Yahoo Tuning Groups Ultimate Backup tuning-math More proposed definitions

algebraic number

Algebraic numbers are the roots of polynomial equations with <integer.htm> coefficients.
A polynomial equation with integer coefficients is
a_0 x^n + a_1 x^{n-1} + ... + a_0
where the a_i. . . are integers.
x is algebraic if and only if it is the solution to such an equation.

algebraic integer

An algebraic number which satisifies a polynomial equation with integer coefficients such that the leading coefficient a_0 is 1.

algebraic number field

If r satisfies an irreducible (non-factoring) polynomial equation with integer coefficients of degree d, then the algebraic number field Q(r) is defined as the set of elements a_0 + a_1 r + ... + a_{d-1} r^{d-1}, where the coefficients a_i are rational numbers. An example of an algebraic number field would be all numbers of the form
a + b r, where r = (1+sqrt(5))/2 is the golden ratio. The elements of an algebraic number field form a field--they may be added, subtracted, multiplied, and divided.

🔗monz <joemonz@yahoo.com>

Gene, I thank you much for all these definitions.
(Won't be able to upload them until tonight at the
earliest.)

But ... isn't it getting to be more about "mathematics"
and less about "tuning"? I hesitate to put all of
these definitions directly into the Tuning Dictionary.
Perhaps there should be a separate "mathematical
supplement"?

Some of you others, please comment on this.

-monz

----- Original Message -----
From: genewardsmith <genewardsmith@juno.com>
To: <tuning-math@yahoogroups.com>
Sent: Friday, January 11, 2002 2:08 PM
Subject: [tuning-math] More proposed definitions

> algebraic number
>
> Algebraic numbers are the roots of polynomial equations with <integer.htm>
coefficients.
> A polynomial equation with integer coefficients is
> a_0 x^n + a_1 x^{n-1} + ... + a_0
> where the a_i. . . are integers.
> x is algebraic if and only if it is the solution to such an equation.
>
> algebraic integer
>
> An algebraic number which satisifies a polynomial equation with integer
coefficients such that the leading coefficient a_0 is 1.
>
> algebraic number field
>
> If r satisfies an irreducible (non-factoring) polynomial equation with
integer coefficients of degree d, then the algebraic number field Q(r) is
defined as the set of elements a_0 + a_1 r + ... + a_{d-1} r^{d-1}, where
the coefficients a_i are rational numbers. An example of an algebraic number
field would be all numbers of the form
> a + b r, where r = (1+sqrt(5))/2 is the golden ratio. The elements of an
algebraic number field form a field--they may be added, subtracted,
multiplied, and divided.
>
>
>
>
>
>
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>

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🔗paulerlich <paul@stretch-music.com>

🔗genewardsmith <genewardsmith@juno.com>

🔗monz <joemonz@yahoo.com>

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, January 11, 2002 2:24 PM
> Subject: [tuning-math] Re: More proposed definitions
>
>

> More to the point would be definitions of vector,
> vector space, lattice, bilinear form, group,
> abelian group, homomophism, kernel, equivalence
> relation, equivalence class, quotient group, graph,
> wedge product, and determinant, but this would definately
> start to look like mathematics.

Yes, well ... I think we might be agreeing that there
should be a separate area in the Dictionary for the
heavy math definitions.

But I already have definitions for vector, lattice,
matrix, and determinant -- so it *is* already beginning
to look a lot like a math dictionary. I just want to
make sure that the focus stays on musical concepts.

I'd like to hear from some others besides Gene and Paul
... should these math terms go directly into the
Tuning Dictionary, or should they live "off-campus"?

