Pure octave TOP-RMS/pesuodinverse/Frobenius tuning

One question one might ask is what the pure octave equivalent to this
tuning is. An obvious approach is to shrink or stretch the tuning so
that it has pure octaves. However, another is the pure-octave
Frobenius tuning I mentioned some time back.

TOP-RMS/Frobenius/pseudoinverse tuning has the bizzare feature,
resulting from the symmetry of the projection matrix for the tuning,
that the eigenmonzos are simply the vals consistent with the tuning,
reinterpreted as monzos. For example, take the TOP-RMS tuning of
septimal meantone, which has <12 19 28 34| as a val, and now
|12 19 28 34> will be an interval which the tuning leaves pure. This
feature, when I came across it in the context of Frobenius tuning,
made me think it was somewhat artificial, though clearly the results
were looking quite reasonable.

However, since the tuning keeps coming up and has various means of
justification, I regard it now as a feature. The tuning can in fact
be defined as the tuning which sends the kernel elements to the
unison, and leaves fixed all the vals of the tuning, when these are
reinterpreted as monzos.

Now suppose we want to keep octaves pure. This entails that 2 is an
eigenmonzo, and one way of defining a tuning which does it, the pure-
octave constrained Frobenius tuning, is as the tuning which sends the
kernel elements to the unison, and leaves fixed 2 and all the vals of
the system which don't involve 2--that is, which send 2 to the unison.
If we are looking at a rank two temperament, we can get such an
eigenmonzo from the OE part of the wedgie. For instance, from
<<1 4 10 4 13 12|| we get |0 1 4 10> as an eigenmonzo; that is,
3 * 5^4 * 7^10 = 529641091875 is, along with 2, left fixed by the
tuning. In this case we get a fifth of 696.884345 cents, which
happens to be what you need to accomplish all of this; this is the
octave-constrained Frobenius septimal meantone tuning. It's close to
31-et and 4/17-comma, and is clearly a reasonable meantone tuning.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> One question one might ask is what the pure octave equivalent to this
> tuning is.

Added a bunch of stuff to the new web page on this.

http://66.98.148.43/~xenharmo/inverses.htm

On 06/03/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@...> wrote:
> >
> > One question one might ask is what the pure octave equivalent to this
> > tuning is.
>
> Added a bunch of stuff to the new web page on this.
>
> http://66.98.148.43/~xenharmo/inverses.htm

Okay, I can get this. Scratch my last message.

Graham