Sensamagic clan: Difference between revisions

{{Main|Bohlen–Pierce–Stearns}}

''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]].

{{mapping|legend=2| 1 1 2 | 0 -2 1 }}

[[Optimal tuning]] ([[POTE]]): ~3 = 1\1edt, ~9/7 = 440.4881

* [[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]]. Pental 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.  

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

* ''[[Pental (temperament)|Pental]]'' (+16807/16384) → [[Pental family #Septimal pental|Pental family]]

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

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

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 146.474

: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.5

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

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.545

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

{{Optimal ET sequence|legend=1| 41, 90e, 131e }}

Optimal tuning (POTE): ~2 = 1\1, ~12/11 = 146.603

: Eigenmonzo (unchanged-interval) basis: 2.5

{{Optimal ET sequence|legend=1| 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.  

Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.828

{{Optimal ET sequence|legend=1| 49, 123ce, 172 }}

Optimal tuning (POTE): ~2 = 1\1, ~77/75 = 48.822

{{Optimal ET sequence|legend=1| 49f, 123ce, 172f, 295ce, 467bccef }}

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~128/105 = 351.049

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.014

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

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 351.025

{{Optimal ET sequence|legend=1| 17, 24, 41, 106df, 147df }}

{{See also| High badness temperaments #Stump }}

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~45/32 = 567.720

{{See also| Shibboleth family }}

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 439.076

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.152

{{Optimal ET sequence|legend=1| 11cd, 30d, 41, 153be, 194be, 235bcee }}

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 439.119

{{Optimal ET sequence|legend=1| 11cdf, 30df, 41 }}

[[Optimal tuning]] ([[POTE]]): ~392/375 = 1\19, ~3/2 = 704.166

Optimal tuning (POTE): ~33/32 = 1\19, ~3/2 = 705.667

{{Optimal ET sequence|legend=1| 19, 76bcd, 95, 114e }}

Optimal tuning (POTE): ~33/32 = 1\19, ~3/2 = 705.801

{{Optimal ET sequence|legend=1| 19, 76bcdf, 95, 114e }}

: ''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). 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]].

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 391.465

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.503

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

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 391.366

{{Optimal ET sequence|legend=1| 46, 233bcff, 279bccff }}

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 704.536

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.554

{{Optimal ET sequence|legend=1| 17, 29c, 46, 109, 264b, 373b, 637bbe }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.571

{{Optimal ET sequence|legend=1| 17, 29c, 46, 63, 109 }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.540

{{Optimal ET sequence|legend=1| 17g, 29cg, 46, 109, 155f, 264bfg }}

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.537

{{Optimal ET sequence|legend=1| 17, 29c, 46, 109g, 155fg, 264bfgg }}