-monz

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🔗unidala <JGill99@imajis.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: genewardsmith <genewardsmith@j...>
> > To: <tuning-math@y...>
> > Sent: Friday, January 11, 2002 2:24 PM
> > Subject: [tuning-math] Re: More proposed definitions
> >
> >
>
> > More to the point would be definitions of vector,
> > vector space, lattice, bilinear form, group,
> > abelian group, homomophism, kernel, equivalence
> > relation, equivalence class, quotient group, graph,
> > wedge product, and determinant, but this would definately
> > start to look like mathematics.
>
>
> Yes, well ... I think we might be agreeing that there
> should be a separate area in the Dictionary for the
> heavy math definitions.
>
> But I already have definitions for vector, lattice,
> matrix, and determinant -- so it *is* already beginning
> to look a lot like a math dictionary. I just want to
> make sure that the focus stays on musical concepts.
>
> I'd like to hear from some others besides Gene and Paul
> ... should these math terms go directly into the
> Tuning Dictionary, or should they live "off-campus"?
>
>
> -monz

J Gill: While it might not be the easiest task to implement,
how about presenting it as non-esoterically as possible in
the main definition, with links (much as Monz allready does)
within that text which (heirarchically) enter the realms of
complexity (deeper and deeper) if the reader is so inclined?

That way the information is pre-compiled in levels of detail.

🔗monz <joemonz@yahoo.com>

> From: unidala <JGill99@imajis.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, January 11, 2002 8:05 PM
> Subject: [tuning-math] Re: More proposed definitions
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > I'd like to hear from some others besides Gene and Paul
> > ... should these math terms go directly into the
> > Tuning Dictionary, or should they live "off-campus"?
> >
> >
> > -monz
>
>
> J Gill: While it might not be the easiest task to implement,
> how about presenting it as non-esoterically as possible in
> the main definition, with links (much as Monz allready does)
> within that text which (heirarchically) enter the realms of
> complexity (deeper and deeper) if the reader is so inclined?
>
> That way the information is pre-compiled in levels of detail.

Hey J, thanks for your input ... and I think, thanks to your
suggestions, that I've already hit on the right way to do this:
simply include Gene's defintions "as is" as individual entries
in the Dictionary, but then have a "see also" link at the bottom
of each, which leads to an elementary tutorial webpage which
explains this brand of tuning math.

Feedback?

-monz

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🔗paulerlich <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > From: unidala <JGill99@i...>
> > To: <tuning-math@y...>
> > Sent: Friday, January 11, 2002 8:05 PM
> > Subject: [tuning-math] Re: More proposed definitions
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >
> > > I'd like to hear from some others besides Gene and Paul
> > > ... should these math terms go directly into the
> > > Tuning Dictionary, or should they live "off-campus"?
> > >
> > >
> > > -monz
> >
> >
> > J Gill: While it might not be the easiest task to implement,
> > how about presenting it as non-esoterically as possible in
> > the main definition, with links (much as Monz allready does)
> > within that text which (heirarchically) enter the realms of
> > complexity (deeper and deeper) if the reader is so inclined?
> >
> > That way the information is pre-compiled in levels of detail.
>
>
> Hey J, thanks for your input ... and I think, thanks to your
> suggestions, that I've already hit on the right way to do this:
> simply include Gene's defintions "as is" as individual entries
> in the Dictionary, but then have a "see also" link at the bottom
> of each, which leads to an elementary tutorial webpage which
> explains this brand of tuning math.
>
> Feedback?

I don't think you'd want to do this for "pitch", "interval", etc.,
though . . .

🔗unidala <JGill99@imajis.com>

Monz: I can sure understand why you might not want to seem
to "chop-up" the words of your contributors, and in doing
so, (perhaps) make their text *less* clear, but ....

From the perspective of the *reader*, if you were new to a
field (or even somewhat familiar) would *you* want to start
with the *least* (immediately) understandable text, and have
to wade through several levels of sub-links before you see
a word that you understand?

This seems (in the extreme case, anyway) to defeat the utility
of the (I should have said "reverse-hierarchical") presentation.

Wouldn't you want to *start* with things that you *do* (at least
somewhat) understand and recognize, and *then* (should one feel
more curious) delve *deeper* into detail (and the corresponding
complexities)?

