Superpyth: Difference between revisions

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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.

27edo is also the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where 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 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 main extension to prime 5 is in fact not superpyth, but ultrapyth.

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 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.
-edo is also the point where superpyth tunes 5/4 to the familiar 400 cents of [[12edo]], and where 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 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 main extension to prime 5 is in fact not superpyth, but ultrapyth.