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.

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

> 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"?

That's exactly what I was thinking.

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

> But ... isn't it getting to be more about "mathematics"

> and less about "tuning"?

Possibly, but I thought you asked me to give you a definition of algebraic number field for your dictionary. Paul already gave a (correct, even though I fiddled with it) definition of algebraic number, and also of transcendental number.

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.

> 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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--- 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.

JG

> 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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Get your free @yahoo.com address at http://mail.yahoo.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 . . .

--- 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?

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 :)

> 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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Get your free @yahoo.com address at http://mail.yahoo.com

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

> 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?

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.

--- 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...)

--- 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?

--- 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...)

--- 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

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.

> 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.

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

--- 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.