Finding useful chromas in tempered systems

The post about the meantone intervals vanishing in 19-EDO and 31-EDO
have led me to consider the problem of finding small useful chromas in
tempered systems in general. These are useful for understanding what
sorts of complex "enharmonic equivalences," from the perspective of
some rank-2 temperament, exist in the various EDOs tempering out the
chroma, and are also useful for understanding the general structure of
the higher enharmonic-sized MOS's of that temperament.

To start picking this apart, we're generally looking for intervals
that have the following characteristics:
1) They're of low complexity
2) They appear often between other common or important intervals
3) They're small

#1 and #2 are the same thing, which is still just "low complexity", so
as the complexity goes up, we care less.

#3 poses a problem: we can't just look for "small" intervals, because
unlike with JI, we don't have a single privileged tuning map to use to
evaluate interval sizes like the JIP. Fortunately, there's a simple
solution: if intervals are small, then when we temper them out (and
bring us back down to rank-1), then the increase in optimal error from
the additional tempering will be small. So rather than worry about
size, we can simply worry about error.

So our three constraints above are very well satisfied by just
worrying about the error and complexity of the tempered chromas - or,
in more familiar terms, the badness of the chromas, which it's now
clear are just commas in tempered systems. The whole thing turns out
to be pretty simple if you approach it using tvals and tmonzos, which
I wrote about here:
http://xenharmonic.wikispaces.com/Tmonzos+and+Tvals

So, simply stated, here's the algorithm to assign any tempered
interval a badness:
1) Pick a tempered interval T, and represent it in tmonzo form under
some basis given by a mapping matrix M
2) Compute T°, the tval which is the dual of T
3) Right-multiply by the mapping matrix to find the unique val that
this tval corresponds to. Let V = T°·M
4) Compute the badness of V

And now you're done. In fact, Graham's temperament finder already does
this - see the EDOs at the top here:
http://x31eq.com/cgi-bin/uv.cgi?uvs=81/80. I'm not sure how they're
being sorted though, as 5 appears before 26 (we can't specify a target
error for the UV search like we can with the general temperament
search).

-Mike

On Tue, Oct 23, 2012 at 1:46 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> So our three constraints above are very well satisfied by just
> worrying about the error and complexity of the tempered chromas - or,
> in more familiar terms, the badness of the chromas, which it's now
> clear are just commas in tempered systems.

One useful corollary of this is that this applies to any temperament
of any rank, not just rank-2. This means it also applies to JI, if
you're looking for the best JI chromas.

This can be useful in notation system design. For instance, say you're
designing a JI notation system a la Sagittal, where you want a set of
accidentals representing alterations by JI interval. But which JI
intervals do you care about? You definitely don't want accidentals
corresponding to intervals like 5/4, and you get diminishing returns
once you start diving into things as complex as 184549376/184528125.

By going through the same thought process as in the last email, it
becomes clear that the set of commas you care about are those lowest
in badness, and hence dual to the set of temperaments lowest in
badness. So for instance, if you want to design a 5-limit JI notation
system, you'll want accidentals corresponding well to this set of
commas:

http://x31eq.com/cgi-bin/more.cgi?r=2&limit=5&error=10.0

I've cranked the error up to 10.0 because we're not intending to
temper these intervals out, so complexity counts more and size counts
less than we're used to.

Of course, practically speaking, any notation system you'd want to
design will have some accidentals that differ from exactly those on
this list, because you'll need to take into account which are
important from the standpoint of your chain of nominals.

-Mike

Mike Battaglia <battaglia01@gmail.com> wrote:

> And now you're done. In fact, Graham's temperament finder
> already does this - see the EDOs at the top here:
> http://x31eq.com/cgi-bin/uv.cgi?uvs=81/80. I'm not sure
> how they're being sorted though, as 5 appears before 26
> (we can't specify a target error for the UV search like
> we can with the general temperament search).

There are "Simpler" and "More accurate" buttons.

Graham