Diesis: Difference between revisions

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(**Is called a diesis in a theory of [[Lériendil]]'s that uses a definition of diesis currently not documented on this page.)

(***Note that we only get one additional diesis, (2 * 10/9) / (7/6)5, so the rest of the dieses only get more ways of reaching them at best, hence we ignore this.)

(***Note that we only get one additional diesis, (2 * 10/9) / (7/6)5, so the rest of the dieses only get more ways of reaching them at best, hence we ignore this. If the interval we want to reach with our stack is in the 7-odd-limit, the set of dieses is the same.)

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.)

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. To tweak with parameters yourself, you can use [[User:Godtone#My Python 3 code|Godtone's code]], which has no dependencies other than Python 3; use <code>diesis</code> with <code>show=True</code>.)

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]])3 }} as it's equal to {{nowrap| 875/864 {{=}} [[25/24|S5]]/[[36/35|S6]] {{=}} ([[7/4]])/([[6/5]])3 }}, so it's in some sense a 7-limit analogue of the 5-limit standard diesis, and might be named based on this.)

@@ Line 32: / Line 32: @@
 (**Is called a diesis in a theory of [[Lériendil]]'s that uses a definition of diesis currently not documented on this page.)
-(***Note that we only get one additional diesis, (2 * 10/9) / (7/6)<sup>5</sup>, so the rest of the dieses only get more ways of reaching them at best, hence we ignore this.)
+(***Note that we only get one additional diesis, (2 * 10/9) / (7/6)<sup>5</sup>, so the rest of the dieses only get more ways of reaching them at best, hence we ignore this. If the interval we want to reach with our stack is in the 7-odd-limit, the set of dieses is the same.)
-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.)
+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. To tweak with parameters yourself, you can use [[User:Godtone#My Python 3 code|Godtone's code]], which has no dependencies other than Python 3; use <code>diesis</code> with <code>show=True</code>.)
 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.)