Superpyth: Difference between revisions

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Without tempered octaves, superpyth is of considerably higher damage than meantone, despite it being seen as the "counterpart" of meantone for sharp fifths and septimal thirds. The vanishing comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5, has a tuning error on 3 and 5 of 4.3{{c}}, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 [[7-odd-limit]] [[tonality diamond]]) has a tuning error on 3 and 7 of 9.1{{c}}, over twice as much. 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 9/7 supermajor third may 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 and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 [[9-odd-limit]], the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-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 and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 [[9-odd-limit]], the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-comma.

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. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} come very close to forming closed systems of 2/7-comma.

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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" ([[1/6-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 is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating 6:7:8 as the fundamental chord of the 2.3.7 subgroup, 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.

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]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, 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.

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.

=== Norm-based tunings ===

@@ Line 184: / Line 184: @@
 Without tempered octaves, superpyth is of considerably higher damage than meantone, despite it being seen as the "counterpart" of meantone for sharp fifths and septimal thirds. The vanishing comma, 64/63, is not only larger than 81/80, but it must be split over only three intervals (one minor seventh and two perfect fifths), rather than five as in meantone (one major third and four fifths). This can be shown by the fact that 1/5-comma meantone, the meantone tuning with the minimum damage to harmonics 3 and 5, has a tuning error on 3 and 5 of 4.3{{c}}, while 1/3-comma superpyth, the superpyth tuning with the minimum damage to harmonics 3 and 7 (the minimax tuning for the no-5 [[7-odd-limit]] [[tonality diamond]]) has a tuning error on 3 and 7 of 9.1{{c}}, over twice as much. 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 9/7 supermajor third may 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 and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 [[9-odd-limit]], the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-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 and 1/4-comma turn out to be logical solutions. In 1/3-comma superpyth, the whole tone leans towards 8/7 so that 3 and 7 are equally sharp and the minor third is tuned to exactly 7/6; 27edo is extremely close to a closed system of 1/3-comma. In 1/4-comma tuning, which is the minimax tuning for the no-5 [[9-odd-limit]], the whole tone is midway between 8/7 and 9/8 so that the 7 is twice as sharp as 3 and that the major third is exactly 9/7; 22edo is very close to a closed circle of 1/4-comma.
 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. 2/7-comma superpyth is particularly notable since it tunes the 7/6 and 9/7 equally sharp and 3/2 twice as sharp as the thirds; [[71edo]] (709.859{{c}}) and [[93edo]] with its sharp fifth of 709.677{{c}} come very close to forming closed systems of 2/7-comma.
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 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" ([[1/6-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 is a similar logic to Zarlino's preference for [[2/7-comma meantone]], treating 6:7:8 as the fundamental chord of the 2.3.7 subgroup, 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.
+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]], treating [[~]][[6:7:8]] as the fundamental chord of the 2.3.7 subgroup, 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.
+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.
 === Norm-based tunings ===