Wilson and recurrences

I've been playing with these, and don't know exactly why. To me the
most interesting thing about the Wilson fifth is that it gives a
meantone brat of -1; however Wilson thought it was important that it
was the limit of a linear recurrence.

He also used a linear recurrence for something called "chopi", which
may or may not be more or less the linear temperament I've been
calling hexidecimal, with commas 36/35 and 135/128. If so, it seems we
should call that one "chopi". He used the characteristic polynomial
x^4-x-2, which gives

C[n] = C[n-2] + 2 C[n-4]

This polynomial is reducible, equalling (x+1)(x^3-x^2+x-2), so we
could also use

C[n] = C[n-1] - C[n-2] + 2 C[n-3]

This is a cubic rather than a quartic recurrence, and so would seem to
be preferable. It may be that since if zero coefficients are ignored
we have a three-term rather than a four-term recurrence, Wilson
preferred his version.

I've two questions:

(1) What musical value attaches to these linear recurrences? As usual,
reading Wilson is difficult for me; he seems to think it is important
but I can't tell why.

(2) Why did he choose the fourth-order recurrence when he could have
used a cubic recurrence?

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

> I've been playing with these, and don't know exactly why. To me the
> most interesting thing about the Wilson fifth is that it gives a
> meantone brat of -1; however Wilson thought it was important that it
> was the limit of a linear recurrence.

the two observations are really one and the same. consider a sequence
of simple harmonic-series "diatonic scales" which have this brat for
each triad (easy to arrange within a harmonic series) but are not
regularly tempered (of course, they're not tempered at all). applying
the recursion to the scale as a whole (say, by dropping the lowest
note and replacing it with the next higher note in the original
sequence, and transposing by a fifth if necessary), the limit is the
regular metameantone.

> He also used a linear recurrence for something called "chopi", which
> may or may not be more or less the linear temperament I've been
> calling hexidecimal, with commas 36/35 and 135/128. If so, it seems
we
> should call that one "chopi".

or "chopic", an extension of "pelogic" (135/128) to the 7-limit.

> (1) What musical value attaches to these linear recurrences? As
usual,
> reading Wilson is difficult for me; he seems to think it is
important
> but I can't tell why.

it is possible to tune by combinational tones when the scale is
generated by a linear recurrence.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > I've been playing with these, and don't know exactly why. To me
the
> > most interesting thing about the Wilson fifth is that it gives a
> > meantone brat of -1; however Wilson thought it was important that
it
> > was the limit of a linear recurrence.

Most temperaments do not give a Perron generator for the generator
with brat -1. Schismic is exceptional in that respect; it might be
worth a systematic look.

> the two observations are really one and the same. consider a
sequence
> of simple harmonic-series "diatonic scales" which have this brat
for
> each triad (easy to arrange within a harmonic series) but are not
> regularly tempered (of course, they're not tempered at all).
applying
> the recursion to the scale as a whole (say, by dropping the lowest
> note and replacing it with the next higher note in the original
> sequence, and transposing by a fifth if necessary), the limit is
the
> regular metameantone.

True of metameantone, but not true in general.

> it is possible to tune by combinational tones when the scale is
> generated by a linear recurrence.

I haven't thought of that. When I asked the question, I still hadn't
realized it was about constructing scales.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > > I've been playing with these, and don't know exactly why. To me
> the
> > > most interesting thing about the Wilson fifth is that it gives a
> > > meantone brat of -1; however Wilson thought it was important
that
> it
> > > was the limit of a linear recurrence.
>
> Most temperaments do not give a Perron generator for the generator
> with brat -1. Schismic is exceptional in that respect; it might be
> worth a systematic look.
>
> > the two observations are really one and the same. consider a
> sequence
> > of simple harmonic-series "diatonic scales" which have this brat
> for
> > each triad (easy to arrange within a harmonic series) but are not
> > regularly tempered (of course, they're not tempered at all).
> applying
> > the recursion to the scale as a whole (say, by dropping the
lowest
> > note and replacing it with the next higher note in the original
> > sequence, and transposing by a fifth if necessary), the limit is
> the
> > regular metameantone.
>
> True of metameantone, but not true in general.

for which wilson "meta-" example is it not true? how about a made-up
example?

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

> > True of metameantone, but not true in general.
>
> for which wilson "meta-" example is it not true? how about a made-
up
> example?

The Wilson fifth is Perron--all of its conjugates are less than it in
absolute value. The schismic fourth with brat -1 is also Perron. The
point is, this is not normally the case, so this meta business does
not normally work.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> > > True of metameantone, but not true in general.
> >
> > for which wilson "meta-" example is it not true? how about a made-
> up
> > example?
>
> The Wilson fifth is Perron--all of its conjugates are less than it
in
> absolute value. The schismic fourth with brat -1 is also Perron.
The
> point is, this is not normally the case, so this meta business does
> not normally work.