You could still (at some level) provide access to the intact
(and more rigorous) definitions of the more technical terms.

J Gill :)

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, January 11, 2002 8:18 PM
> Subject: [tuning-math] Re: More proposed definitions
>
>

> > Hey J, thanks for your input ... and I think, thanks to your
> > suggestions, that I've already hit on the right way to do this:
> > simply include Gene's defintions "as is" as individual entries
> > in the Dictionary, but then have a "see also" link at the bottom
> > of each, which leads to an elementary tutorial webpage which
> > explains this brand of tuning math.
> >
> > Feedback?
>
> I don't think you'd want to do this for "pitch", "interval", etc.,
> though . . .

Well ... right, Paul, this is exactly what's on my mind.
Terms like "pitch" and "interval", which are common currency
among all musicians, certainly belong, but most of Gene's
terms obviously lean much more towards pure mathematics.

For those of you who are more familiar with algebra (and
specifically, multilinear algebra) than I am, I guess what
I'm really driving at with this is: are these terms mainly
quite personal to Gene and his work, or are they more
widely accepted among other mathematically-savvy tuning
theorists? If the former, then are there terms used by
others which are equivalent to Gene's, and therefore which
also need to be included in the Dictionary as synonyms?

This is really important to me, because I get a sense (and
an eyewitness account from Paul) that the last three months
or so on the tuning-math list (i.e., since Gene's arrival)
have borne some of the most comprehensive and most
widely-applicable concepts and algorithms in the history
of tuning theory, and it's high time that me and the rest
of us join the party.

-monz

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🔗genewardsmith <genewardsmith@juno.com>

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

The list I recently gave of other things which might go into a tuning dictionary was all standard mathematical terminology. Some else I would like to have available is that terminology adapted to the particular uses of it I've found in music--terms like "val" and "wedgie" are of that latter type. However, I see no difference in principle between this and terms such as "unique" and "consistent" which people have been coming up with. I've also come up with rather musically specialized definitions, such as for "notation".

Some of them are proposals, put out there for consideration. I'm in effect asking "what do you think of this as a mathematically precise definition of scale?" or "what do you think of using 'tone group' in this way?"

If the former, then are there terms used by
> others which are equivalent to Gene's, and therefore which
> also need to be included in the Dictionary as synonyms?

If there are synonyms, I'd like to know.

> This is really important to me, because I get a sense (and
> an eyewitness account from Paul) that the last three months
> or so on the tuning-math list (i.e., since Gene's arrival)
> have borne some of the most comprehensive and most
> widely-applicable concepts and algorithms in the history
> of tuning theory, and it's high time that me and the rest
> of us join the party.

I'm not an authority on the history of tuning-theory, but I would presume this has been more evolutionary than revolutionary. It is clear there are certain (so far, not particularly difficult ones, so things could be much worse!) mathematical concepts which it would be very good for people to know who work in this area. Just as you would be severely limited not to know anything about logarithms or determinants, a little knowledge of groups and multilinear algebra would go a long way.

🔗jonszanto <jonszanto@yahoo.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> I'd like to hear from some others besides Gene and Paul
> ... should these math terms go directly into the
> Tuning Dictionary, or should they live "off-campus"?

The Tuning Dictionary is invaluable, and I would be the last one to
know *exactly* where to draw the line as to where the definitions are
becoming too much like a math dictionary.

That said, it's value will be greatly enhanced by the degree which is
serves it's main purpose: to be an online resource for the
definitions of *tuning* terminology. If within those definitions a
clarification should be made of a mathematical term, then it should
point to a *separate* dictionary, and the link should open a new
window so as to not lose one's place - easily done.

Cheers,
Jon (who notes that you *did* ask for other comments, and I'm
directly responsible for links to the dictionary on 3 or 4 sites...)

🔗genewardsmith <genewardsmith@juno.com>

🔗jonszanto <jonszanto@yahoo.com>

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> What about tuning terminology of a mathematical nature?

