The wedge invariant commas

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 f�r 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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