I'm not a mathematician.

If, at some point, the "lattice" wraps around

and becomes redundant, how can we assign a single

point of reference?

If it's not a lattice, then what is it?

---- Original message ----

>Date: Mon, 21 Oct 2002 22:50:46 -0000

>From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

>Subject: [tuning-math] [tuning] Re: Everyone Concerned

>To: tuning-math@yahoogroups.com

>

>--- In tuning-math@y..., "Gene Ward Smith"

<genewardsmith@j...> wrote:

>> --- In tuning-math@y..., "wallyesterpaulrus"

><wallyesterpaulrus@y...> wrote:

>>

>> > > You should remember that many self-respecting

mathematicians

>would

>> > >not call something a lattice unless it inherited a

group

>structure

>> > >from R^n.

>> >

>> > any examples of one that doesn't?

>>

>> The hexagonal tiling of chords in the 5-limit, for one.

>

>well, it's still a lattice in the crystallographic sense,

or even the

>geometric sense: a regular array of points, that is, an

array of

>points in which every point has exactly the same

relationships with

>its neighbors as any other point does with *its* neighbors.

>

>

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--- In tuning-math@y..., <Josh@o...> wrote:

> I'm not a mathematician.

>

> If, at some point, the "lattice" wraps around

> and becomes redundant, how can we assign a single

> point of reference?

why would we want to?

> If it's not a lattice, then what is it?

i think it's still a lattice!

Maybe I misunderstood the issue.

Where I come from, if it looks like a lattice

and it quacks like a lattice, we call it a lattice.

What is the reason people are objecting to this?

---- Original message ----

>Date: Mon, 21 Oct 2002 23:25:45 -0000

>From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

>Subject: Fwd: [tuning-math] [tuning] Re: Everyone

Concerned

>To: tuning-math@yahoogroups.com

>

>--- In tuning-math@y..., <Josh@o...> wrote:

>> I'm not a mathematician.

>>

>> If, at some point, the "lattice" wraps around

>> and becomes redundant, how can we assign a single

>> point of reference?

>

>why would we want to?

>

>> If it's not a lattice, then what is it?

>

>i think it's still a lattice!

>

>

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>

--- In tuning-math@y..., <Josh@o...> wrote:

> Maybe I misunderstood the issue.

>

> Where I come from, if it looks like a lattice

> and it quacks like a lattice, we call it a lattice.

>

> What is the reason people are objecting to this?

in the field of mathematics known as abstract algebra, the

term "lattice" is used to signify something completely different.

i recall a certain list member who at first insisted that all words

should be understood according to their "math" definitions, ignoring

that fact that many of the same words already had "music" definitions

which should take precedence at least on the tuning list.

of course this is somewhat different, few musicians use the

term "lattice" except of course just intonation theorists and the

like . . . so i guess there already is a "music" definition of

lattice and the issue should be moot.

that's why i found it odd that monz was bringing it up at this late

date!

Speaking as a musician, I have no objection to being

corrected by mathematicians or other people for misuse

of terms, provided an effective alternative is provided.

using pc set theory, I once attempted to construct

a 12tet all-trichord isohedron that could be used to

produce useful non-serial row forms.

I quickly discovered that this was impossible.

I eventually started toying with the idea of concentric

imperfect isohedrons linked by their respective loose ends.

The mathematicians I asked about this didn't seem to have

any terminology to explain the problem, and I was

even unable to get a clear answer on whether the problem

could or should be approached topologically.

Maybe if I paid them, they'd have done better?

In the meantime, they're welcome to correct my terms

for free if they don't like the way I describe things.

---- Original message ----

>Date: Tue, 22 Oct 2002 00:20:25 -0000

>From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

>Subject: Fwd: [tuning-math] [tuning] Re: Everyone

Concerned

>To: tuning-math@yahoogroups.com

>

>--- In tuning-math@y..., <Josh@o...> wrote:

>

>> Maybe I misunderstood the issue.

>>

>> Where I come from, if it looks like a lattice

>> and it quacks like a lattice, we call it a lattice.

>>

>> What is the reason people are objecting to this?

>

>in the field of mathematics known as abstract algebra, the

>term "lattice" is used to signify something completely

different.

>

>i recall a certain list member who at first insisted that

all words

>should be understood according to their "math" definitions,

ignoring

>that fact that many of the same words already had "music"

definitions

>which should take precedence at least on the tuning list.

