Superpyth: Difference between revisions

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== Tunings ==

=== Tuning considerations and optima ===

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, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of ~~full 7-limit~~ 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, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.

Despite being seen as the "counterpart" of meantone for sharp fifths, superpyth is actually of considerably higher damage than meantone, since the error accumulated to represent the 7th harmonic is split over only 2 generator steps, rather than 4 as in meantone, in addition to the tempered comma being slightly larger. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing for others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the interval 9/7 may also be more important than it looks from the bare math.

If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma, 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, emerges as a logical solution, due to being the [[minimax]] ~~tuning~~ for the [[tonality diamond]] ~~formed by odds 3 and 7~~; 27edo is very close to a closed system of 1/3-comma ~~archy~~. In general, however, we would want to treat 3 somewhat more importantly than 7; in meantone, similar principles imply than an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]], and in archy, these place it in between 1/3-comma and 1/4-comma. This is the most common approach to optimizing archy, ~~and is supported by the standard [[CTE]] and [[CWE]] metrics. From that lens, 22edo can be viewed as a closed form of~~ 1/4-comma ~~archy~~, 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; ~~it is, furthermore~~, slightly sharp ~~of 1/4-comma~~ (though still flat of the CTE optimum) and ~~therefore pushes in~~ the ~~correct direction given~~ the ~~above discussion~~.

If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma tuning, 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, emerges as a logical solution, due to being the [[minimax tuning]] for the no-5 [[7-odd-limit]] [[tonality diamond]]; 27edo is very close to a closed system of 1/3-comma. Another common approach to optimizing archy would pit 7 against 9, from which we have 1/4-comma tuning, 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, which is the minimax tuning for the no-5 [[9-odd-limit]]; 22edo can be viewed as a closed form thereof, but very slightly sharp (though still flat of the [[CTE]] optimum). Between 1/3-comma and 1/4-comma is the region supported by the standard CTE and [[CWE]] metrics. In meantone, similar principles would imply an optimum sharp of 1/4-comma, and flat of [[1/5-comma meantone|1/5-comma]].

27edo is also the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where in sharper tunings, different mappings of 5/4 arise 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.

~~However, a~~ case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6, a similar logic to Zarlino's preference for [[2/7-comma meantone]]. This would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning; reading off the comma 64/63, both 3 and 7 must be sharp, and because the comma has two 3's and one 7 in the denominator, 3 is tuned twice as sharp as 7 in the CEE tuning. In this range of archy, the best extension to prime 5 is in fact not superpyth, but ultrapyth.

A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6, a similar logic to Zarlino's preference for [[2/7-comma meantone]]. This would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning; reading off the comma 64/63, both 3 and 7 must be sharp, and because the comma has two 3's and one 7 in the denominator, 3 is tuned twice as sharp as 7 in the CEE tuning. In this range of archy, the best extension to prime 5 is in fact not superpyth, but ultrapyth.

Finally, it may be noted that 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 an extremely sharp variety of archy. This can be explained since archy 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 138: / Line 138: @@
 == Tunings ==
 === Tuning considerations and optima ===
-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, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of full 7-limit 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, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of superpyth's optimal range.
 Despite being seen as the "counterpart" of meantone for sharp fifths, superpyth is actually of considerably higher damage than meantone, since the error accumulated to represent the 7th harmonic is split over only 2 generator steps, rather than 4 as in meantone, in addition to the tempered comma being slightly larger. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing for others. An additional consideration is the use of tertian triads in conventional diatonic harmony, whereby the interval 9/7 may also be more important than it looks from the bare math.
-If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma, 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, emerges as a logical solution, due to being the [[minimax]] tuning for the [[tonality diamond]] formed by odds 3 and 7; 27edo is very close to a closed system of 1/3-comma archy. In general, however, we would want to treat 3 somewhat more importantly than 7; in meantone, similar principles imply than an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]], and in archy, these place it in between 1/3-comma and 1/4-comma. This is the most common approach to optimizing archy, and is supported by the standard [[CTE]] and [[CWE]] metrics. From that lens, 22edo can be viewed as a closed form of 1/4-comma archy, 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; it is, furthermore, slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the correct direction given the above discussion.
+If we focus purely on the 2.3.7 subgroup for now, and as a starting point adopt an approach based on the example of [[quarter-comma meantone]], treating archy's harmonic 7 as analogous to 5 in meantone, 1/3-comma tuning, 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, emerges as a logical solution, due to being the [[minimax tuning]] for the no-5 [[7-odd-limit]] [[tonality diamond]]; 27edo is very close to a closed system of 1/3-comma. Another common approach to optimizing archy would pit 7 against 9, from which we have 1/4-comma tuning, 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, which is the minimax tuning for the no-5 [[9-odd-limit]]; 22edo can be viewed as a closed form thereof, but very slightly sharp (though still flat of the [[CTE]] optimum). Between 1/3-comma and 1/4-comma is the region supported by the standard CTE and [[CWE]] metrics. In meantone, similar principles would imply an optimum sharp of 1/4-comma, and flat of [[1/5-comma meantone|1/5-comma]].
 edo is also the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where in sharper tunings, different mappings of 5/4 arise 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.
-However, a case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6, a similar logic to Zarlino's preference for [[2/7-comma meantone]]. This would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning; reading off the comma 64/63, both 3 and 7 must be sharp, and because the comma has two 3's and one 7 in the denominator, 3 is tuned twice as sharp as 7 in the CEE tuning. In this range of archy, the best extension to prime 5 is in fact not superpyth, but ultrapyth.
+A case can also be made for tuning archy even sharper than 27edo, which involves the notion of splitting the error of 4/3 into that of 8/7 and 7/6, a similar logic to Zarlino's preference for [[2/7-comma meantone]]. This would imply 2/5-comma archy, where [[49/48]] is tuned justly, and 8/7 and 7/6 are both 1/5 a septimal comma off, and which is closely approximated by [[32edo]]. Unlike in the case of meantone, [[CEE]] optimization agrees with the notion of such a sharp tuning; reading off the comma 64/63, both 3 and 7 must be sharp, and because the comma has two 3's and one 7 in the denominator, 3 is tuned twice as sharp as 7 in the CEE tuning. In this range of archy, the best extension to prime 5 is in fact not superpyth, but ultrapyth.
 Finally, it may be noted that 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 an extremely sharp variety of archy. This can be explained since archy 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

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