Superpyth: Difference between revisions
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"Superpythagorean" more commonly means any tuning with sharp of just fifths |
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'''Superpyth''' | '''Superpyth''' is a [[temperament]] where [[~]][[3/2]] is a [[generator]], and the septimal comma ([[64/63]]) is [[tempering out|tempered out]], so that a stack of two perfect fifths [[octave reduction|octave-reduced]] gives a major whole tone that represents both [[9/8]] and [[8/7]] (likewise, two perfect fourths give a minor seventh that represents both [[7/4]] and [[16/9]], so intervals such as A–G and C–B♭ are harmonic sevenths). Since 3/2 is a generator we can use the same standard [[chain-of-fifths notation]] that is also used for [[meantone]] and [[12edo]], 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. [[17edo|10\17]], [[22edo|13\22]], and [[27edo|16\27]] are typical tunings of the generator. | ||
Such a temperament without the 5th harmonic is also called '''archy'''. If intervals of 5 are desired, | Such a temperament without the [[5/1|5th harmonic]] is also called '''archy'''. If intervals of 5 are desired, 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ is 5/4. So superpyth is the "opposite" of septimal meantone in several different ways: meantone (including [[12edo]]) has 3/2 tuned flat so that the 5th harmonic's intervals are simple and the 7th harmonic's intervals are complex, while superpyth has 3/2 tuned sharp so that the 7th harmonic's intervals are simple while the 5th harmonic's intervals are complex. | ||
If intervals of 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second ( | 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]]. | ||
If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth ( | If intervals of 13 are desired, 13/8 is mapped to +13 generators, or a doubly augmented fourth (C–F𝄪), by tempering out [[31213/31104]]. | ||
[[ | [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22. | ||
For more technical data, see [[Archytas clan#Superpyth]]. | For more technical data, see [[Archytas clan #Superpyth]]. | ||
== Interval chains == | == Interval chains == | ||