These seem to have the interesting property of giving powers of one

simple interval as equivalent to another. Incidentally, I'm

normalizing the wedge invariant by making the 5-limit comma >= 1;

some normalization is needed to use with computers.

Here's an example from Miracle:

(1) 3^7 5^6 2^(-25) ==> (16/15)^6 ~ 3/2

(2) 2^20 3^(-2) 7^(-6); taking the square root gives

2^10 3^(-1) 7^(3) ==> (8/7)^3 ~ 3/2

(3) 2^15 5^2 7^(-7) ==> (7/4)^7 ~ 50

(4) 3^15 5^20 7^(-25); taking the fifth root gives

3^3 5^4 7^(-5) ==> (7/5)^4 ~ 27/7 and (7/5)^5 ~ 27/5

Can you guys please explain what you've been discussing

here for about the past two months? I'm totally lost.

-monz

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

> Can you guys please explain what you've been discussing

> here for about the past two months? I'm totally lost.

>

>

> -monz

Hi Monz,

There is little hope of having a full and rigorous understanding of

everything Gene is doing without some serious undergraduate and

graduate abstract algebra courses. Apparently, he himself didn't

realize how many of the important mathematical concepts he was

familiar with (torsion, multilinear algebra, . . .) actually could be

important in music theory until he got here.

But basically, the whole field of periodicity blocks and regular

temperaments seems to be on a much more solid mathematical foundation

than before. This means that all kinds of difficult particular

questions can be answered, deeper relationships between structures

discerned, and comprehensive survey conducted (now being done for the

linear temperament, octave-equivalent, 7-limit case).

Perhaps it would be best if you went back to the archives from when

you last were active here, and tried to follow as much as you could

from there, working your way to the present as slowly, and with as

many questions, as you need to.

Good luck

-Paul

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> Apparently, he himself didn't

> realize how many of the important mathematical concepts he was

> familiar with (torsion, multilinear algebra, . . .) actually could

be

> important in music theory until he got here.

It's been educational, for sure. I worked in isolation from the late

sixities to the mid eighties, and I didn't know how much I didn't

know. I mentioned that I first saw ets purely in terms of accuracy of

tuning, and then learned that they had to be understood in terms of

homomorphisms and kernels. When I got here, I found I *still* didn't

understand certain basic things, for example that ets must also be

understood in terms of associated linear temperaments.

I've also found out, as Paul says, that certain kinds of math I knew

but had not related to music were in fact relevant, so I've messed

things up around here by introducing multilinear algebra, Baker's

theorem and what-not, as well as something I (and Pierre) saw as

relevant already, namely abelian groups (or Z-modules, as Pierre

prefers to call them), and quadratic forms in connection with

lattices.

Hi Gene,

> From: ideaofgod <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, December 05, 2001 5:08 PM

> Subject: [tuning-math] Re: The wedge invariant commas

>

>

> ... so I've messed

> things up around here by introducing multilinear algebra, Baker's

> theorem and what-not, as well as something I (and Pierre) saw as

> relevant already, namely abelian groups (or Z-modules, as Pierre

> prefers to call them), and quadratic forms in connection with

> lattices.

I'm having lots of trouble understanding what's been discussed

on this list since you joined. But this bit of your post jumped

out at me, and I thought you'd find this profitable:

Mark Lindley & Ronald Turner-Smith. 1993.

_Mathematical Models of Musical Scales: A New Approach_.

Orpheus-Schriftenreihe zu Grundfragen der Musik vol. 66,

Verlag fï¿½r systematische Musikwissenschaft, Bonn-Bad Godesberg.

Lindley, Mark and Ronald Turner-Smith.

"An Algebraic Approach to Mathematical Models of Scales",

Music Theory Online vol. 0 no. 3, June 1993.

http://boethius.music.ucsb.edu/mto/issues/mto.93.0.3/mto.93.0.3.lindley.art

Lindley/Turner-Smith view tuning systems as abelian groups.

(see especially paragraph [5] of the latter article)

love / peace / harmony ...

-monz

http://www.monz.org

"All roads lead to n^0"

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