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.

If intervals of [[5/1|5]] are desired, the 5th harmonic is canonically 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 [[7/1|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex. This mapping works between 22edo and [[5edo]] (though it rapidly becomes inaccurate past 27edo), as tunings flatter than 22edo map [[7/5]] wider than [[10/7]], and tunings sharper than 5edo map 8/7 wider than 7/6.

If intervals of [[5/1|5]] are desired, the 5th harmonic is canonically 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 [[7/1|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex. This mapping works between 22edo and [[5edo]] (though it rapidly becomes inaccurate past 27edo), as tunings flatter than 22edo map [[7/5]] wider than [[10/7]], and tunings sharper than 5edo map 8/7 wider than 7/6 (among many other things).

For a tuning between 27edo and [[32edo]], the 5th harmonic can be mapped to -18 generators, resulting in [[quasiultra]], or for a tuning sharper than 32edo, +14 generators, resulting in [[ultrapyth]]. For a tuning flat of 22edo, the 5th harmonic can be mapped to -13 generators to get [[quasisuper]].

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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.
-If intervals of [[5/1|5]] are desired, the 5th harmonic is canonically 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 [[7/1|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex. This mapping works between 22edo and [[5edo]] (though it rapidly becomes inaccurate past 27edo), as tunings flatter than 22edo map [[7/5]] wider than [[10/7]], and tunings sharper than 5edo map 8/7 wider than 7/6.
+If intervals of [[5/1|5]] are desired, the 5th harmonic is canonically 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 [[7/1|7]] are complex, while superpyth has the fifth tuned sharp so that intervals of 7 are simple while intervals of 5 are complex. This mapping works between 22edo and [[5edo]] (though it rapidly becomes inaccurate past 27edo), as tunings flatter than 22edo map [[7/5]] wider than [[10/7]], and tunings sharper than 5edo map 8/7 wider than 7/6 (among many other things).
 For a tuning between 27edo and [[32edo]], the 5th harmonic can be mapped to -18 generators, resulting in [[quasiultra]], or for a tuning sharper than 32edo, +14 generators, resulting in [[ultrapyth]]. For a tuning flat of 22edo, the 5th harmonic can be mapped to -13 generators to get [[quasisuper]].