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

 

 

 

 

 

 

 

 

 

 

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

 

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

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

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

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

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

[[Badness]] (Sintel): 1.73

Subgroup: 2.3.5.7.11

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

 

* 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

{{Optimal ET sequence|legend=0| 8d, …, 41, 90e, 131e }}

Badness (Sintel): 1.12

 

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

 

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

 

 

 

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 tunings:  

* 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

Subgroup: 2.3.5.7.11.13

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

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

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

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

Badness (Sintel): 3.39

: ''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 }}

[[Optimal tuning]]s:  

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

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

{{Optimal ET sequence|legend=1| 17, 19, 55c, 74cd, 93cdd }}

[[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

: mapping generators: ~2, ~9/5

: [[error map]]: {{val| +1.086 -0.020 +5.507 -9.898 }}

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

: error map: {{val| 0.000 -0.976 +4.424 -11.748 }}

[[Badness]] (Sintel): 2.25

: ''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]].

[[Subgroup]]: 2.3.5.7

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

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

[[Optimal tuning]]s:  

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

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

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

[[Badness]] (Sintel): 2.74

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

 

 

 

 

 

 

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

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}}

{{Optimal ET sequence|legend=0| 3de, 43de, 46 }}

Badness (Sintel): 1.78

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 }}

: [[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 }}

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

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

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

Badness (Sintel): 3.36

=== 13-limit ===

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}}

{{Optimal ET sequence|legend=0| 19, 76bcdf, 95, 114e, 209bcef }}

Badness (Sintel): 2.20

: ''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]]s:  

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

{{Optimal ET sequence|legend=1| 11cd, 30d, 41 }}

[[Badness]] (Sintel): 3.53

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 }}

* WE: ~2 = 1199.5116{c}}, ~9/7 = 438.9734{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 439.1362{{c}}

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

Badness (Sintel): 2.34

=== 13-limit ===

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}}

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

: ''Not to be confused with scales produced by leap week calendars such as [[Symmetry454]].''

Leapweek may be described as the {{nowrap| 46 & 63 }} temperament, generated by a perfect fifth and being a strong extension of [[leapfrog]]. [[109edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

: mapping generators: ~2, ~3

* [[WE]]: ~2 = 1199.6301{{c}}, ~3/2 = 704.3191{{c}}

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

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5387{{c}}

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

[[Badness]] (Sintel): 3.56

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 tunings:  

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

{{Optimal ET sequence|legend=0| 17, 46, 109, 264b }}

Badness (Sintel): 1.68

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}}

{{Optimal ET sequence|legend=0| 17, 46, 63, 109 }}

Badness (Sintel): 1.35

==== 17-limit ====

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 }}

Optimal tunings:  

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

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

{{Optimal ET sequence|legend=0| 17g, 46, 109 }}

Badness (Sintel): 1.34

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 169/168, 221/220, 245/243, 364/363

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 17, 46, 109g, 155fg }}

Badness (Sintel): 1.36

[[Category:Sensamagic clan| ]] <!-- main article -->

[[Category:Temperament clans]]

[[Category:Catalogs of rank-2 temperaments]]