Diesis: Difference between revisions
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Therefore, if we are interested only in how a stack of 2 to 4<ref group="note">Setting it to 5 only gives us one additional diesis, (2⋅(10/9))/(7/6)<sup>5</sup>, and the rest of the dieses only get more ways of reaching them at best. If the interval we want to reach with our stack is in the 7-odd-limit, the set of dieses is the same.</ref> of a 9-odd-limit interval differs from another 9-odd-limit interval under these constraints, we get the following list of [[7-limit]] dieses, with new things categorized as such linked: | Therefore, if we are interested only in how a stack of 2 to 4<ref group="note">Setting it to 5 only gives us one additional diesis, (2⋅(10/9))/(7/6)<sup>5</sup>, and the rest of the dieses only get more ways of reaching them at best. If the interval we want to reach with our stack is in the 7-odd-limit, the set of dieses is the same.</ref> of a 9-odd-limit interval differs from another 9-odd-limit interval under these constraints, we get the following list of [[7-limit]] dieses, with new things categorized as such linked: | ||
* 20000/19683, [[3645/3584]] = (9/8)<sup>3</sup>/(7/5)<ref group="note">Which might be the tritonic diesis by contrast with the septimal tritonic diesis of 50/49 | * 20000/19683, [[3645/3584]] = (9/8)<sup>3</sup>/(7/5) <ref group="note">Which might be the tritonic diesis by contrast with the septimal tritonic diesis of 50/49, as the latter is more clearly specific to the 7-limit tritones while this is the difference between the tritone and 7/5.</ref>, 3125/3072, 50/49, [[5103/5000]] = (7/5)/(10/9)<sup>3</sup> = (14/9)/(10/9)<sup>4</sup> <ref group="note">Called a diesis in a theory of [[Lériendil]]'s that uses yet another definition of diesis.</ref>, 49/48, [[12288/12005]] = (8/7)<sup>4</sup>/(5/3), 128/125, [[33614/32805]] = (2⋅(7/5))/(9/7)<sup>4</sup>, [[19683/19208]] = (9/7)<sup>4</sup>/(4/3), [[16807/16384]] = (7/4)/(8/7)<sup>4</sup>, 36/35, 250/243 | ||
A few definitions conveniently happen to give an equivalent list; the set of LCJI intervals we are 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 [https://www.python.org/ Python 3]; use <code>diesis</code> with <code>show=True</code>.) | A few definitions conveniently happen to give an equivalent list; the set of LCJI intervals we are 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 [https://www.python.org/ Python 3]; use <code>diesis</code> with <code>show=True</code>.) | ||
Finally, in regards to the specific set of equivalent definitions discussed, it should be noted that they are also equivalent without the requirement on a minimum size in [[cent]]s for the comma, instead allowing the minimum damage to impose a minimum size. 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 | Finally, in regards to the specific set of equivalent definitions discussed, it should be noted that they are also equivalent without the requirement on a minimum size in [[cent]]s for the comma, instead allowing the minimum damage to impose a minimum size. 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 seventeen 7-limit dieses. In regards to 875/864, one might note that it is similar to the standard diesis of 128/125 (([[2/1]])/([[5/4]])<sup>3</sup>, S-expression [[16/15|S4]]/[[25/24|S5]]) as it is equal to ([[7/4]])/([[6/5]])<sup>3</sup> and has the S-expression [[25/24|S5]]/[[36/35|S6]], so it is 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 == | == As a diatonic interval category == | ||