Diesis: Difference between revisions

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== As an interval region ==

As an interval region, the diesis is a small melodic unit of about an augmented comma's size. The specific range varies considerably among musicians, but is generally agreed to be roughly 30–60 cents.

As an interval region, the diesis is a small melodic unit of about an augmented comma's size. The specific range varies considerably among musicians, but is generally agreed to be roughly 30–60 cents. In [[Sagittal notation]], a diesis is specifically defined as between half of the [[17-comma]] {{monzo| 27 -17 }} and half of the [[19-comma]] {{monzo| -30 19 }}, about 33.4{{c}} to 68.6{{c}}<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.

=== Just intervals ===

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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 being slightly larger than 250/243), 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 historical 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 ''at most'', one is 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.)

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.)

== As a diatonic interval category ==

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== External links ==

* [http://tonalsoft.com/enc/d/diesis.aspx Tonalsoft Encyclopedia | ''Diesis'']

== References ==

[[Category:Terms]]

[[Category:MOS scale]]

[[Category:Interval size measures]]

@@ Line 6: / Line 6: @@
 == As an interval region ==
-As an interval region, the diesis is a small melodic unit of about an augmented comma's size. The specific range varies considerably among musicians, but is generally agreed to be roughly 30–60 cents.
+As an interval region, the diesis is a small melodic unit of about an augmented comma's size. The specific range varies considerably among musicians, but is generally agreed to be roughly 30–60 cents. In [[Sagittal notation]], a diesis is specifically defined as between half of the [[17-comma]] {{monzo| 27 -17 }} and half of the [[19-comma]] {{monzo| -30 19 }}, about 33.4{{c}} to 68.6{{c}}<ref>[https://sagittal.org/sagittal.pdf ''Sagittal – A Microtonal Notation System''] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>.
 === Just intervals ===
@@ 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 being slightly larger than 250/243), 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 historical 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 ''at most'', one is 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.)
+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 ==
@@ Line 41: / Line 59: @@
 == External links ==
 * [http://tonalsoft.com/enc/d/diesis.aspx Tonalsoft Encyclopedia | ''Diesis'']
+== References ==
+<references/>
 [[Category:Terms]]
 [[Category:MOS scale]]
 [[Category:Interval size measures]]