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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