torsion

Torsion describes a condition wherein an independent set of n unison vectors (<uvector.htm>) fails to define a periodicity block of dimension n, because of the existence of torsion elements, meaning intervals which are not products of the proposed set of unison vectors, but some power of which are.

Torsion may be tested by forming the n by n+1 matrix whose rows correspond to the unison vectors, and calculating the gcd(<http://mathworld.wolfram.com/GreatestCommonDivisor.html>) of the minors (<http://mathworld.wolfram.com/Minor.html>)

of the matrix. If the rows are linearly independent but the gcd is not one, we have torsion.

The term comes from mathematical usage, see

<http://mathworld.wolfram.com/TorsionGroup.html>.

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

> torsion

>

> Torsion describes a condition wherein an independent set of n

unison vectors (<uvector.htm>) fails to define a periodicity block of

dimension n, because of the existence of torsion elements, meaning

intervals which are not products of the proposed set of unison

vectors, but some power of which are.

>

> Torsion may be tested by forming the n by n+1 matrix whose rows

correspond to the unison vectors, and calculating the gcd

(<http://mathworld.wolfram.com/GreatestCommonDivisor.html>) of the

minors (<http://mathworld.wolfram.com/Minor.html>)

> of the matrix. If the rows are linearly independent but the gcd is

not one, we have torsion.

>

> The term comes from mathematical usage, see

> <http://mathworld.wolfram.com/TorsionGroup.html>.

This is awesome!!

Should you say "field of unison vectors" or "ring of unison vectors"

or some such as opposed to "set of unison vectors" above? The idea is

to eliminate the ambiguity that arises from the two usages of "unison

vector" that you brought up -- a member of the basis for the kernel,

or a member of the kernel.

P.S. An early example of torsion:

/tuning/topicId_9694.html#9694

Kees' corrections:

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

> Should you say "field of unison vectors" or "ring of unison vectors"

> or some such as opposed to "set of unison vectors" above?

You could say "group generated by the unison vectors", but I thought I made it clear with "set" that I was talking about a basis for the kernel, not the kernel itself. It seems to me that makes fewer demands on the reader by way of knowledge of mathematics.

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

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

>

> > Should you say "field of unison vectors" or "ring of unison

vectors"

> > or some such as opposed to "set of unison vectors" above?

>

> You could say "group generated by the unison vectors", but I

>thought I made it clear with "set" that I was talking about a basis

>for the kernel, not the kernel itself.

Well then your definition doesn't seem to work, because if the basis

is the diesis and the schisma, the syntonic comma squared is in the

kernel, but not in the basis.

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

> torsion

>

> Torsion describes a condition wherein an independent set of n unison vectors (<uvector.htm>) fails to define a periodicity block of dimension n...

No agreement has been reached on what peridicity block means, so this could also read "defines an anomalous periodicity block".

Properties which might or might not be included:

(1) Epimorphic

(2) Convex

(3) Connected

(4) Linf "Max" norm, with paralelipiped blocks.

Clearly (4) ==> (2), but I don't think there are any other implications, so this leaves a lot up in the air. I was going to try to nail it down, but people thought I was getting too technical for a general math dictionary; however it seems to me some precise definitions should be agreed to.

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

> Well then your definition doesn't seem to work, because if the basis

> is the diesis and the schisma, the syntonic comma squared is in the

> kernel, but not in the basis.

That's why I said "not products of the proposed set of unison vectors" instead of "not members of the proposed set of unision vectors".

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

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

> > torsion

> >

> > Torsion describes a condition wherein an independent set of n

unison vectors (<uvector.htm>) fails to define a periodicity block of

dimension n...

>

> No agreement has been reached on what peridicity block means, so

this could also read "defines an anomalous periodicity block".

That would be preferable.

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

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

>

> > Well then your definition doesn't seem to work, because if the

basis

> > is the diesis and the schisma, the syntonic comma squared is in

the

> > kernel, but not in the basis.

>

> That's why I said "not products of the proposed set of unison

>vectors" instead of "not members of the proposed set of unision

>vectors".

Oops! I read it wrong, somehow. But "products" might still fail to

capture a case like a^2/b where a and b are in the basis. Can't we

think of a better terminology?

