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 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 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♯, 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.

~~Alternatively, for~~ a ~~sharper~~ tuning, the 5th harmonic can be mapped to +14 generators, resulting in [[ultrapyth]]. For a tuning flat of 22edo, the 5th harmonic can be mapped to -13 generators to get [[quasisuper]].

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]].

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 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 two mappings unite on [[22edo]]. Note that ~~the~~ only ~~reasonable tuning for suprapyth is~~ 22edo, as sharper tunings swap the sizes of [[11/10]] and [[12/11]], ~~and~~ flatter tunings ~~swap 11/10 and [[10/9]], as well~~ as ~~[[7/5]] and [[10/7]]~~. However, by keeping the 2.3.7.11 mapping of suprapyth (simply called [[supra]]) and using the quasisuper mapping of 5, we get [[quasisupra]], which has a flexible tuning range.

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]]. 27edo is now a definitive upper bound, as sharper tunings map 11/8 wider than 7/5. 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 two mappings unite on [[22edo]]. Note that suprapyth only works in 22edo, as sharper tunings swap the sizes of [[11/10]] and [[12/11]], while flatter tunings don't work as discussed above. However, by keeping the 2.3.7.11 mapping of suprapyth (simply called [[supra]]) and using the quasisuper mapping of 5, we get [[quasisupra]], which has a flexible tuning range.

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]]. In practice, however, this mapping only works in [[27edo]], as flatter tunings swap the sizes of [[13/12]] and [[14/13]]. An alternative mapping is -14 generators, or a doubly diminished octave (C–C𝄫), by tempering out [[9604/9477]]. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo. A more practical option is to split the sharp ~3/2 into two ~[[16/13]]'s, which results in [[beatles]], and has an alternative mapping of primes 5 and 11. Alternatively, one can keep the superpyth mappings of 5 and 11 to get [[archytas clan #Thomas|thomas]].

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]]. In practice, however, this mapping only works in [[27edo]], as flatter tunings swap the sizes of [[13/12]] and [[14/13]]. An alternative mapping is -14 generators (known as [[uberpyth]]), or a doubly diminished octave (C–C𝄫), by tempering out [[9604/9477]]. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo. A more practical option is to split the sharp ~3/2 into two ~[[16/13]]'s, which results in [[beatles]], and has an alternative mapping of primes 5 and 11. Alternatively, one can keep the superpyth mappings of 5 and 11 to get [[archytas clan #Thomas|thomas]].

[[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.

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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 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 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♯, 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.
-Alternatively, for a sharper tuning, the 5th harmonic can be mapped to +14 generators, resulting in [[ultrapyth]]. For a tuning flat of 22edo, the 5th harmonic can be mapped to -13 generators to get [[quasisuper]].
+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]].
-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 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 two mappings unite on [[22edo]]. Note that the only reasonable tuning for suprapyth is 22edo, as sharper tunings swap the sizes of [[11/10]] and [[12/11]], and flatter tunings swap 11/10 and [[10/9]], as well as [[7/5]] and [[10/7]]. However, by keeping the 2.3.7.11 mapping of suprapyth (simply called [[supra]]) and using the quasisuper mapping of 5, we get [[quasisupra]], which has a flexible tuning range.
+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]]. 27edo is now a definitive upper bound, as sharper tunings map 11/8 wider than 7/5. 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 two mappings unite on [[22edo]]. Note that suprapyth only works in 22edo, as sharper tunings swap the sizes of [[11/10]] and [[12/11]], while flatter tunings don't work as discussed above. However, by keeping the 2.3.7.11 mapping of suprapyth (simply called [[supra]]) and using the quasisuper mapping of 5, we get [[quasisupra]], which has a flexible tuning range.
-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]]. In practice, however, this mapping only works in [[27edo]], as flatter tunings swap the sizes of [[13/12]] and [[14/13]]. An alternative mapping is -14 generators, or a doubly diminished octave (C–C𝄫), by tempering out [[9604/9477]]. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo. A more practical option is to split the sharp ~3/2 into two ~[[16/13]]'s, which results in [[beatles]], and has an alternative mapping of primes 5 and 11. Alternatively, one can keep the superpyth mappings of 5 and 11 to get [[archytas clan #Thomas|thomas]].
+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]]. In practice, however, this mapping only works in [[27edo]], as flatter tunings swap the sizes of [[13/12]] and [[14/13]]. An alternative mapping is -14 generators (known as [[uberpyth]]), or a doubly diminished octave (C–C𝄫), by tempering out [[9604/9477]]. This has a more flexible range, but the 13 tends to be tuned very sharp except in 27edo. A more practical option is to split the sharp ~3/2 into two ~[[16/13]]'s, which results in [[beatles]], and has an alternative mapping of primes 5 and 11. Alternatively, one can keep the superpyth mappings of 5 and 11 to get [[archytas clan #Thomas|thomas]].
 [[Mos scale]]s of superpyth have cardinalities of 5, 7, 12, 17, or 22.