Diesis: Difference between revisions
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Therefore, if we are interested only in how a stack of 2 to 4 (or 2 to 5***) 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 (or 2 to 5***) 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) (*), 3125/3072, 50/49, [[5103/5000]] = (7/5) / (10/9)<sup>3</sup> = (14/9) / (10/9)<sup>4</sup> (**), 49/48, [[12288/12005]] = (8/7)<sup>4</sup> / (5/3), 128/125, [[33614/32805]] = ( | 20000/19683, [[3645/3584]] = (9/8)<sup>3</sup> / (7/5) (*), 3125/3072, 50/49, [[5103/5000]] = (7/5) / (10/9)<sup>3</sup> = (14/9) / (10/9)<sup>4</sup> (**), 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 | ||
(*Which might be the tritonic diesis by contrast with the septimal tritonic diesis of 50/49.) | (*Which might be the tritonic diesis by contrast with the septimal tritonic diesis of 50/49.) | ||
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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.) | (**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, ( | (***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.) | ||
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.) | ||