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 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 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.

Alternatively, for a sharper tuning, the 5th harmonic can be mapped to +14 generators, resulting in [[ultrapyth]].

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== Tunings ==

The fifth of superpyth is ~~supposed to be~~ tuned sharp of just. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, close to [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.

The fifth of superpyth is accurately tuned sharp of just. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, close to [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.

22edo can be viewed as a closed form of 1/4-comma superpyth, where the whole tone is midway between 8/7 and 9/8, so that the 7 is as sharp as the 9 and that the 9/7 major third is tuned just. 27edo can be viewed as a closed form of 1/3-comma superpyth, where the whole tone leans towards 8/7 a bit, so that the 7 is as sharp as the 3 and that the 7/6 minor third is tuned just.

27edo is also the point where 5/4 is tuned to the familiar 400 cents of [[12edo]], and in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.

These flat tunings being common is the reason the mapping of 5/4 at +9 generators is chosen as the canonical extension; 27edo is also the point where 5/4 is tuned to the familiar 400 cents of [[12edo]], and in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to 19edo (which represents 1/3-comma meantone and is on the edge between septimal meantone and flattone). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.

The {{w|plastic number}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes a variety of superpyth. This can be explained since superpyth equates [[21/16]] and [[4/3]], making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap|~9 + ~12 {{=}} ~21}} the generator becomes the plastic number.

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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 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 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.
 Alternatively, for a sharper tuning, the 5th harmonic can be mapped to +14 generators, resulting in [[ultrapyth]].
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 == Tunings ==
-The fifth of superpyth is supposed to be tuned sharp of just. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, close to [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.
+The fifth of superpyth is accurately tuned sharp of just. Roughly speaking, it ranges from as flat as [[Pythagorean tuning|Pythagorean]] (where 3 is tuned just) to 1/2-comma (where 7 is tuned just, close to [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.
 edo can be viewed as a closed form of 1/4-comma superpyth, where the whole tone is midway between 8/7 and 9/8, so that the 7 is as sharp as the 9 and that the 9/7 major third is tuned just. 27edo can be viewed as a closed form of 1/3-comma superpyth, where the whole tone leans towards 8/7 a bit, so that the 7 is as sharp as the 3 and that the 7/6 minor third is tuned just.
-edo is also the point where 5/4 is tuned to the familiar 400 cents of [[12edo]], and in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.
+These flat tunings being common is the reason the mapping of 5/4 at +9 generators is chosen as the canonical extension; 27edo is also the point where 5/4 is tuned to the familiar 400 cents of [[12edo]], and in sharper tunings, there are different mappings of 5/4 with more accuracy (see [[quasiultra]] and [[ultrapyth]]), somewhat analogous to 19edo (which represents 1/3-comma meantone and is on the edge between septimal meantone and flattone). The same goes for flatter tunings than 22edo (see [[quasisuper]] and [[dominant (temperament)|dominant]]). Furthermore, the 11-limit extension works strictly within 22edo and 27e-edo, with 22edo conflating 11/10 with 12/11, and 27e-edo conflating 11/8 with 7/5. There is an alternative extension, suprapyth, that works for tunings in the range of 17edo to 22edo, however.
 The {{w|plastic number}} has a value of ~486.822 cents, which, taken as a generator (~4/3) and assuming a pure-octave period, constitutes a variety of superpyth. This can be explained since superpyth equates [[21/16]] and [[4/3]], making the 9:12:16:21 chord evenly spaced by ~4/3, and when keeping {{nowrap|~9 + ~12 {{=}} ~21}} the generator becomes the plastic number.