2.5 subgroup: Difference between revisions

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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)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.

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)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.

[[Category:Just intonation subgroups|#]]

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

[[Category:5-limit|#]]

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 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. 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.
 {{Stub}}
 [[Category:Just intonation subgroups|#]]
 [[Category:Rank-2 temperaments|#]]
 [[Category:5-limit|#]]

2.5 subgroup: Difference between revisions

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