Superpyth: Difference between revisions

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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 ratio}} 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 ===

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 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 ratio}} 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 ===