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]].

[[Subgroup]]: 3.5.7

[[Comma list]]: 245/243

: mapping generators: ~3, ~9/7

[[Optimal tuning]]s:  

* [[WE]]: ~3 = 1903.7398{{c}}, ~9/7 = 440.9014{{c}}

: [[error map]]: {{val| +1.785 -0.771 -2.248 }}

* [[CWE]]: ~3 = 1901.9550{{c}}, ~9/7 = 440.6646{{c}}

: error map: {{val| 0.000 -3.030 -5.580 }}

[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]], [[367edt|b367cdd]], [[462edt|b462cdd]]

[[Badness]] (Sintel): 0.0659

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

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

* [[Godzilla]] (+49/48 or 81/80) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]

* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]

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

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

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

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

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

 

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

 

 

== Bohpier ==

{{Main| Bohpier }}

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

 

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]].

 

 

[[Subgroup]]: 2.3.5.7

 

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

 

: mapping generators: ~2, ~27/25

 

[[Optimal tuning]]s:  

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

: [[error map]]: {{val| -0.003 +2.203 -3.314 +0.068 }}

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

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

 

* [[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 }}

 

{{Optimal ET sequence|legend=1| 8d, …, 41, 131, 172, 213c }}

 

 

=== 11-limit ===

Subgroup: 2.3.5.7.11

 

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

 

 

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

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

 

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

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

 

 

 

==== 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 tunings:  

* WE: ~2 = 1198.5478{{c}}, ~12/11 = 146.4252{{c}}

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

 

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

: unchanged-interval (eigenmonzo) basis: 2.5

 

{{Optimal ET sequence|legend=0| 8d, …, 41, 90ef }}

 

 

=== 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

 

 

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

: mapping generators: ~2, ~77/75

 

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

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

 

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

 

Badness (Sintel): 5.38

 

==== 13-limit ====

 

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

 

 

* WE: ~2 = 1199.9962{{c}}, ~77/75 = 48.8219{{c}}

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

 

 

Badness (Sintel): 3.39

 

== Pycnic ==

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''

 

Pycnic is related to [[triton]], but its mapping differs for the [[7/1|7th harmonic]]. It is also related to [[liese]], from which its mapping differs for the [[5/1|5th harmonic]].

 

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.

 

[[Subgroup]]: 2.3.5.7

 

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

 

{{Mapping|legend=1| 1 0 6 -3 | 0 3 -7 11 }}

: mapping generators: ~2, ~64/45

 

* [[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 }}

 

 

[[Badness]] (Sintel): 1.87

 

== Xenia ==

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''

 

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]].

 

[[Subgroup]]: 2.3.5.7

 

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

 

{{Mapping|legend=1| 1 -6 -12 -9 | 0 9 17 14 }}

: 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}}

 

{{Optimal ET sequence|legend=1| 19, 70d, 89d }}

 

 

== Magus ==

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

 

 

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

 

 

[[Optimal tuning]]s:  

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

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

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.4129{{c}}

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

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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