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.  

* ''[[Father]]'', {16/15, 28/27} → [[Father family #Father|Father family]]

* [[Rodan]], {245/243, 1029/1024} → [[Gamelismic clan #Rodan|Gamelismic clan]]

* ''[[Shrutar]]'', {245/243, 2048/2025} → [[Diaschismic family #Shrutar|Diaschismic family]]

* ''[[Clyde]]'', {245/243, 3136/3125} → [[Kleismic family #Clyde|Kleismic family]]

Tempering out 245/243 alone in the full 7-limit leads to a [[Planar temperament|rank-3 temperament]], [[sensamagic]], for which [[283edo|283EDO]] is the [[optimal patent val]].

The ''BPS'', for ''Bohlen–Pierce–Stearns'', is the 3.5.7 subgroup temperament tempering out 245/243. This subgroup temperament was formerly called as ''lambda temperament'', which was named after [[4L 5s (tritave-equivalent)|lambda scale]].

 

Subgroup: 3.5.7

[[Sval]] [[mapping]]: [{{val| 1 1 2 }}, {{val| 0 -2 1 }}]

 

 

 

 

 

 

 

 

[[Mapping]]: [{{val| 1 -1 -1 -2 }}, {{val| 0 7 9 13 }}]

 

 

 

 

{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Subgroup: 2.3.5.7.11.13

 

 

Mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]

 

 

Optimal GPV sequence: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}

 

Badness: 0.020789

 

 

 

 

 

 

 

 

 

Mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 11 }}]

POTE generator: ~9/7 = 443.506

Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}

Badness: 0.023258

=== Hemisensi ===

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 245/242

Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]

POTE generator: ~25/22 = 221.605

Optimal GPV sequence: {{Val list| 27e, 65, 157de, 222cde }}

Badness: 0.048714

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Bohpier]].''

'''[[Bohpier]]''' is named after its [[Relationship between Bohlen-Pierce and octave-ful temperaments|interesting relationship with the non-octave Bohlen-Pierce equal temperament]].

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 13 19 23 }}]

 

[[POTE generator]]: ~27/25 = 146.474

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

: Eigenmonzos (unchanged intervals): 2, 5/4

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

: Eigenmonzos (unchanged intervals): 2, 4/3

{{Val list|legend=1| 41, 131, 172, 213c }}

Mapping: [{{val| 1 0 0 0 2 }}, {{val| 0 13 19 23 12 }}]

POTE generator: ~12/11 = 146.545

: Eigenmonzos (unchanged intervals): 2, 11/9

Optimal GPV sequence: {{Val list| 41, 90e, 131e }}

Mapping: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]

POTE generator: ~12/11 = 146.603

: Eigenmonzos (unchanged intervals): 2, 5/4

Optimal GPV sequence: {{Val list| 41, 90ef, 131ef, 221bdeff }}

'''Triboh''' is named after "[[39edt|Triple Bohlen-Pierce scale]]", which divides each step of the [[13edt|equal-tempered]] [[Bohlen-Pierce]] scale into three equal parts.  

Mapping: [{{val| 1 0 0 0 0 }}, {{val| 0 39 57 69 85 }}]

POTE generator: ~77/75 = 48.828

Optimal GPV sequence: {{Val list| 49, 123ce, 172 }}

Mapping: [{{val| 1 0 0 0 0 0 }}, {{val| 0 39 57 69 85 91 }}]

POTE generator: ~77/75 = 48.822

Optimal GPV sequence: {{Val list| 49f, 123ce, 172f, 295ce, 467bccef }}

{{see also| Schismatic family }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 1 7 -1 }}, {{val| 0 2 -16 13 }}]

{{Multival|legend=1| 2 -16 13 -30 15 75 }}

 

{{Val list|legend=1| 17, 24, 41, 106d, 147d, 188cd, 335cd }}

Mapping: [{{val| 1 1 7 -1 2 }}, {{val| 0 2 -16 13 5 }}]

POTE generator: ~11/9 = 351.014

Optimal GPV sequence: {{Val list| 17, 24, 41, 106d, 147d }}

Mapping: [{{val| 1 1 7 -1 2 4 }}, {{val| 0 2 -16 13 5 -1 }}]

POTE generator: ~11/9 = 351.025

Optimal GPV sequence: {{Val list| 17, 24, 41, 106df, 147df }}

{{see also| High badness 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

[[Mapping]]: [{{val| 1 3 -1 8 }}, {{val| 0 -3 7 -11 }}]

{{Multival|legend=1| 3 -7 11 -18 9 45 }}

 