Yes, I understand that. I am sure that there is no clear line where
musical leaves off and mathematical begins; I leave that to the more
learned minds here. I would wonder: who is the audience of said
dictionary? If oriented to musicians, then reference and tread
lightly on math; if to mathematicians, I suppose it's fair game, but
then wouldn't they already be familiar?

The other big question - again, not for me to answer (since most of
it is foreign to me) - is whether or not the terms are necessary,
whether they are accepted (or bound to be) by a larger group than the
few here, or whether any further 'advancements' in tuning (and the
understanding thereof) are impossible without their use.

Just some thoughts,
Jon
(who spent an enjoyable 2nd half of a concert last night - Beethoven
3rd - seated next to a visiting professor (Dr.) of mathematics of
some sort from Princeton, in town for a national convention of
mathematicians...)

🔗unidala <JGill99@imajis.com>

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "jonszanto" <jonszanto@y...> wrote:
>
> > That said, it's value will be greatly enhanced by the degree which is
> > serves it's main purpose: to be an online resource for the
> > definitions of *tuning* terminology.
>
> What about tuning terminology of a mathematical nature?

Monz (#2606): "...it's high time that me and the rest
of us join the party."

Monz (#2603): "simply include Gene's defintions "as is"
as individual entries in the Dictionary, but then have a
"see also" link at the bottom of each, which leads to an
elementary tutorial webpage which explains this brand of
tuning math."

Gill (#2605): "From the perspective of the *reader*,
if you were new to a field (or even somewhat familiar)
would *you* want to start with the *least* (immediately)
understandable text, and have to wade through several levels
of sub-links before you see a word that you understand?
Wouldn't you want to *start* with things that you *do* (at least
somewhat) understand and recognize, and *then* (should one feel
more curious) delve *deeper* into detail (and the corresponding
complexities)?
You could still (at some level) provide access to the intact
(and more rigorous) definitions of the more technical terms.

When I get "snowed" from the git-go, I tend to lose interest
(since it is often unclear *where one would begin* in order
to decipher that which one does *not comprehend*). What would
be useful to the unititiated is *not* to have to learn all of
the aspects (in total) in order to emerge from the experience
having learned something new.

Yet, if I am presented with concepts which I *do* comprehend,
and the resources are (by hyperlink) readily available whereby
I could learn more, then (even if one does not master *all*
of the elements in order to comprehend something *in total*),
one could learn *some* things about the relationships, and
emerge "wisened" (at least somewhat) to the issues, as opposed
to *no less mystified*, due to an approach which starts with
the most esoteric knowledge, and forces one to follow "links"
which might leave me with a comprehensible concept - but only
out on a given "branch" of a "tree" of related information,
the "whole" of which I am, as a result, hardly less confused
about. That would seem more like an exercise in admiring the
resultant complexity of (but not understanding) the concepts.
One can do that at present, as it stands. And, that's the
problem (one would, it seems) be attempting to here address!

Smith (#2609): "What about tuning terminology of a mathematical nature?".

Some "branches" (and the terms and phrases surrounding them)
start "farther out on the tree", and deserve an independent
access point (as well as being linked to more familiar concepts
by a system which can access the *details* via the *core*).

There are plenty of things in Monz's dictionary at present
which are very esoteric. Great! But the problem (as I see it)
is - that people come staggering out of searching the archives
with a few new esoteric terms to look up, where little exists
which attempts to facilitate the process of determining *how*
these "factoids" inter-relate, and which are of the greatest
*importance* in mastering a concept which may include many of
such individual concepts.

A "users manual" is much harder to compose than is
a "reference manual". Which would you consult first
if you were relatively unfamiliar with the "machine"?

It seems that if folks *want* others to comprehend their
thoughts, that there is no substitute for compiling them
in a way which *encourages* (rather than *discourages*)
further curiousity. This is no small feat. I typed pages
and pages about my ideas for Monz, but he didn't understand
my writings, so he did not benefit from the experience.

