Optimized comma of a patent val

I've already remarked that given an et val, there's a height-minimal
interval which is "reversed", in the sense that it's mapped below the
unison. Here's another goofball comma associated to a patent val--the
height-minimal comma for which the patent val provides the most nearly
optimal tuning.

For instance, for 12-et in the 7-limit, the reversed comma is
1728/1715, and the optimized comma is 360/343. What these say about 12
I don't know. I don't recall ever giving any attention to 360/343.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> For instance, for 12-et in the 7-limit, the reversed comma is
> 1728/1715, and the optimized comma is 360/343. What these say about 12
> I don't know. I don't recall ever giving any attention to 360/343.

One thing to say about these optimized commas is that they rend to be
poster children for the proposition that if you temper out a comma,
you might as well temper out reasonable commas whose product is the
given comma.

For example, 360/343 = 36/35 50/40. Hence, we "might as well" temper out
36/35 and 50/49, which leads to diminished temperament. And factoring
36/35 as 64/63 81/80, we "might as well" temper out 12-et.

Similarly, for 22-et we have 3200/3087 = 64/63 50/49, so we "might as
well" go to pajara. However, another product, 250/243 * 1728/1715.
;eads to a cheeseball temperament: <<9 15 4 3 -19 33||.