Pure octave tunings

The POTE tuning for any temperament is the TE tuning, stretched
linearly so that octaves are equal. So for instance, the POTE tuning
for 5-limit meantone is <1200 1896.238 2784.954|.

There's another relevant pure-octave TE tuning, given by the tuning
map <1200 a b c ...| that directly minimizes the T2 distance to the
JIP. I'll call this bizarro-POTE. For 5-limit meantone, this is <1200
1897.214 2788.857|.

Everyone goes with POTE instead of bizarro-POTE. Paul's been known to
advocate strongly against bizarro-POTE, under the claim that
bizarro-POTE isn't optimal for "all intervals" in the same sense that
POTE is. However, this latter point doesn't seem to be true. For
5-limit meantone, among tuning maps starting with <1200 ...|,
bizarro-POTE minimizes the T2 distance to the JIP. This means that it
minimizes err(m)/||m|| for all monzos m. Normal POTE, in fact, doesn't
have this property at all.

What normal POTE does, instead is to minimize the angle from the JIP.
But why does this matter? What music theoretic properties does
angle-minimization have? I've only heard some intuitionistic geometric
reasoning from Paul on this, or some assertions that since TE is
optimal, it remains optimal if you stretch the octave, but this is NOT
true algebraically, at least not in the sense that we mean optimal: by
minimizing weighted error over all intervals. So why do it this way
instead of the bizarro way?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> However, this latter point doesn't seem to be true. For
> 5-limit meantone, among tuning maps starting with <1200 ...|,
> bizarro-POTE minimizes the T2 distance to the JIP. This means that it
> minimizes err(m)/||m|| for all monzos m. Normal POTE, in fact, doesn't
> have this property at all.
>

I've been saying this for probably 6 months now, but everyone thought I was crazy! Of course the thing we call "POTE" doesn't minimize anything.

Consider what would happen if we stretched the octave even further, such as to 1260 cents. Stretching all of the other intervals proportionally doesn't give us the best tuning around this quasi-octave. So why should it give us the best tuning when we stretch it to 1200 cents?

Ryan

On Mon, Sep 10, 2012 at 1:33 PM, Ryan Avella <domeofatonement@yahoo.com>
wrote:
>
> I've been saying this for probably 6 months now, but everyone thought I
> was crazy! Of course the thing we call "POTE" doesn't minimize anything.

It minimizes the angle around the JIP, and Paul gave a sort of
convincing sounding argument for why that would be a good thing. But,
algebraically, it doesn't hold up.

I suspect that the real reason for doing POTE this way is because it
deliberately skews things to be more in line with Kees expressibility,
so that you weight intervals by odd-limit rather than Tenney height. I
think this might just be a coincidence or general trend, though.

Anyway, the real question is, now that we have an actual Weil tuning,
we can come up with two different notions of the tuning that
"minimizes the max odd-limit weighted error over all intervals," which
has to be the holy grail of tuning optimization, at least for
pure-octave intervals. So which one is right? Is it the Weil tuning,
stretched linearly so the octave becomes 1200 cents? Or is it the
tuning map on the line of maps starting with <1200 ... | that's
supported by the temperament and directly minimizes the W1 distance to
the JIP?

Once I finish figuring Keenan's TIPTOP algorithm out, we can test both
of these with actual numbers. (Hell, we can do that with the W2 norm
right now, even though I haven't formally introduced that yet. But,
you all already know what it is, it's just the L2 norm of the vector
|a b c ... ; (a+b+c+...)>. Gene and I have it already coded up too
with the pseudoinverse.)

-Mike