Superpyth: Difference between revisions

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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. The tempered comma, 64/63, is not only larger than 81/80, but superpyth must split the error accumulated to represent the 7th harmonic over two fifths, rather than four as in meantone. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others~~, shown in part by the fact that there are considerably fewer EDOs that support superpyth than meantone, and many of them require alternate mappings for 3 and 7~~. 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. The tempered comma, 64/63, is not only larger than 81/80, but superpyth must split the error accumulated to represent the 7th harmonic over two fifths, rather than four as in meantone. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing 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. ~~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 that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma ~~(for example, where the fifth is tuned to 709.745{{c}}~~, which ~~is closely approximated by the sharp fifth of 55\[[93edo]], and the 7/6 and 9/7 are sharp by the same amount, giving us 2/7-comma archy). This is the most common approach to optimizing archy, and~~ is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is 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.

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 approach is 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 in general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is 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 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.

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 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. The tempered comma, 64/63, is not only larger than 81/80, but superpyth must split the error accumulated to represent the 7th harmonic over two fifths, rather than four as in meantone. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing others, shown in part by the fact that there are considerably fewer EDOs that support superpyth than meantone, and many of them require alternate mappings for 3 and 7. 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. The tempered comma, 64/63, is not only larger than 81/80, but superpyth must split the error accumulated to represent the 7th harmonic over two fifths, rather than four as in meantone. Therefore, tuning superpyth can be a somewhat contentious matter, as some intervals have to be essentially sacrificed for the sake of optimizing 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. 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 that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma (for example, where the fifth is tuned to 709.745{{c}}, which is closely approximated by the sharp fifth of 55\[[93edo]], and the 7/6 and 9/7 are sharp by the same amount, giving us 2/7-comma archy). This is the most common approach to optimizing archy, and is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is 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.
+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 approach is 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 in general, we would want to consider 3 somewhat more important than 7, and 7 somewhat more important than 9; in meantone, similar principles imply that an optimum is to be found sharp of 1/4-comma, though flat of [[1/5-comma meantone|1/5-comma]]. In archy, these place it sharper than 1/4-comma but flatter than 1/3-comma, which is supported by the standard [[CTE]] and [[CWE]] metrics. In fact, 22edo is 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 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.