Superpyth: Difference between revisions

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Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo|13\22]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.

Tuning superpyth is challenging because [[9/7]] and [[7/4]] are both best approximated by different tunings (around 22 and 57b respectively), so tuning one accurately means compromising the other. For example, in 22edo, 9/7 is tuned nearly perfectly, but 7/4 is sharp by about 13 cents, and in 57b, 7/4 is tuned nearly perfectly, but 9/7 is sharp by almost 30 cents. A compromise may be found at [[32edo]], which tunes 7/4 sharp about 7 cents and 9/7 sharp about 15 cents. However, the flatter tunings are easier to extend to other intervals:

If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so 5/4 is an augmented second (e.g. C–D♯, a limma-flat major third). Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.

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 Since the generator is a perfect fifth, superpyth can be notated using the same standard [[chain-of-fifths notation]] that is also used for [[meantone]], with the understanding that sharps are sharper than flats (for example, A♯ is sharper than B♭) just like in [[Pythagorean tuning]], in contrast to meantone where sharps are flatter than or equal to the corresponding flats. [[22edo|13\22]] (~1/4 septimal comma) and [[27edo|16\27]] (~1/3 septimal comma) are the most common tunings of the generator.
-Tuning superpyth is challenging because [[9/7]] and [[7/4]] are both best approximated by different tunings (around 22 and 57b respectively), so tuning one accurately means compromising the other. For example, in 22edo, 9/7 is tuned nearly perfectly, but 7/4 is sharp by about 13 cents, and in 57b, 7/4 is tuned nearly perfectly, but 9/7 is sharp by almost 30 cents. A compromise may be found at [[32edo]], which tunes 7/4 sharp about 7 cents and 9/7 sharp about 15 cents. However, the flatter tunings are easier to extend to other intervals:
 If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so 5/4 is an augmented second (e.g. C–D♯, a limma-flat major third). Therefore superpyth is the "opposite" of meantone in several different ways: most notably, meantone (including [[12edo]]) has the fifth tuned flat so that intervals of harmonic 5 are simple while intervals of 7 are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.