Diesis: Difference between revisions

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This is not to be confused with the related sense of the same term introduced next, for which the [[648/625|major diesis (648/625)]] was named while being way wider than the "maximal diesis".

==== Generalization ====

For someone looking for what sets these (and a few others) apart from other commas in the size range, it might be worth noting be noted that with the exception of 648/625, almost all just intervals commonly called dieses have a few properties in common that might be used to derive a definition that expands the set of commas called dieses to something closer to the spirit of the term as a whole:

* Being an awkward xenmelodic size (as characteristic of intervals of a size between that of the minimal diesis (27.66{{c}}) and maximal diesis (49.17{{c}})). Relatedly:

* Equating a ''short'' stack of one LCJI interval with some other LCJI interval (and it appears it's never more than 4 or 5, as suggested by the unusual "diesis" of 256/243 = ([[4/3]])5 / [[4/1|4]] = ([[4/3]])4 / [[3/1|3]]. Note that due to the minimal and maximal diesis both using a stack of [[~]][[10/9]]'s, one could argue that one is intuitively looking at how a short stack of some [[9-odd-limit]] interval relates to some other simple interval of interest.

* Due to the last two constraints, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a not-unnoticeable amount of damage on the interval being stacked*. This is arguably what truly makes them feel awkward in JI, as they are also small enough to feel like potential commas without being very efficient to temper out. *Notably, some of these are more debatable than others in terms of damage, so one should clarify that the minimum damage logically is that of the minimal diesis (6.9{{c}}), as more than 7 cents of damage is not insignificant for most intervals and is essentially a flexibility afforded by LCJI's temperability.

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)3 / (7/5) (*), 3125/3072, 50/49, [[5103/5000]] = (7/5) / (10/9)3 = (14/9) / (10/9)4 (**), 49/48, [[12288/12005]] = (8/7)4 / (5/3), 128/125, [[19683/19208]] = (9/7)4 / (4/3) = (9/7)5 / (12/7), [[16807/16384]] = (7/4) / (8/7)4 = (2/1) / (8/7)5, 36/35, 250/243

(*which might be the tritonic diesis by contrast with the septimal tritonic diesis of 50/49)

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

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.)

== As a diatonic interval category ==

@@ Line 17: / Line 17: @@
 This is not to be confused with the related sense of the same term introduced next, for which the [[648/625|major diesis (648/625)]] was named while being way wider than the "maximal diesis".
+==== Generalization ====
+For someone looking for what sets these (and a few others) apart from other commas in the size range, it might be worth noting be noted that with the exception of 648/625, almost all just intervals commonly called dieses have a few properties in common that might be used to derive a definition that expands the set of commas called dieses to something closer to the spirit of the term as a whole:
+* Being an awkward xenmelodic size (as characteristic of intervals of a size between that of the minimal diesis (27.66{{c}}) and maximal diesis (49.17{{c}})). Relatedly:
+* Equating a ''short'' stack of one LCJI interval with some other LCJI interval (and it appears it's never more than 4 or 5, as suggested by the unusual "diesis" of 256/243 = ([[4/3]])<sup>5</sup> / [[4/1|4]] = ([[4/3]])<sup>4</sup> / [[3/1|3]]. Note that due to the minimal and maximal diesis both using a stack of [[~]][[10/9]]'s, one could argue that one is intuitively looking at how a short stack of some [[9-odd-limit]] interval relates to some other simple interval of interest.
+* Due to the last two constraints, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a not-unnoticeable amount of damage on the interval being stacked*. This is arguably what truly makes them feel awkward in JI, as they are also small enough to feel like potential commas without being very efficient to temper out. *Notably, some of these are more debatable than others in terms of damage, so one should clarify that the minimum damage logically is that of the minimal diesis (6.9{{c}}), as more than 7 cents of damage is not insignificant for most intervals and is essentially a flexibility afforded by LCJI's temperability.
+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:
+/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, [[19683/19208]] = (9/7)<sup>4</sup> / (4/3) = (9/7)<sup>5</sup> / (12/7), [[16807/16384]] = (7/4) / (8/7)<sup>4</sup> = (2/1) / (8/7)<sup>5</sup>, 36/35, 250/243
+(*which might be the tritonic diesis by contrast with the septimal tritonic diesis of 50/49)
+(**called a diesis in a theory of [[Lériendil]]'s that uses a definition of diesis currently not documented on this page)
+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.)
 == As a diatonic interval category ==