Superparticularity

I need to get back up to speed --- so i was looking at the blues scale, and the V7/I scale, and then started to play with this a bit, and found that, in the 7-limit at least, all my commas ended up being superparticular (epimoric), in situations where I work with a linear chain of values, like 12,13,14,15,16,17,18,19,20,21,22,23,24 and then approximate them in the same logarithmic division (here good ol' 12-EDO)

For example, 12,14,15,16,18,20 are right on in the 7-limit, as would would expect (this is trivial), and I found every single other approximation for 13,17,19,22 and 23 are superparticular (except that in the case of 22 or 23 you have to chose the best one) ---

For example, in the 7-limit, 13 has a comma of 27/26, 17 can be
136/135, 256/255, 85/84, 19 is 96/95, 57/56, 22 is 45/44, 23 is 46/45, and every single one is superparticular --- this would be a fun easy proof to write, has it been done? Is this true in all limits (like 3-limit, 5-limit) and in all linear splits of the octave, I wonder....

PGH

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:

> For example, in the 7-limit, 13 has a comma of 27/26, 17 can be
> 136/135, 256/255, 85/84, 19 is 96/95, 57/56, 22 is 45/44, 23 is 46/45, and every single one is superparticular --- this would be a fun easy proof to write, has it been done? Is this true in all limits (like 3-limit, 5-limit) and in all linear splits of the octave, I wonder....

Are you asking if a patent val can always be characterized in terms of superparticular commas?

On Tue, Nov 27, 2012 at 1:14 PM, Paul <phjelmstad@msn.com> wrote:
>
> For example, 12,14,15,16,18,20 are right on in the 7-limit, as would would
> expect (this is trivial), and I found every single other approximation for
> 13,17,19,22 and 23 are superparticular (except that in the case of 22 or 23
> you have to chose the best one) ---
>
> For example, in the 7-limit, 13 has a comma of 27/26, 17 can be
> 136/135, 256/255, 85/84, 19 is 96/95, 57/56, 22 is 45/44, 23 is 46/45, and
> every single one is superparticular --- this would be a fun easy proof to
> write, has it been done? Is this true in all limits (like 3-limit, 5-limit)
> and in all linear splits of the octave, I wonder....

What val are you using for 17-EDO?

All of these vals also temper out lots of non-superparticular commas.
Are you asking if it's guaranteed that they also temper out
superparticular ones?

-Mike