Diesis: Difference between revisions

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For someone looking for what sets these just intervals apart from others in the size range, it might be observed that almost all just intervals that have been 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]]{{idio}}, as characteristic of intervals of a size between that of the minimal diesis (27.66{{c}}) and maximal diesis (49.17{{c}}).

* Equating a ''short'' stack of one [[LCJI]] interval with some other LCJI interval (never more than 4 or 5). 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.

As a result, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a noticeable amount of damage on the interval being stacked~~<ref group="note">~~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 too much for most complex intervals and is essentially a flexibility afforded by LCJI's temperability~~.</ref>~~. 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.

As a result, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a noticeable amount of damage on the interval being stacked. 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 too much for most complex intervals and is essentially a flexibility afforded by LCJI's temperability. 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.

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)5, 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:

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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>.)

(Also interesting is that the maximal diesis of 250/243 isn't only maximum in terms of size in cents but in terms of damage implied in a tuning that makes the target (6/5) pure, meaning half the damage is on each of two 10/9's.)

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]])3, S-expression [[16/15|S4]]/[[25/24|S5]]) as it is equal to ([[7/4]])/([[6/5]])3 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.

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 For someone looking for what sets these just intervals apart from others in the size range, it might be observed that almost all just intervals that have been 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]]{{idio}}, as characteristic of intervals of a size between that of the minimal diesis (27.66{{c}}) and maximal diesis (49.17{{c}}).
 * Equating a ''short'' stack of one [[LCJI]] interval with some other LCJI interval (never more than 4 or 5). 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.
-As a result, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a noticeable amount of damage on the interval being stacked<ref group="note">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 too much for most complex intervals and is essentially a flexibility afforded by LCJI's temperability.</ref>. 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.
+As a result, when tempered out and in a tuning that makes the other simple interval of interest pure, all dieses incur a noticeable amount of damage on the interval being stacked. 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 too much for most complex intervals and is essentially a flexibility afforded by LCJI's temperability. 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.
 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:
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 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>.)
+(Also interesting is that the maximal diesis of 250/243 isn't only maximum in terms of size in cents but in terms of damage implied in a tuning that makes the target (6/5) pure, meaning half the damage is on each of two 10/9's.)
 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.