The analog of the Wilson fifth for the schismic temperament

It turns out there is something closely analogous to the Wilson fifth
for the schismic temperament--in this case it is a fourth. The Wilson
fifth is the positive real root of x^4 - 2x - 2. It gives brats of -1,
which is a particularly interesting value, and is a Perron number,
meaning all of its conjugates are less than it in absolute value. It
is also a monic polynomial with all coefficients zero or +- a power of
two. Since it is Perron the ratios of the corresponding linear
recurrences converge, and from this and the power of two property we
have metameantone.

Very similarly, x^9 - 4x - 8 gives us a Perron number which is a
fourth which works well as a generator for schismic, and it also gives
a brat of -1. It's just barely Perron, but it could be used for
metaschismic if one were so inclined.

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

> Very similarly, x^9 - 4x - 8 gives us a Perron number which is a
> fourth which works well as a generator for schismic, and it also
gives
> a brat of -1. It's just barely Perron, but it could be used for
> metaschismic if one were so inclined.

I forgot to add that this fourth is a poptimal generator, and in this
respect bests the Wilson fifth.

on 2/9/03 6:28, Gene Ward Smith at gwsmith@svpal.org wrote:

> It turns out there is something closely analogous to the Wilson fifth
> for the schismic temperament--in this case it is a fourth. The Wilson
> fifth is the positive real root of x^4 - 2x - 2. It gives brats of -1,
> which is a particularly interesting value, and is a Perron number,
> meaning all of its conjugates are less than it in absolute value. It
> is also a monic polynomial with all coefficients zero or +- a power of
> two. Since it is Perron the ratios of the corresponding linear
> recurrences converge, and from this and the power of two property we
> have metameantone.
>
> Very similarly, x^9 - 4x - 8 gives us a Perron number which is a
> fourth which works well as a generator for schismic, and it also gives
> a brat of -1. It's just barely Perron, but it could be used for
> metaschismic if one were so inclined.
>

I knew that : - )

--- In tuning@yahoogroups.com, Alison Monteith
<alison.monteith3@w...> wrote:
> on 2/9/03 6:28, Gene Ward Smith at gwsmith@s... wrote:
>
> > It turns out there is something closely analogous to the Wilson
fifth
> > for the schismic temperament--in this case it is a fourth. The
Wilson
> > fifth is the positive real root of x^4 - 2x - 2. It gives brats
of -1,
> > which is a particularly interesting value, and is a Perron number,
> > meaning all of its conjugates are less than it in absolute value.
It
> > is also a monic polynomial with all coefficients zero or +- a
power of
> > two. Since it is Perron the ratios of the corresponding linear
> > recurrences converge, and from this and the power of two property
we
> > have metameantone.
> >
> > Very similarly, x^9 - 4x - 8 gives us a Perron number which is a
> > fourth which works well as a generator for schismic, and it also
gives
> > a brat of -1. It's just barely Perron, but it could be used for
> > metaschismic if one were so inclined.
> >
>
> I knew that : - )

seriously, gene, please restrict posts like this to the tuning-math
list (with a mere by-line here if you want to attract attention), and
when you do post them there, g . . . o . . . . . . . . . . . m . . .
o . . . r . . . e . . . . . . . . . . s . . . l . . . o . . .
w . . . l . . . y.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> seriously, gene, please restrict posts like this to the tuning-math
> list (with a mere by-line here if you want to attract attention),
and
> when you do post them there, g . . . o . . . . . . . . . . .
m . . .
> o . . . r . . . e . . . . . . . . . . s . . . l . . . o . . .
> w . . . l . . . y.

I thought the topic would be of more general interest.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> > seriously, gene, please restrict posts like this to the tuning-
math
> > list (with a mere by-line here if you want to attract attention),
> and
> > when you do post them there, g . . . o . . . . . . . . . . .
> m . . .
> > o . . . r . . . e . . . . . . . . . . s . . . l . . . o . . .
> > w . . . l . . . y.
>
> I thought the topic would be of more general interest.

maybe if you fully explained each step in what you were saying. at it
stood, i seriously doubt anyone here understood it. seriously.