Hi Gene,

Two things.

1) I love your new definition of "torsion". What exactly

should I replace in my old definition? Everything? Please

be as specific as possible.

2) The last post on this subject was this one from Paul,

with a question to which you have not yet responded:

> From: paulerlich <paul@stretch-music.com>

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

> Sent: Wednesday, January 23, 2002 4:05 PM

> Subject: [tuning-math] Re: Proposed dictionary entry: torsion

>

>

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

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

> >

> > > Well then your definition doesn't seem to work, because if the

> basis

> > > is the diesis and the schisma, the syntonic comma squared is in

> the

> > > kernel, but not in the basis.

> >

> > That's why I said "not products of the proposed set of unison

> >vectors" instead of "not members of the proposed set of unision

> >vectors".

>

> Oops! I read it wrong, somehow. But "products" might still fail to

> capture a case like a^2/b where a and b are in the basis. Can't we

> think of a better terminology?

I'd appreciate your help in getting the definiton webpage

of "torsion" finished. Thanks.

-monz

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torsion

Torsion describes a condition wherein an independent set of n unison vectors {u1, u2, ..., un} (<uvector.htm>) defines a non-epimophic (epimorphic.htm>) periodicity block, because of the existence a torsion element, meaning an interval which is not the product

u1^e1 u2^e2 ... un^en

of the set of unison vectors raised to (positive, negative or zero) integral powers, but some integer power of which is. An example would be a block defined by 648/625 and 2048/2025;

here 81/80 is not a product of these commas, but

(81/80)^2 = (648/625) (2048/2025)^(-1).

Torsion may be tested by forming the n by n+1 matrix whose rows correspond to the unison vectors, and calculating the

gcd(<<http://mathworld.wolfram.com/GreatestCommonDivisor.html>>)

of the minors

(<<http://mathworld.wolfram.com/Minor.html>>)

of the matrix. If the rows are linearly independent but the gcd is not one, we have torsion.

The term comes from mathematical usage, see

<<http://mathworld.wolfram.com/TorsionGroup.html>>.

Hi Gene,

> From: genewardsmith <genewardsmith@juno.com>

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

> Sent: Friday, January 25, 2002 11:46 AM

> Subject: [tuning-math] Proposed dictionary entry: torsion (revised)

>

>

> torsion

>

> Torsion describes a condition wherein an

> independent set of n unison vectors {u1, u2, ..., un}

> (<uvector.htm>) defines a non-epimophic (epimorphic.htm>)

> periodicity block, because of the existence a torsion

> element, meaning an interval which is not the product

>

> u1^e1 u2^e2 ... un^en

>

> of the set of unison vectors raised to (positive,

> negative or zero) integral powers, but some integer

> power of which is. An example would be a block

> defined by 648/625 and 2048/2025; here 81/80 is

> not a product of these commas, but

> (81/80)^2 = (648/625) (2048/2025)^(-1).

Thanks for the revised definition!

http://www.ixpres.com/interval/dict/torsion.htm

-24 0 0

-38 -2 4

-56 4 4

Is there some special reason to use the ...

UVs =

<648/625, 2048/2025> = [3 4 -4], [11 -4 -2]

adj =

[-24 0 0]

[-38 -2 4]

[-56 4 4]

... PB as an example, instead of the one I already put into

the definition? -- that one is also the same one which Paul

used as an illustration when the torsion discussion first

began on this list:

UVs =

<128/125, 32805/32768> = [7 0 -3], [-15 8 1]

adj =

[24 0 0]

[38 1 3]

[56 -8 0]

My website already has several webpages and lattice diagrams

of this PB, and I'd like to link to them. For example, see

the first graphic at:

http://www.ixpres.com/interval/td/gill/duodene.htm

If I use your PB in the definition, I'll have to create new

diagrams for it. ... Not that I don't want to do that anyway ...

but since I already have diagrams of a torsional PB, I'd

like to employ them right away as illustration.

Also, please tell me if I should keep anything that appears

below the row of asterisks in the definition. Otherwise it's

trash, but I don't fully understand torsion yet, so I'm being

careful and only deleting what you tell me to delete.