{{Val list|legend=1| 17, 19, 55c, 74cd, 93cdd }}

== Cohemiripple ==

{{see also| Ripple family }}

Subgroup: 2.3.5.7

[[Comma list]]: 245/243, 1323/1250

[[Mapping]]: [{{val| 1 -3 -5 -5 }}, {{val| 0 10 16 17 }}]

{{Multival|legend=1| 10 16 17 2 -1 -5 }}

[[POTE generator]]: ~7/5 = 549.944

 

[[Badness]]: 0.190208

Comma list: 77/75, 243/242, 245/242

Mapping: [{{val| 1 -3 -5 -5 -8 }}, {{val| 0 10 16 17 25 }}]

POTE generator: ~7/5 = 549.945

Optimal GPV sequence: {{Val list| 11cdee, 13cdee, 24 }}

Badness: 0.082716

Comma list: 66/65, 77/75, 147/143, 243/242

Mapping: [{{val| 1 -3 -5 -5 -8 -5 }}, {{val| 0 -10 -16 -17 -25 -19 }}]

POTE generator: ~7/5 = 549.958

Optimal GPV sequence: {{Val list| 11cdeef, 13cdeef, 24 }}

Badness: 0.049933

== Superthird ==

Subgroup: 2.3.5.7

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

[[Mapping]]: [{{val| 1 -5 -5 -10 }}, {{val| 0 18 20 35 }}]

{{Multival|legend=1| 18 20 35 -10 5 25 }}

[[POTE generator]]: ~9/7 = 439.076

 

[[Badness]]: 0.139379

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

Mapping: [{{val| 1 -5 -5 -10 2 }}, {{val| 0 18 20 35 4 }}]

POTE generator: ~9/7 = 439.152

Optimal GPV sequence: {{Val list| 11cd, 30d, 41, 153be, 194be, 235bcee }}

Badness: 0.070917

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

Mapping: [{{val| 1 -5 -5 -10 2 -8 }}, {{val| 0 18 20 35 4 32 }}]

POTE generator: ~9/7 = 439.119

Optimal GPV sequence: {{Val list| 11cdf, 30df, 41 }}

Badness: 0.052835

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Magus]].''

Magus temperament tempers out 50331648/48828125 (salegu) in the 5-limit. This temperament can be described as 46&49 temperament, which tempers out the sensamagic and 28672/28125 (sazoquingu). Alternative extension [[Starling temperaments #Amigo|amigo]] (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.

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 -2 2 -6 }}, {{val| 0 11 1 27 }}]

 

{{Multival|legend=1| 11 1 27 -24 12 60 }}

[[POTE generator]]: ~5/4 = 391.465

{{Val list|legend=1| 46, 95, 141bc, 187bc, 328bbcc }}

Mapping: [{{val| 1 -2 2 -6 -6 }}, {{val| 0 11 1 27 29 }}]

POTE generator: ~5/4 = 391.503

Optimal GPV sequence: {{Val list| 46, 95, 141bc }}

Mapping: [{{val| 1 -2 2 -6 -6 5 }}, {{val| 0 11 1 27 29 -4 }}]

POTE generator: ~5/4 = 391.366

Optimal GPV sequence: {{Val list| 46, 233bcff, 279bccff }}

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 1 17 -6 }}, {{val| 0 1 -25 15 }}]

[[POTE generator]]: ~3/2 = 704.536

{{Val list|legend=1| 17, 29c, 46, 109, 155, 264b, 419b }}

Mapping: [{{val| 1 1 17 -6 -3 }}, {{val| 0 1 -25 15 11 }}]

POTE generator: ~3/2 = 704.554

Optimal GPV sequence: {{Val list| 17, 29c, 46, 109, 264b, 373b, 637bbe }}

Mapping: [{{val| 1 1 17 -6 -3 -1 }}, {{val| 0 1 -25 15 11 8 }}]

POTE generator: ~3/2 = 704.571

Optimal GPV sequence: {{Val list| 17, 29c, 46, 63, 109 }}

Mapping: [{{val| 1 1 17 -6 -3 -1 -10 }}, {{val| 0 1 -25 15 11 8 24 }}]

POTE generator: ~3/2 = 704.540

Optimal GPV sequence: {{Val list| 17g, 29cg, 46, 109, 155f, 264bfg }}

Mapping: [{{val| 1 1 17 -6 -3 -1 17 }}, {{val| 0 1 -25 15 11 8 -22 }}]

POTE generator: ~3/2 = 704.537

Optimal GPV sequence: {{Val list| 17, 29c, 46, 109g, 155fg, 264bfgg }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Regular temperament theory]]

[[Category:Temperament clan]]