Superpyth: Difference between revisions

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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 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]]~~.

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. In general, however, we would want to treat 3 somewhat more importantly than 7. It is also justifiable to 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 as we would want to consider 7 less important than 3, likewise we would consider 9 less important 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. In fact, 22edo is very slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion.

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.

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 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 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]].
+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. In general, however, we would want to treat 3 somewhat more importantly than 7. It is also justifiable to 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 as we would want to consider 7 less important than 3, likewise we would consider 9 less important 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. In fact, 22edo is very slightly sharp of 1/4-comma (though still flat of the CTE optimum) and therefore pushes in the more accurate direction given the above discussion.
 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.