-monz

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----- Original Message -----

From: monz <joemonz@yahoo.com>

To: <tuning-math@yahoogroups.com>

Sent: Saturday, January 26, 2002 1:09 PM

Subject: Re: [tuning-math] Proposed dictionary entry: torsion (revised)

> Thanks for the revised definition!

> http://www.ixpres.com/interval/dict/torsion.htm

>

>

>

> -24 0 0

> -38 -2 4

> -56 4 4

>

> Is there some special reason to use the ...

Oops ... my bad. That matrix didn't belong there,

it means nothing where it is, and should have been deleted.

Sorry.

-monz

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

> Is there some special reason to use the ...

>

> UVs =

> <648/625, 2048/2025> = [3 4 -4], [11 -4 -2]

>

> adj =

> [-24 0 0]

> [-38 -2 4]

> [-56 4 4]

>

> ... PB as an example, instead of the one I already put into

> the definition?

I wanted an example, and I cooked this one up, that's all. The only advantage of it I can see is that it uses simpler commas.

Hey Gene,

> From: genewardsmith <genewardsmith@juno.com>

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

> Sent: Saturday, January 26, 2002 8:04 PM

> Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised)

>

>

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

>

> > Is there some special reason to use the ...

> >

> > UVs =

> > <648/625, 2048/2025> = [3 4 -4], [11 -4 -2]

> >

> > adj =

> > [-24 0 0]

> > [-38 -2 4]

> > [-56 4 4]

> >

> > ... PB as an example, instead of the one I already put into

> > the definition?

>

> I wanted an example, and I cooked this one up, that's all.

> The only advantage of it I can see is that it uses simpler commas.

OK, fair enough. I decided to go ahead and make the lattice

diagram of your example after all. Here's the latest definition:

http://www.ixpres.com/interval/dict/torsion.htm

I'd like to leave in the bit which explains how to calculate

the torsion factor from the gcd of the determinants of the minors.

Can you integrate that into the "good" definition in the top

part of the page? Then I can delete all the other old junk

in the bottom part. Thanks.

-monz

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Hi Gene,

> From: monz <joemonz@yahoo.com>

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

> Sent: Saturday, January 26, 2002 10:37 PM

> Subject: Re: [tuning-math] Re: Proposed dictionary entry: torsion

(revised)

>

>

> Can you integrate that into the "good" definition in the top

> part of the page? Then I can delete all the other old junk

> in the bottom part. Thanks.

Just thought I'd mention ...

Even tho I really still don't understand it, because of what

I see on the lattice I can intuitively sense how torsion works.

And my intuition tells me that torsion is a very important

part of getting a better focus on my model of "finity":

http://www.ixpres.com/interval/dict/finity.htm

I'm thinking that the patterns of unison-vectors that one

can see within a torsional block mean something, and this

can be modeled mathematically.

So I'd really like to keep exploring it until I understand it

fully, and to correspondingly expand the Dictionary webpage.

I've had the idea to create a book full of 5-limit

periodicity- and torsional-blocks, and many of these

can go into the webpage.

Please, Gene and the others here who do understand torsion,

feel free to comment profusely, with lots of 5- and higher-limit

examples. I'll try to diagram all of them and include them

in the definition, or perhaps I'll make a separate analytical

page about torsion.

-monz

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> From: monz <joemonz@yahoo.com>

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

> Sent: Sunday, January 27, 2002 1:00 AM

> Subject: Re: [tuning-math] Re: Proposed dictionary entry: torsion

(revised)

>

>

> Even tho I really still don't understand it, because of what

> I see on the lattice I can intuitively sense how torsion works.

> And my intuition tells me that torsion is a very important

> part of getting a better focus on my model of "finity":

>

> http://www.ixpres.com/interval/dict/finity.htm

>

>

> I'm thinking that the patterns of unison-vectors that one

> can see within a torsional block mean something, and this

> can be modeled mathematically.

Well, OK ... actually I can already see that the unison-vectors

inside the torsional-block on my lattice diagram here

http://www.ixpres.com/interval/dict/torsion.htm

are exactly the same pair as the bounding vectors of the Duodene

http://www.ixpres.com/interval/dict/duodene.htm

... a *real* periodicity-block, and apparently one whose

12 pitches can "stand in" for the 24 of this torsional-block?

