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. (In regards to 875/864, one might note that according to [[S-expression]]s, it's similar to the standard diesis of {{nowrap| 128/125 {{=}} [[16/15|S4]]/[[25/24|S5]] {{=}} ([[2/1|6/3]])/([[5/4]])<sup>3</sup> }} as it's equal to {{nowrap| 875/864 {{=}} [[25/24|S5]]/[[36/35|S6]] {{=}} ([[7/4]])/([[6/5]])<sup>3</sup> }}, so it's in some sense a 7-limit analogue of the 5-limit standard diesis, and might be named based on this.)

== 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. (In regards to 875/864, one might note that according to [[S-expression]]s, it's similar to the standard diesis of {{nowrap| 128/125 {{=}} [[16/15|S4]]/[[25/24|S5]] {{=}} ([[2/1|6/3]])/([[5/4]])<sup>3</sup> }} as it's equal to {{nowrap| 875/864 {{=}} [[25/24|S5]]/[[36/35|S6]] {{=}} ([[7/4]])/([[6/5]])<sup>3</sup> }}, so it's in some sense a 7-limit analogue of the 5-limit standard diesis, and might be named based on this.)
 == As a diatonic interval category ==