2.5 subgroup: Difference between revisions

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== MOS scales ==

As it is a [[rank-2]] system, stacking 5/4 with octave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces a version of the pentatonic, diatonic and chromatic scales. An important structural feature of the 2.5 subgroup is that [[5/4]]<~~sub~~>3</~~sub~~> = [[125/64]], which is only a diesis ([[128/125]]) short of an octave. This means that the cube root of 2, or the familiar [[3edo]]/[[12edo]] major third, is a decent approximation of 5/4. It also means that most small MOSes generated by the 2.5 subgroup are of the form 3L ''x''s, and are extremely [[hard]] (high L:s ratio) which makes them difficult to use.

As it is a [[rank-2]] system, stacking 5/4 with octave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces a version of the pentatonic, diatonic and chromatic scales. An important structural feature of the 2.5 subgroup is that [[5/4]]3 = [[125/64]], which is only a diesis ([[128/125]]) short of an octave. This means that the cube root of 2, or the familiar [[3edo]]/[[12edo]] major third, is a decent approximation of 5/4. It also means that most small MOSes generated by the 2.5 subgroup are of the form 3L ''x''s, and are extremely [[hard]] (high L:s ratio) which makes them difficult to use. It takes all the way up to a 31-tone MOS (28L 3s) to break out of the 3L ''x''s chain, which corresponds to how [[31edo]] is the smallest EDO to have an approximation of 5/4 in-between just 5/4 and the cube root of 2.

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, 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]]

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 == MOS scales ==
-As it is a [[rank-2]] system, stacking 5/4 with octave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces a version of the pentatonic, diatonic and chromatic scales. An important structural feature of the 2.5 subgroup is that [[5/4]]<sub>3</sub> = [[125/64]], which is only a diesis ([[128/125]]) short of an octave. This means that the cube root of 2, or the familiar [[3edo]]/[[12edo]] major third, is a decent approximation of 5/4. It also means that most small MOSes generated by the 2.5 subgroup are of the form 3L ''x''s, and are extremely [[hard]] (high L:s ratio) which makes them difficult to use.
+As it is a [[rank-2]] system, stacking 5/4 with octave-reduction can produce [[MOS scale]]s, just like how Pythagorean tuning famously produces a version of the pentatonic, diatonic and chromatic scales. An important structural feature of the 2.5 subgroup is that [[5/4]]<sup>3</sup> = [[125/64]], which is only a diesis ([[128/125]]) short of an octave. This means that the cube root of 2, or the familiar [[3edo]]/[[12edo]] major third, is a decent approximation of 5/4. It also means that most small MOSes generated by the 2.5 subgroup are of the form 3L ''x''s, and are extremely [[hard]] (high L:s ratio) which makes them difficult to use. It takes all the way up to a 31-tone MOS (28L 3s) to break out of the 3L ''x''s chain, which corresponds to how [[31edo]] is the smallest EDO to have an approximation of 5/4 in-between just 5/4 and the cube root of 2.
 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, 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]]
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2.5 subgroup: Difference between revisions

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