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 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~~.

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.

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, where 3 is twice as sharp as 7. ~~Flatter tunings than 1/4-comma archy such as 1/5-comma~~, ~~1/6-comma, … are analogous~~ to ~~the historical [[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]]~~, ~~… tunings of meantone~~.

Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical [[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], … tunings of meantone, as they value a more just 3/2 over accurate septimal harmony. The alternative 11-limit extension, suprapyth, and an alternative extension to 5, quasisuper, work best for tunings in the range of 17edo to 22edo.

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, where 3 is twice as sharp as 7. In this range, 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.

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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 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.
+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.
-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, where 3 is twice as sharp as 7. Flatter tunings than 1/4-comma archy such as 1/5-comma, 1/6-comma, … are analogous to the historical [[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], … tunings of meantone.
+Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical [[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], … tunings of meantone, as they value a more just 3/2 over accurate septimal harmony. The alternative 11-limit extension, suprapyth, and an alternative extension to 5, quasisuper, work best for tunings in the range of 17edo to 22edo.
+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, where 3 is twice as sharp as 7. In this range, 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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