Superpyth: Difference between revisions

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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, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of full 7-limit 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 7th harmonic ~~appears at~~ only 2 generator steps, rather than 4, 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.

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.

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* [[DKW theory|DKW]] (2.3.5 superpyth): ~2 = 1200.000, ~3/2 = 709.758

* DKW (2.3.7 archy): ~2 = 1200.000, ~3/2 = 712.585

== Scales ==

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 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, between [[52edo|52b-edo]] and [[57edo|57b-edo]]), with 22edo and 27edo being typical endpoints of full 7-limit 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 7th harmonic appears at only 2 generator steps, rather than 4, 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.
+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.
@@ Line 352: / Line 352: @@
 * [[DKW theory|DKW]] (2.3.5 superpyth): ~2 = 1200.000, ~3/2 = 709.758
 * DKW (2.3.7 archy): ~2 = 1200.000, ~3/2 = 712.585
 == Scales ==