Superpyth: Difference between revisions
Intro section describes archy, not necessarily superpyth Tags: Manual revert Visual edit |
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| Pergen = (P8, P5) | | Pergen = (P8, P5) | ||
| Color name = Ruti | | Color name = Ruti | ||
|MOS scales=[[2L 3s]], [[5L 2s]], [[5L 7s]]|Subgroups=2.3.7, 2.3.5.7|Title=Archy; superpyth|Odd limit 1=(2.3.7) 9|Mistuning 1=?|Odd limit 2=9|Mistuning 2=?|Complexity 1=12|Complexity 2=27}}''' | |MOS scales=[[2L 3s]], [[5L 2s]], [[5L 7s]]|Subgroups=2.3.7, 2.3.5.7|Title=Archy; superpyth|Odd limit 1=(2.3.7) 9|Mistuning 1=?|Odd limit 2=9|Mistuning 2=?|Complexity 1=12|Complexity 2=27}}'''Superpyth,''' often called '''archy''' in the no-fives subgroup, is a [[regular temperament|temperament]] where the [[generator]] is [[4/3]], tuned flat so that stacking two of them gives the interval [[7/4]]. This means that intervals such as A–G and C–B♭ (notated in chain-of-fifths notation) are harmonic sevenths. Equivalently, two [[3/2]] perfect fifths [[octave reduction|octave-reduced]] gives [[8/7]]. Three fourths reach a subminor third that approximates [[7/6]], while four fifths reach a supermajor third that approximates [[9/7]]. This means that the septimal comma ([[64/63]]) is [[tempering out|tempered out]]. | ||
Since the generator is a perfect fourth or perfect fifth, archy 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]] and [[27edo|16\27]] are typical tunings of the generator. | Since the generator is a perfect fourth or perfect fifth, archy 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]] and [[27edo|16\27]] are typical tunings of the generator. | ||
If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ (an augmented second or limma-flat major third) is 5/4, leading to | If intervals of 5 are desired, the 5th harmonic is mapped to +9 generators through tempering out [[245/243]], so C–D♯ (an augmented second or limma-flat major third) is 5/4, leading to the full 7-limit superpyth temperament. 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–Dx), 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 11 are desired, the canonical way is to map 11/8 to +16 generators, or a doubly augmented second (C–Dx), 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]]. | ||