In fairness (to myself and others), this kind of "technical
writing" (using esoteric terms in communicating to what is
an esoteric audience) is about as challenging as it gets...

Anyone attempting to so compile such knowledge possessed
by others needs (and deserves) the support of the authors
in the process of making *explanations* out of *presentations*.

One can watch the "titans" spar from afar, and glorious it is,
but what good does such "pedestrianship" do for one,
if at the end of the day, one has not a clue as to
what it means to me and you when we sit down to play?

J Gill

🔗graham@microtonal.co.uk

monz wrote:

> Well ... right, Paul, this is exactly what's on my mind.
> Terms like "pitch" and "interval", which are common currency
> among all musicians, certainly belong, but most of Gene's
> terms obviously lean much more towards pure mathematics.

But those terms should have their common definitions, not Gene's
mathematically precise ones. If Gene wants to express musical concepts in
mathematical terms that's fine. But it shouldn't be in a general tuning
dictionary. If Gene wants space to write his own dictionary, he can have
it at microtonal.co.uk. If Joe wants to include these definitions, I
suggest he puts a link at the bottom of the general definition saying
something like "see Gene Ward Smith's mathematical definition".

> For those of you who are more familiar with algebra (and
> specifically, multilinear algebra) than I am, I guess what
> I'm really driving at with this is: are these terms mainly
> quite personal to Gene and his work, or are they more
> widely accepted among other mathematically-savvy tuning
> theorists? If the former, then are there terms used by
> others which are equivalent to Gene's, and therefore which
> also need to be included in the Dictionary as synonyms?

You don't need to worry on that account. The problem I see is that your
dictionary would turn into a dictionary of multilinear algebra, instead of
tuning theory. If that's what you want, I have some comments:

Please make the two separate. I know from experience that people are bad
at filtering out mathematical detail they don't need to know. If they
encounter such detail in you dictionary, they might give up on the whole
thing.

Be prepared for the mathematical dictionary to become the most popular
thing on your site. My page introducing matrices (without mathematical
precision, BTW) has this status.

Make sure you aren't duplicating effort. If a mathematical dictionary
already exists, aimed at the right audience, contribute and link to that
instead.

A simple (not mathematically precise) introduction to groups and rings may
be useful. Also something about wedge products, because they aren't that
prominent in linear algebra (certainly don't come up in web searches) but
aren't that difficult to understand and do simplify some of what we've
been talking about.

Really, Gene seems to be aiming at either creating a new branch of
mathematics relating to tuning theory, or defining aspects of tuning
theory in such a way that they become isomorphic to an existing branch of
mathematics. Either way, it's quite exciting. But explaining it all,
with full mathematical precision, to a reader without a mathematical
background is going to be a huge undertaking.

I suggest two new dictionaries -- one containing mathematically precise
definitions for tuning theory, either of new or recycled terms, with links
to and from Monzo's main dictionary. And another that gives
non-mathematicians enough of an idea what the mathematical terms mean that
they can follow what's going on.

Graham

🔗genewardsmith <genewardsmith@juno.com>

--- In tuning-math@y..., graham@m... wrote:

If Gene wants to express musical concepts in
> mathematical terms that's fine. But it shouldn't be in a general tuning
> dictionary. If Gene wants space to write his own dictionary, he can have
> it at microtonal.co.uk.

Actually, I've thought of doing that, but I couldn't get it to work--I could upload files, but could not find them afterwards.

However, some of the concepts, eg val, are of the sort that a word ought to be coined for it, and made a part of the tuning vocabulary. It doesn't need to be my word, but it should be some word.

> Really, Gene seems to be aiming at either creating a new branch of
> mathematics relating to tuning theory, or defining aspects of tuning
> theory in such a way that they become isomorphic to an existing branch of
> mathematics.

The latter--it's applied math. Of couse, as with eg coding theory, that *can* introduce new mathematical objects for consideration.