Re: [tuning] Re: fractional exponents of prime-factors

Hi Jay,

You posted this question to the Tuning List, where I
sent an abbreviated reply. But I feel it's more appropriate
to send the full reply here (tuning-math, with a copy sent
to justmusic).

> 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.

Can anyone provide the math for this? (see my formula below)

>
> 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?

Unfortunately, Jay, I myself don't understand algebra well
enough to answer this one. I'm even more confused about it
than you are.

The Visual Basic code for my lattice formula is here
/justmusic/topicId_unknown.html#48

Basically, this is it:

'----------- BEGIN CODE ------------

'The apostrophe ' indicates a comment.

' DECLARE VARIABLES
'------------------

p 'prime-factor
ar(p) 'angle (in radians) of p
exp(p) 'exponent of p (+ or -)

op(x) 'horizontal origin-point
op(y) 'vertical origin-point

rp(x) 'horizontal ratio plot
rp(y) 'vertical ratio plot

'CALCULATE ANGLE OF PRIME
'-------------------------

for p=2, 3, 5, 7, 11, 13 '...etc., the prime series

ar(p)=((3-(LOG(p)*12/LOG(2)))*PI)/6

next p

'CALCULATE RATIO-PLOT
'---------------------

rp(x)=op(x)
rp(y)=op(y)

for p=2, 3, 5, 7, 11, 13 '...etc., the prime series

start-vector(p)=rp(x),rp(y)

rp(x)=rp(x)+((cos(ar(p))*rp(x))*(exp_num(p)/exp_denom(p)))*p
rp(y)=rp(x)+((sin(ar(p))*rp(y))*(exp_num(p)/exp_denom(p)))*p

end-vector(p)=rp(x),rp(y)

next p

'--------------- END CODE ----------------

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

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> 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

Not a torus, but a cylinder. A torus is created if you further temper
out an additional unison vector, creating a closed temperament.

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, July 20, 2001 8:42 PM
> Subject: [tuning-math] [tuning] Re: fractional exponents of prime-factors
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > 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
>
> Not a torus, but a cylinder. A torus is created if you further temper
> out an additional unison vector, creating a closed temperament.

Oops... my bad! Thanks for catching that, Paul.
I knew that, but got my shapes mixed up with the
wrong types of temperaments.

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

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