Diesis: Difference between revisions

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A few definitions conveniently happen to give an equivalent list; the set of LCJI intervals we're interested in the stack being near to could be the 7-odd-limit instead, and whether we choose a 2 to 4 or 2 to 5 range only changes the number of expressions for some of the dieses, so this appears to be an algorithmically significant result at the very least, evidencing a possible computational basis for the intuitive properties of the notion. (A more general parametrization might only use the 2 to 5 range to look for alternate expressions but 2 to 4 to avoid overcomplex expressions, while having some stack of 9-odd-limit equal a 13-odd-limit interval, but it might be preferred to use definitions that keep the set elegant.)

Finally, in regards to the specific set of 4 equivalent definitions discussed, it should be noted that they are also equivalent if we don't require a minimum size in [[cent]]s for the comma, instead allowing the minimum damage to impose a minimum size (which is arguably more relevant). This causes [[81/80]], [[64/63]], [[875/864]] and [[245/243]] to also be considered dieses, which arguably is not so unexpected as they all share the intuitively-motivated properties discussed above, for a total of 16 7-limit dieses.

== As a diatonic interval category ==

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 A few definitions conveniently happen to give an equivalent list; the set of LCJI intervals we're interested in the stack being near to could be the 7-odd-limit instead, and whether we choose a 2 to 4 or 2 to 5 range only changes the number of expressions for some of the dieses, so this appears to be an algorithmically significant result at the very least, evidencing a possible computational basis for the intuitive properties of the notion. (A more general parametrization might only use the 2 to 5 range to look for alternate expressions but 2 to 4 to avoid overcomplex expressions, while having some stack of 9-odd-limit equal a 13-odd-limit interval, but it might be preferred to use definitions that keep the set elegant.)
+Finally, in regards to the specific set of 4 equivalent definitions discussed, it should be noted that they are also equivalent if we don't require a minimum size in [[cent]]s for the comma, instead allowing the minimum damage to impose a minimum size (which is arguably more relevant). This causes [[81/80]], [[64/63]], [[875/864]] and [[245/243]] to also be considered dieses, which arguably is not so unexpected as they all share the intuitively-motivated properties discussed above, for a total of 16 7-limit dieses.
 == As a diatonic interval category ==