Superpyth: Difference between revisions

Line 152:

== Tunings ==

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.

The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. 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.

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. Even sharper tunings may be used if your music requires more accurate prime 7 than 3.

~~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.

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.

@@ Line 152: / Line 152: @@
 == Tunings ==
-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.
+The fifth of superpyth is supposed to be tuned sharp of just for the accuracy of the overall temperament. 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 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. Even sharper tunings may be used if your music requires more accurate prime 7 than 3.
-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.
+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]]), 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.

Superpyth: Difference between revisions

FloraC (talk | contribs)