Diaschismic extensions: Difference between revisions

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{{Infobox Regtemp

| Title = Srutal; srutal archagall

| Subgroups = 2.3.5, 2.3.5.17

| Comma basis = [[15625/15552]] (2.3.5); <br> [[325/324]], [[625/624]] (2.3.5.13)

| Edo join 1 = 12 | Edo join 2 = 22

| Generator = 16/15 | Generator tuning = 104.898 | Optimization method = POTE

| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]

| Mapping = 2; 1 -2 1

| Odd limit 1 = 5 | Mistuning 1 = ??? | Complexity 1 = 22

| Odd limit 2 = (2.3.5.17) 17 | Mistuning 2 = ??? | Complexity 2 = 22

}}

'''Srutal''' ~~and~~ '''diaschismic''', ~~both discussed at~~ [[~~diaschismic family~~]], ~~are~~ two ~~different temperaments in~~ the ~~7-limit~~. ~~This page compares~~ and ~~contrasts them in detail~~.

'''Srutal''', known interchangeably as '''diaschismic''' in the [[5-limit]], is a [[regular temperament]] defined by [[tempering out]] the comma [[2048/2025]], the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and therefore as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]].

~~Temperaments of diaschismic family has a half~~-~~octave period and tempers out 2048/2025,~~ [[~~diaschisma~~]]~~. Not only the fifth or fourth, but also the diatonic semitone~~ (~~~16/15~~) ~~can be used as a generator. Extending~~ diaschismic ~~temperament to the 7-limit~~, there are ~~several~~ mappings that are comparable in complexity and error: [[pajara]] (12&22), keen (22&34~~), srutal (34d&46) and diaschismic (46&58~~).

== 7-limit extensions ==

The two alternative names for this temperament are assigned to different strong extensions to the [[7-limit]]: srutal (34d&46) and diaschismic (46&58), though there are other mappings that are comparable in complexity and error: [[pajara]] (12&22) and keen (22&34).

In srutal, 7/4 is represented by 15 diatonic semitones minus half octave, or five 6/5s minus half octave.

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 {{URWTC}}
+{{Infobox Regtemp
+| Title = Srutal; srutal archagall
+| Subgroups = 2.3.5, 2.3.5.17
+| Comma basis = [[15625/15552]] (2.3.5); <br> [[325/324]], [[625/624]] (2.3.5.13)
+| Edo join 1 = 12 | Edo join 2 = 22
+| Generator = 16/15 | Generator tuning = 104.898 | Optimization method = POTE
+| MOS scales = [[2L 8s]], [[10L 2s]], [[12L 10s]]
+| Mapping = 2; 1 -2 1
+| Odd limit 1 = 5 | Mistuning 1 = ??? | Complexity 1 = 22
+| Odd limit 2 = (2.3.5.17) 17 | Mistuning 2 = ??? | Complexity 2 = 22
+}}
-'''Srutal''' and '''diaschismic''', both discussed at [[diaschismic family]], are two different temperaments in the 7-limit. This page compares and contrasts them in detail.
+'''Srutal''', known interchangeably as '''diaschismic''' in the [[5-limit]], is a [[regular temperament]] defined by [[tempering out]] the comma [[2048/2025]], the diaschisma. The octave is split into two periods, each representing [[~]][[45/32]]~[[64/45]]; and the [[generator]] can be considered to be a perfect fifth (~[[3/2]]), or a perfect fifth less a period, which is a diatonic semitone of ~[[16/15]]. Tempering out the diaschisma implies that two of these semitones are equated to [[9/8]], and therefore as [[9/8]] = ([[18/17]])([[17/16]]), ~[[16/15]] can very naturally be equated to 17/16 and 18/17 as well, producing a 2.3.5.17 [[subgroup]] extension known as '''srutal archagall''', whose commas are [[136/135]] and [[256/255]].
-Temperaments of diaschismic family has a half-octave period and tempers out 2048/2025, [[diaschisma]]. Not only the fifth or fourth, but also the diatonic semitone (~16/15) can be used as a generator. Extending diaschismic temperament to the 7-limit, there are several mappings that are comparable in complexity and error: [[pajara]] (12&amp;22), keen (22&amp;34), srutal (34d&amp;46) and diaschismic (46&amp;58).
+== 7-limit extensions ==
+The two alternative names for this temperament are assigned to different strong extensions to the [[7-limit]]: srutal (34d&amp;46) and diaschismic (46&amp;58), though there are other mappings that are comparable in complexity and error: [[pajara]] (12&amp;22) and keen (22&amp;34).
 In srutal, 7/4 is represented by 15 diatonic semitones minus half octave, or five 6/5s minus half octave.