>

>of course this is somewhat different, few musicians use the

>term "lattice" except of course just intonation theorists

and the

>like . . . so i guess there already is a "music" definition

of

>lattice and the issue should be moot.

>

>that's why i found it odd that monz was bringing it up at

this late

>date!

>

>

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

> i think it's still a lattice!

If you take a lattice (p-limit, let's say) and then take a sublattice (defined by commas, lets say) of the same rank n, if you mod out R^n by the sublattice you get a discrete group on the compact quotient. As I remarked, you should not assume every mathematican would call this a lattice! Of couse the rule is that you can change definitions as long as you make it clear what you are doing.

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

> i recall a certain list member who at first insisted that all words

> should be understood according to their "math" definitions, ignoring

> that fact that many of the same words already had "music" definitions

> which should take precedence at least on the tuning list.

Eh, I think you remember wrong. There are two completely different *math* definitions in very common use in mathematics. Obviously, I wouldn't suggest using both of them, or the wrong one.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

>

> > i think it's still a lattice!

>

> If you take a lattice (p-limit, let's say) and then take a

>sublattice (defined by commas, lets say) of the same rank n, if you

>mod out R^n by the sublattice you get a discrete group on the

>compact quotient. As I remarked, you should not assume every

>mathematican would call this a lattice! Of couse the rule is that

>you can change definitions as long as you make it clear what you are

>doing.

how about the term "point lattice", which is what mathworld suggests?

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

>

> > i recall a certain list member who at first insisted that all

words

> > should be understood according to their "math" definitions,

ignoring

> > that fact that many of the same words already had "music"

definitions

> > which should take precedence at least on the tuning list.

>

> Eh, I think you remember wrong. There are two completely different

>*math* definitions in very common use in mathematics. Obviously, I

>wouldn't suggest using both of them, or the wrong one.

i wasn't talking about "lattice" in the paragraph above. did you

think i was?

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

>

> > i think it's still a lattice!

>

> If you take a lattice (p-limit, let's say) and then take a

>sublattice (defined by commas, lets say) of the same rank n, if you

>mod out R^n by the sublattice you get a discrete group on the

>compact quotient. As I remarked, you should not assume every

>mathematican would call this a lattice!

what's the musical analogue? i'm lost.

>Of couse the rule is that >you can change definitions as long as you

>make it clear what you are >doing.

how about the term "point lattice", which mathworld suggests?

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

> > If you take a lattice (p-limit, let's say) and then take a

> >sublattice (defined by commas, lets say) of the same rank n, if you

> >mod out R^n by the sublattice you get a discrete group on the

> >compact quotient. As I remarked, you should not assume every

> >mathematican would call this a lattice!

>

> what's the musical analogue? i'm lost.

A 5-limit octave-reduced Fokker block painted on a donut, hopefully using something edible.

> >Of couse the rule is that >you can change definitions as long as you

> >make it clear what you are >doing.

>

> how about the term "point lattice", which mathworld suggests?

Why not? It doesn't get used that much, but we needn't let that worry us. Will we become confused by the question of whether an array of points is a point lattice if it is not a subgroup of R^n?

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

>

> > > If you take a lattice (p-limit, let's say) and then take a

> > >sublattice (defined by commas, lets say) of the same rank n, if

you

> > >mod out R^n by the sublattice you get a discrete group on the

> > >compact quotient. As I remarked, you should not assume every

> > >mathematican would call this a lattice!

> >

> > what's the musical analogue? i'm lost.

>

> A 5-limit octave-reduced Fokker block painted on a donut, hopefully

using something edible.

but if you unroll this onto an infinite bingo card, which hopefully

you've seen examples of lately, then it *is* a lattice. and what if

you consider the donut to be merely a flawed representation of a non-

euclidean closed universe with constant curvature everywhere?

wouldn't the block then qualify as a lattice, even though it's finite?

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

> but if you unroll this onto an infinite bingo card, which hopefully

> you've seen examples of lately, then it *is* a lattice.

It's a quotient of a lattice.

and what if

> you consider the donut to be merely a flawed representation of a non-

> euclidean closed universe with constant curvature everywhere?

> wouldn't the block then qualify as a lattice, even though it's finite?

Topologically impossible.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

>

> > but if you unroll this onto an infinite bingo card, which

hopefully

> > you've seen examples of lately, then it *is* a lattice.

