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.

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♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. 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.

If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group | ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]].

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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.
+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♯). This mapping equates the pythagorean limma, [[256/243]], to the syntonic [[81/80]], tempering out [[20480/19683]], so that 5/4 is mapped to a major third minus a limma. 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.
 If intervals of 11 are desired, the canonical way is to map [[11/8]] to +16 generators, or a doubly augmented second (C–D𝄪), tempering out [[100/99]]. A simpler but less accurate way to map it is to −6 generators, or a diminished fifth (C–G♭), by tempering out [[99/98]]. The latter is called '''suprapyth''', a name coined by [[Mike Battaglia]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96882.html#96895 Yahoo! Tuning Group | ''A few full 11-limit 896/891 temperaments'']</ref>. The [[2.3.7.11 subgroup]] restriction of suprapyth, known as [[supra]], is also notable. The two mappings unite on [[22edo]].