Sensamagic clan: Difference between revisions

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

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

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

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== Pycnic ==

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

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

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

Magus temperament tempers out [[50331648/48828125]] ~~(salegu)~~ in the 5-limit. This temperament can be described as {{nowrap| 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 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 ~~(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]].

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

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== Leapweek ==

: ''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

@@ Line 60: / Line 60: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Bohpier]].''
-Bohpier is named after its interesting [[relationship between Bohlen–Pierce and octave-ful temperaments|relationship with the non-octave Bohlen–Pierce equal temperament]].
+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
@@ Line 208: / Line 210: @@
 == Pycnic ==
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Stump]].''
+Pycnic is related to [[triton]], but its mapping differs for the [[5/1|5th harmonic]]. It is also related to [[liese]], from which its mapping differs for the [[7/1|7th 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 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.
@@ Line 331: / Line 335: @@
 : ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''
-Magus temperament tempers out [[50331648/48828125]] (salegu) in the 5-limit. This temperament can be described as {{nowrap| 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 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 (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]])<sup>3</sup>, 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)<sup>2</sup> 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]].
+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]])<sup>3</sup>, 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)<sup>2</sup> 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
@@ Line 383: / Line 387: @@
 == Leapweek ==
 : ''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