Superpyth: Difference between revisions

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As a consequence of the fifth being tuned sharp, three fourths reach a subminor third that approximates 7/6 instead of 6/5, while four fifths reach a supermajor third that approximates 9/7.

Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ is 5/4. So 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 ~~fift~~ tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.

Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ is 5/4. So 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.

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. Yet a simpler but reasonable way is to map it to −6 generators, or a diminished fifth (C–G♭), by tempering out 99/98. The latter is called '''supra''', or '''suprapyth'''. The two mappings unite on [[22edo]].

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 As a consequence of the fifth being tuned sharp, three fourths reach a subminor third that approximates 7/6 instead of 6/5, while four fifths reach a supermajor third that approximates 9/7.
-Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ is 5/4. So 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 fift tuned sharp so that intervals of 7 are simple while intervals of 5 are complex.
+Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ is 5/4. So 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.
 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. Yet a simpler but reasonable way is to map it to −6 generators, or a diminished fifth (C–G♭), by tempering out 99/98. The latter is called '''supra''', or '''suprapyth'''. The two mappings unite on [[22edo]].