>

> It's a quotient of a lattice.

why? it's just R^2, isn't it?

> > and what if

> > you consider the donut to be merely a flawed representation of a

non-

> > euclidean closed universe with constant curvature everywhere?

> > wouldn't the block then qualify as a lattice, even though it's

finite?

>

> Topologically impossible.

i don't see the big deal -- a lot of video games work this way.

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

> i don't see the big deal -- a lot of video games work this way.

Video games work on manifolds??

Genus 0 corresponds to Riemann sphere, which is to say the complex projective line; positive curvature.

Genus 1, the donut or algebraically an elliptic curve, zero curvature.

Genus > 1, negative curvature.

Relevance to tuning theory--to be determined.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> --- In tuning-math@y..., "wallyesterpaulrus"

<wallyesterpaulrus@y...> wrote:

>

> > i don't see the big deal -- a lot of video games work this way.

>

> Video games work on manifolds??

the edges of the screen wrap around and meet one another, hence a

donut topology.

> Genus 0 corresponds to Riemann sphere, which is to say the complex

projective line; positive curvature.

>

> Genus 1, the donut or algebraically an elliptic curve, zero

curvature.

i did say constant curvature, didn't i? last time i checked, zero was

a constant . . . :)

---- Original message ----

>Date: Wed, 23 Oct 2002 21:09:08 -0000

>From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

...

>the edges of the screen wrap around and meet one another,

hence a

>donut topology.

That's a little misleading.

The video signal is linear, but gets bent around so

that it appears to take on other shapes.

Choosing to recognize only part of the process

as relevant is opportunistic.

Not that I actually care...

--- In tuning-math@y..., <Josh@o...> wrote:

> ---- Original message ----

> >Date: Wed, 23 Oct 2002 21:09:08 -0000

> >From: "wallyesterpaulrus" <wallyesterpaulrus@y...>

> ...

> >the edges of the screen wrap around and meet one another,

> hence a

> >donut topology.

>

> That's a little misleading.

>

> The video signal is linear, but gets bent around so

> that it appears to take on other shapes.

>

> Choosing to recognize only part of the process

> as relevant is opportunistic.

>

> Not that I actually care...

i have no idea what you're talking about.

have you ever played Asteroids?

Oh, you're talking about the game content, not

the mechanics of the CRT!

Your use of the term "screen" confused me.

The property you're describing varies from game

to game. While it defies basic spatial properties

in some unobvious ways that are probably more

important than the obvious ones, I think it may

be a useful analogy for the quasi-spatial modeling

of non-strictly-spatial mathematical properties.

My question about "lattices" would be whether it's relevant

if they're physically impossible, as long as the spatial

aspect is modeling non-spatial mathematical relationships.

Anyone?

---- Original message ----

>Date: Thu, 24 Oct 2002 05:08:37 -0000

>From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>

>Subject: Fwd: [tuning-math] [tuning] Re: Everyone

Concerned

>To: tuning-math@yahoogroups.com

>

>--- In tuning-math@y..., <Josh@o...> wrote:

>> ---- Original message ----

>> >Date: Wed, 23 Oct 2002 21:09:08 -0000

>> >From: "wallyesterpaulrus" <wallyesterpaulrus@y...>

>> ...

>> >the edges of the screen wrap around and meet one

another,

>> hence a

>> >donut topology.

>>

>> That's a little misleading.

>>

>> The video signal is linear, but gets bent around so

>> that it appears to take on other shapes.

>>

>> Choosing to recognize only part of the process

>> as relevant is opportunistic.

>>

>> Not that I actually care...

>

>i have no idea what you're talking about.

>

>have you ever played Asteroids?

>

>

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>

>

--- In tuning-math@y..., <Josh@o...> wrote:

> Oh, you're talking about the game content, not

> the mechanics of the CRT!

right.

> Your use of the term "screen" confused me.

sorry.

> The property you're describing varies from game

> to game.

didn't mean to suggest otherwise.

> While it defies basic spatial properties

> in some unobvious ways that are probably more

> important than the obvious ones, I think it may

> be a useful analogy for the quasi-spatial modeling

> of non-strictly-spatial mathematical properties.

>

> My question about "lattices" would be whether it's relevant

> if they're physically impossible, as long as the spatial

> aspect is modeling non-spatial mathematical relationships.

>

> Anyone?

i don't understand any of the above.