Hmmm...

Very curious,

-monz

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

>

> OK, fair enough. I decided to go ahead and make the lattice

> diagram of your example after all. Here's the latest definition:

>

> http://www.ixpres.com/interval/dict/torsion.htm

What happened to the really nice definition Gene gave??????????????

This should be _at the top_, rather than omitted entirely!!!!!!!!!!!

> From: paulerlich <paul@stretch-music.com>

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

> Sent: Sunday, January 27, 2002 10:01 PM

> Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised)

>

>

> > ... Here's the latest definition:

> >

> > http://www.ixpres.com/interval/dict/torsion.htm

>

> What happened to the really nice definition Gene gave??????????????

>

> This should be _at the top_, rather than omitted entirely!!!!!!!!!!!

The definition I now have at the top (apparently the same one

you commented on here) is the latest one Gene sent to the

tuning-math list:

Message 2973

From: "genewardsmith" <genewardsmith@j...>

Date: Fri Jan 25, 2002 2:46 pm

Subject: Proposed dictionary entry: torsion (revised)

/tuning-math/message/2973

(... did you click "refresh/reload"?)

-monz

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

>

> > From: paulerlich <paul@s...>

> > To: <tuning-math@y...>

> > Sent: Sunday, January 27, 2002 10:01 PM

> > Subject: [tuning-math] Re: Proposed dictionary entry: torsion

(revised)

> >

> >

> > > ... Here's the latest definition:

> > >

> > > http://www.ixpres.com/interval/dict/torsion.htm

> >

> > What happened to the really nice definition Gene

gave??????????????

> >

> > This should be _at the top_, rather than omitted

entirely!!!!!!!!!!!

>

>

>

> The definition I now have at the top (apparently the same one

> you commented on here) is the latest one Gene sent to the

> tuning-math list:

>

>

> Message 2973

> From: "genewardsmith" <genewardsmith@j...>

> Date: Fri Jan 25, 2002 2:46 pm

> Subject: Proposed dictionary entry: torsion (revised)

> /tuning-math/message/2973

>

>

> (... did you click "refresh/reload"?)

>

>

>

> -monz

Who is going to understand that definition? You, Monz? I meant the

definition in

but possibly changing "products of the proposed set of unison

vectors" to "members of the group generated by the unison vectors".

> From: paulerlich <paul@stretch-music.com>

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

> Sent: Tuesday, January 29, 2002 12:27 PM

> Subject: [tuning-math] Re: Proposed dictionary entry: torsion (revised)

>

>

> > > > ... Here's the latest definition:

> > > >

> > > > http://www.ixpres.com/interval/dict/torsion.htm

> > >

> >

> > The definition I now have at the top (apparently the same one

> > you commented on here) is the latest one Gene sent to the

> > tuning-math list:

> >

> >

> > Message 2973

> > From: "genewardsmith" <genewardsmith@j...>

> > Date: Fri Jan 25, 2002 2:46 pm

> > Subject: Proposed dictionary entry: torsion (revised)

> > /tuning-math/message/2973

> >

> >

> > (... did you click "refresh/reload"?)

> >

> >

> >

> > -monz

>

> Who is going to understand that definition? You, Monz? I meant the

> definition in

>

> /tuning-math/message/2937

>

> but possibly changing "products of the proposed set of unison

> vectors" to "members of the group generated by the unison vectors".

OK, well, the definition I previously put at the top is

still the latest revision from Gene. I haven't heard from

him on this for a while now, so the Dictionary entry for

"torsion" continues to grow and get messier.

I've now included Paul's favorite definition at the top,

and labeled it as such, with everything else still intact

below it.

Any comments on the lattice diagram and description of it

which I added to illustrate Gene's example? I found it

interesting that the torsion element appears as a line

which divides the PB in half. Is this typical? Omnipresent?

-monz

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

> Is this typical?

Yes -- try the 24-tone {diesis, schisma} case.

> Omnipresent?

No -- it depends which UVs you use to construct the parellelepiped.

For example, you could restate your 24-tone {diesis, schisma} one

with a {6561:6400, 128:125} basis, and then the syntonic-comma-

squared will _not_ go from one diagonal of the block to the other.

Try it!