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. 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.

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 sharper than 1/4-comma but flatter than 1/3-comma (for example, if the fifth is tuned to 709.745{{c}}, which is closely approximated by 55 ~~steps of~~ [[93edo]] ~~at 709.677{{c}}~~, the 7/6 and 9/7 are sharp by the same amount, ~~a similar logic to Zarlino's preference for [[~~2/7-comma ~~meantone]]~~). 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. 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 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.

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.

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

Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/5-comma meantone|1/5-comma]], [[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of [[3/2]] more than the accuracy of 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. 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 ultrapyth.

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. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], and in this case 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 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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 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.
-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 sharper than 1/4-comma but flatter than 1/3-comma (for example, if the fifth is tuned to 709.745{{c}}, which is closely approximated by 55 steps of [[93edo]] at 709.677{{c}}, the 7/6 and 9/7 are sharp by the same amount, a similar logic to Zarlino's preference for [[2/7-comma meantone]]). 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. 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 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.
 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.
-Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/5-comma meantone|1/5-comma]], [[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of 3 more than 7. The alternative 11-limit extension, suprapyth, and an alternative extension to 5, quasisuper, work best for tunings in the range of 17edo to 22edo.
+Tunings flatter than 1/4-comma archy, such as 1/5-comma (close to [[39edo]]), 1/6-comma, … are analogous to the historical "modified meantones" ([[1/5-comma meantone|1/5-comma]], [[1/6-comma meantone|1/6-comma]], [[1/7-comma meantone|1/7-comma]], …), as they prioritize the tuning of [[3/2]] more than the accuracy of 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. 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 ultrapyth.
+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. This is a similar logic to Zarlino's preference for [[2/7-comma meantone]], and in this case 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 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.