

Subgroup: 2.3.5.7.11.13.17

 

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 }}

 

* WE: ~2 = 1199.8670{{c}}, ~3/2 = 704.4620{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5395{{c}}

 

 

Badness (Sintel): 1.34

 

Subgroup: 2.3.5.7.11.13.17

 

 

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

 

* WE: ~2 = 1200.1737{{c}}, ~3/2 = 704.6390{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5364{{c}}

 

 

 

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