2.5 subgroup: Difference between revisions

(9 intermediate revisions by the same user not shown)

Line 8:

The non-trivial MOS scales that are produced are: [[1L 2s]], [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]], [[3L 16s]], [[3L 19s]], [[3L 22s]], [[3L 25s]], [[28L 3s]], [[28L 31s]], [[59L 28s]], [[59L 87s]], etc.

Equating 5/4 to the cube root of 2, or [[tempering out]] [[128/125]], results in the [[augmented]] temperament. There are regular temperaments generated by ~5/4, which have a one-to-one correspondence to the intervals of 2.5 subgroup the same way [[meantone]] does to [[Pythagorean tuning]]. These temperaments often exploit the 128/125 diesis, even if they don't temper it out: for example, [[magic]] temperament has a flat 5/4 of about 380 cents to equate (5/4)<~~sub~~>5</~~sub~~> octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/(128/125) to 3/2. [[Wurschmidt]] has a more accurate 5/4 and equates (5/4)<~~sub~~>8</~~sub~~> octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/(([128/125]])2) to 3/2.

Equating 5/4 to the cube root of 2, or [[tempering out]] [[128/125]], results in the [[augmented]] temperament, which has a [[period]] of ~5/4 and [[generator]] of ~3/2. This temperament may be familiar because [[12edo]] [[support]]s it, and 12edo's tuning of the Augmented[6] [[3L 3s]] scale is the {{w|augmented scale}} of standard music theory. There are regular temperaments generated by ~5/4, which have a one-to-one correspondence to the intervals of 2.5 subgroup the same way [[meantone]] does to [[Pythagorean tuning]]. These temperaments often exploit the 128/125 diesis, even though they don't temper it out: for example, [[magic]] temperament has a flat 5/4 of about 380 cents to equate (5/4)5 octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/(128/125) to 3/2. [[Wurschmidt]] has a more accurate 5/4 and equates (5/4)8 octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/((128/125)2) to 3/2. These can be seen as [[cluster temperament]]s using 128/125 as a versatile chroma.

== See also ==

* [[3-limit]] (2.3)

* [[3.5 subgroup]]

[[Category:Just intonation subgroups|#]]

[[Category:Rank-2 temperaments|#]]

[[Category:5-limit|#]]

@@ Line 8: / Line 8: @@
 The non-trivial MOS scales that are produced are: [[1L 2s]], [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]], [[3L 16s]], [[3L 19s]], [[3L 22s]], [[3L 25s]], [[28L 3s]], [[28L 31s]], [[59L 28s]], [[59L 87s]], etc.
-Equating 5/4 to the cube root of 2, or [[tempering out]] [[128/125]], results in the [[augmented]] temperament. There are regular temperaments generated by ~5/4, which have a one-to-one correspondence to the intervals of 2.5 subgroup the same way [[meantone]] does to [[Pythagorean tuning]]. These temperaments often exploit the 128/125 diesis, even if they don't temper it out: for example, [[magic]] temperament has a flat 5/4 of about 380 cents to equate (5/4)<sub>5</sub> octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/(128/125) to 3/2. [[Wurschmidt]] has a more accurate 5/4 and equates (5/4)<sub>8</sub> octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/(([128/125]])<sup>2</sup>) to 3/2.
+Equating 5/4 to the cube root of 2, or [[tempering out]] [[128/125]], results in the [[augmented]] temperament, which has a [[period]] of ~5/4 and [[generator]] of ~3/2. This temperament may be familiar because [[12edo]] [[support]]s it, and 12edo's tuning of the Augmented[6] [[3L 3s]] scale is the {{w|augmented scale}} of standard music theory. There are regular temperaments generated by ~5/4, which have a one-to-one correspondence to the intervals of 2.5 subgroup the same way [[meantone]] does to [[Pythagorean tuning]]. These temperaments often exploit the 128/125 diesis, even though they don't temper it out: for example, [[magic]] temperament has a flat 5/4 of about 380 cents to equate (5/4)<sup>5</sup> octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/(128/125) to 3/2. [[Wurschmidt]] has a more accurate 5/4 and equates (5/4)<sup>8</sup> octave-reduced to [[3/2]], which is the same thing as equating ([[25/16]])/((128/125)<sup>2</sup>) to 3/2. These can be seen as [[cluster temperament]]s using 128/125 as a versatile chroma.
+== See also ==
+* [[3-limit]] (2.3)
+* [[3.5 subgroup]]
 {{Stub}}
 [[Category:Just intonation subgroups|#]]
 [[Category:Rank-2 temperaments|#]]
 [[Category:5-limit|#]]

2.5 subgroup: Difference between revisions

CompactStar (talk | contribs)