Re: [tuning] fractional exponents of prime-factors

Joe wrote:
>The main use I have for it is that I can create lattices
>of tempered tunings which plot on top of my JI lattices.

>I'm intrigued by this because Paul Erlich has said that
>prime-factoring tempered ratios isn't useful, because
>the fractional exponents do not give a unique factorization.

The consequence of this is that the same tone can show up in
multiple places in the lattice. That's the reason that the
III-V lattice of 12-tET gets warped into a torus shape.

Manuel

> From: <manuel.op.de.coul@eon-benelux.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, July 18, 2001 7:55 AM
> Subject: Re: [tuning] fractional exponents of prime-factors
>
>
>
> Joe wrote:
> >The main use I have for it is that I can create lattices
> >of tempered tunings which plot on top of my JI lattices.
>
> >I'm intrigued by this because Paul Erlich has said that
> >prime-factoring tempered ratios isn't useful, because
> >the fractional exponents do not give a unique factorization.
>
> The consequence of this is that the same tone can show up in
> multiple places in the lattice. That's the reason that the
> III-V lattice of 12-tET gets warped into a torus shape.

Thanks, Manuel. Right, I do understand that. The lattices
I'm creating showing both the JI pitches and the tempered
ones on top of them would be better if I could add the
math to my lattice formula so that they would be drawn as
a torus. Any ideas on that?

My lattice formula itself is in the JustMusic code, of course.
I've been looking for the tuning list post where I quote
my lattice formula and couldn't find it yet. I'll have
to resume tonight. It would take too much time right now
to generalize the variables I use in my actual Excel formula.

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: <manuel.op.de.coul@e...>
> > To: <tuning@y...>
> > Sent: Wednesday, July 18, 2001 7:55 AM
> > Subject: Re: [tuning] fractional exponents of prime-factors
> >
> >
> >
> > Joe wrote:
> > >The main use I have for it is that I can create lattices
> > >of tempered tunings which plot on top of my JI lattices.
> >
> > >I'm intrigued by this because Paul Erlich has said that
> > >prime-factoring tempered ratios isn't useful, because
> > >the fractional exponents do not give a unique factorization.

> > Manuel wrote:
> > The consequence of this is that the same tone can show up in
> > multiple places in the lattice. That's the reason that the
> > III-V lattice of 12-tET gets warped into a torus shape.
>
> Joe wrote:
> The lattices
> I'm creating showing both the JI pitches and the tempered
> ones on top of them would be better if I could add the
> math to my lattice formula so that they would be drawn as
> a torus. Any ideas on that?

What is the meaning of such a periodicity (wrapping of the "III-V
lattice" (of 12-tones scales) lattice into a "torus" shape?

Is this the same thing as a tone appearing in "multiple places" in a
lattice?

How could one (algebraically) solve for "where" in a (non-integer
power) lattice the tone would appear (since multiple combinations of
primes to various powers could result in the same interval as a
result), where no algebraic process exists to do so. Iteration?

Maybe I am misinterpreting or missing something here?

Respectfully, J Gill

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> From: J Gill <JGill99@imajis.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, July 20, 2001 5:06 AM
> Subject: [tuning] Re: fractional exponents of prime-factors
>
>
> > > Manuel wrote:
> > > The consequence of this is that the same tone can show up in
> > > multiple places in the lattice. That's the reason that the
> > > III-V lattice of 12-tET gets warped into a torus shape.
>
> What is the meaning of such a periodicity (wrapping of the "III-V
> lattice" (of 12-tones scales) lattice into a "torus" shape?
>
> Is this the same thing as a tone appearing in "multiple places" in a
> lattice?

Yes. Manuel is referring to the fact that in 1/4-comma meantone,
the lattice-point representing the "major 3rd" must fall at the
5:4 position, but must also represent 4 tempered 3:2s. Thus,
the lattice has to be twisted into a torus so that each 3:2
goes 1/4 of the way towards 5^1 while 5:4 goes towards 3^4.

-monz
http://www.monz.org
"All roads lead to n^0"

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