ADVERTISEMENT

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> > > True of metameantone, but not true in general.
> >
> > for which wilson "meta-" example is it not true? how about a made-
> up
> > example?
>
> The Wilson fifth is Perron--all of its conjugates are less than it
in
> absolute value. The schismic fourth with brat -1 is also Perron.
The
> point is, this is not normally the case, so this meta business does
> not normally work.

hopefully i won't have to say this again, but please explain this
step-by-step for dummies (by conjugates you presumably don't mean
complex conjugates, since those have the same absolute value, so
explain what you do mean and why -- that sort of thing) and/or post
it to the tuning-math list -- preferably *and*.

now, i asked for an example. does it work for all the "meta-"
examples that kraig brought up? if not, you should have given one of
them as an example. otherwise, you were supposed to make up an
example. you had a long streak of never actually answering the
questions you were responding to. apparently you're reverting to this
behavior. :(

also, if you could avoid mentioning brats and the like in your
explanation, that would be best, because as i understand kraig's
portrayal of erv the process in general has nothing to do with
beating or brats, but is instead founded on combinational tones, as
in general the timbres may have inharmonic partials or none
whatsoever.

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

> hopefully i won't have to say this again, but please explain this
> step-by-step for dummies (by conjugates you presumably don't mean
> complex conjugates, since those have the same absolute value, so
> explain what you do mean and why -- that sort of thing) and/or post
> it to the tuning-math list -- preferably *and*.

Here's the World of Math scoop on algebraic conjugates:

http://mathworld.wolfram.com/ConjugateElements.html

It is not necessary to read or absorb this, since we can put the
matter in another way. We have characteristic polynomials for
recurrences. If the largest real root is positive, and larger in
absolute value than all of the other roots, then we have the property
we desire, for which I don't know a name (hence my invocation of
"Perron", which is related.) We could call it metastable or something,
I suppose.

> also, if you could avoid mentioning brats and the like in your
> explanation, that would be best, because as i understand kraig's
> portrayal of erv the process in general has nothing to do with
> beating or brats, but is instead founded on combinational tones, as
> in general the timbres may have inharmonic partials or none
> whatsoever.

This business has a lot to do with brats, and b=-1 in particular, so
this is a bad idea.

The most obvious example of something which isn't "metastable" is
perhaps the Wilson fourth, for which all of the conjugates are larger
in absolute value. Here is this and some other b=-1 examples:

Wilson fifth (metastable): x^4-2*x-2 = 0

Example recurrence:

d[1, 1, 1, 1, 4, 4, 4, 10, 16, 16, 28, 52, 64, 88, 160, 232, 304, 496,
784, 1072, 1600]

The ratios converge to the Wilson fifth

Wilson fourth (not metastable): y^4+2*y^3-8 = 0

Example recurrence:

[1, 1, 1, 1, 6, 6, 6, -4, 36, 36, 56, -104, 216, 176, 656, -1264,
1376, 96, 7776, -12864, 10816]

The ratios do not converge

Semisixths generator (metastable): z^9-z^7-4 = 0

[1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 9, 9, 13, 13, 17, 17, 21, 37, 41, 73]

Orwell generator (not metastable): w^10+2*w^3-8 = 0

Example recurrence:

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 6, 6, 6, -4, -4, -4, 36]

Orwell generator (metastable): x^10-2*x^7-128

Example recurrence:

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 130, 130, 130, 388, 388, 388, 904, 904,
904, 1936, 18448]

The degree of the minimal polynomial for these generators equals the
5-limit Graham complexity, by the way.

>We have characteristic polynomials for
>recurrences.

The root(s) of which are the convergents of
the recurrence?

or...

>If the largest real root is positive, and larger in
>absolute value than all of the other roots, then we have
>the property we desire,

For the record, what is that property? You refer to it
earlier as "meta business".

>The most obvious example of something which isn't "metastable" is
>perhaps the Wilson fourth, for which all of the conjugates are larger
>in absolute value. Here is this and some other b=-1 examples:
>
>Wilson fifth (metastable): x^4-2*x-2 = 0
>
>Example recurrence:
>
>d[1, 1, 1, 1, 4, 4, 4, 10, 16, 16, 28, 52, 64, 88, 160, 232, 304, 496,
>784, 1072, 1600]
>
>The ratios converge to the Wilson fifth
>
>Wilson fourth (not metastable): y^4+2*y^3-8 = 0
>
>Example recurrence:
>
>[1, 1, 1, 1, 6, 6, 6, -4, 36, 36, 56, -104, 216, 176, 656, -1264,
>1376, 96, 7776, -12864, 10816]
>
>The ratios do not converge
>
>
>Semisixths generator (metastable): z^9-z^7-4 = 0
>
>[1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 9, 9, 13, 13, 17, 17, 21, 37, 41, 73]
>
>
>Orwell generator (not metastable): w^10+2*w^3-8 = 0
>
>Example recurrence:
>
>[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 6, 6, 6, -4, -4, -4, 36]
>
>Orwell generator (metastable): x^10-2*x^7-128
>
>Example recurrence:
>
>[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 130, 130, 130, 388, 388, 388, 904, 904,
>904, 1936, 18448]
>
>The degree of the minimal polynomial for these generators equals the
>5-limit Graham complexity, by the way.

Hrm...

-Carl