Characteristic polynomials and inverse matriies of interval matricies

It seems these might be worth looking at, especially for a
normalization of intervals to the octave. For

1--9/8--5/4--4/3--3/2--5/3--15/8

I find that the characteristic polynomial is

2*x^7 - (x+1)^7 = x^7-7*x^6-21*x^5-35*x^4-35*x^3-21*x^2-7*x-1

The inverse matrix has -1 along the main diagonal, with the steps of
the scale in a circulating diagonal below it--9/8, 10/9, 16/15, 9/8,
10/9, 9/8 plus a 16/15 in the upper right-hand corner to complete the
circulating diagonal.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> It seems these might be worth looking at, especially for a
> normalization of intervals to the octave. For
>
> 1--9/8--5/4--4/3--3/2--5/3--15/8
>
> I find that the characteristic polynomial is
>
> 2*x^7 - (x+1)^7 = x^7-7*x^6-21*x^5-35*x^4-35*x^3-21*x^2-7*x-1

Unfortunately, this is merely a consequence of the fact that it is an
interval matrix; it tells us nothing about the scale.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > It seems these might be worth looking at, especially for a
> > normalization of intervals to the octave. For
> >
> > 1--9/8--5/4--4/3--3/2--5/3--15/8
> >
> > I find that the characteristic polynomial is
> >
> > 2*x^7 - (x+1)^7 = x^7-7*x^6-21*x^5-35*x^4-35*x^3-21*x^2-7*x-1
>
> Unfortunately, this is merely a consequence of the fact that it is
an
> interval matrix; it tells us nothing about the scale.

I would be interested in learning how to derive characteristic
polynomials from interval matrices or scales. Thanks!

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> I would be interested in learning how to derive characteristic
> polynomials from interval matrices or scales. Thanks!

It's just the usual characteristic polynomial of a matrix, but
unfortunately does not seem to be telling us anything new.