new Dictionary entry: "torsion"

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

Feedback appreciated.

(and thanks for the helpful criticisms, Paul)

-monz

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

>

> new Dictionary entry: "torsion"

>

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

>

>

> Feedback appreciated.

>

> (and thanks for the helpful criticisms, Paul)

>

>

>

> -monz

Why hyphenate periodicity-block? You don't hyphenate vacuum-cleaner,

do you?

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

>

> new Dictionary entry: "torsion"

I would suggest adding something to explain why it is called "torsion"--namely, the existence of torsion elements. If you decide to temper out 6561/6250 and 128/125, then 81/80 is not a unison, but (81/80)^2 is. It is therefore an element of finite order, which is called a torsion element.

Here is what Mathworld has to say, which you could link to:

new Dictionary entry: "torsion"

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

> I would suggest adding something to explain why it is called "torsion"

Done. Thanks, Gene!

-monz

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

> >

> > new Dictionary entry: "torsion"

> >

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

> >

> >

> > Feedback appreciated.

> >

> > (and thanks for the helpful criticisms, Paul)

Monz, this might be confusing because even the 12-tone Duodene can be

split in this way (into two augmented scale PBs a 9:8 apart) and yet

it does not possess torsion. The key point is that some unison vector

(or combination thereof, depending on how you define UV) is an

integer power (or multiple, in cents) of some interval within the

block. In the particular example, the "unison vector" in question

goes from one corner of the block to the opposite corner, and is the

syntonic comma squared.

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

> Done. Thanks, Gene!

Here are more useful mathworld definitions:

http://mathworld.wolfram.com/Group.html

http://mathworld.wolfram.com/AbelianGroup.html

http://mathworld.wolfram.com/FreeAbelianGroup.html

http://mathworld.wolfram.com/RankGroup.html

http://mathworld.wolfram.com/CyclicGroup.html

http://mathworld.wolfram.com/GroupHomomorphism.html

http://mathworld.wolfram.com/IsomorphicGroups.html

http://mathworld.wolfram.com/Nullspace.html

http://mathworld.wolfram.com/RankMatrix.html

http://mathworld.wolfram.com/PointLattice.html

http://mathworld.wolfram.com/QuadraticForm.html

http://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html

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

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

> Sent: Wednesday, January 23, 2002 12:21 PM

> Subject: [tuning-math] Re: new Dictionary entry: "torsion"

>

>

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

> >

> > new Dictionary entry: "torsion"

> >

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

> >

>

>

> Monz, this might be confusing because even the 12-tone Duodene can be

> split in this way (into two augmented scale PBs a 9:8 apart) and yet

> it does not possess torsion. The key point is that some unison vector

> (or combination thereof, depending on how you define UV) is an

> integer power (or multiple, in cents) of some interval within the

> block. In the particular example, the "unison vector" in question

> goes from one corner of the block to the opposite corner, and is the

> syntonic comma squared.

Try it now.

I included Paul's post verbatim under my description of the

"classic example", but also changed a bit in the first paragraph

(i.e., the actual definition part). Is that first paragraph

good enough now?

-monz

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

>

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

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

> > Sent: Wednesday, January 23, 2002 12:21 PM

> > Subject: [tuning-math] Re: new Dictionary entry: "torsion"

> >

> >

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

> > >

> > > new Dictionary entry: "torsion"

> > >

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

> > >

> >

> >

> > Monz, this might be confusing because even the 12-tone Duodene

can be

> > split in this way (into two augmented scale PBs a 9:8 apart) and

yet

> > it does not possess torsion. The key point is that some unison

vector

> > (or combination thereof, depending on how you define UV) is an

> > integer power (or multiple, in cents) of some interval within the

> > block. In the particular example, the "unison vector" in question

> > goes from one corner of the block to the opposite corner, and is

the

> > syntonic comma squared.

>

>

> Try it now.

>

> I included Paul's post verbatim under my description of the

> "classic example", but also changed a bit in the first paragraph

> (i.e., the actual definition part). Is that first paragraph

> good enough now?

I don't see the change as addressing my objection above in any way.

Do you?

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

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

> >

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

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

> > > Sent: Wednesday, January 23, 2002 12:21 PM

> > > Subject: [tuning-math] Re: new Dictionary entry: "torsion"

> > >

> > >

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

> > > >

> > > > new Dictionary entry: "torsion"

> > > >

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

> > > >

> > >

> > >

> > > Monz, this might be confusing because even the 12-tone Duodene

> can be

> > > split in this way (into two augmented scale PBs a 9:8 apart)

and

> yet

> > > it does not possess torsion. The key point is that some unison

> vector

> > > (or combination thereof, depending on how you define UV) is an

> > > integer power (or multiple, in cents) of some interval within

the

> > > block. In the particular example, the "unison vector" in

question

> > > goes from one corner of the block to the opposite corner, and

is

> the

> > > syntonic comma squared.

> >

> >

> > Try it now.

> >

> > I included Paul's post verbatim under my description of the

> > "classic example", but also changed a bit in the first paragraph

> > (i.e., the actual definition part). Is that first paragraph

> > good enough now?

>

> I don't see the change as addressing my objection above in any way.

> Do you?

Sorry, Monz, I didn't hit refresh.

Well, the definition now is quite incorrect even as a description

(let alone a definition), as you can see by comparing what you say

with the example you give.

"Torsion describes a condition where a set of pitch-classes appears

to create a periodicity-block, when in reality that apparent

periodicity-block is actually composed of two or more (call it n)

identical smaller periodicity-blocks, each exactly 1/n as large as

the apparent one, whose pitch-classes are separated by some interval

which acts as a unison-vector and which is an integer power of an

interval included within the actual smaller periodicity-block itself."

You won't find sets of pitch classes separated by such an interval.

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

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

> Sent: Wednesday, January 23, 2002 12:54 PM

> Subject: [tuning-math] Re: new Dictionary entry: "torsion"

>

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

>

> > new Dictionary entry: "torsion"

> >

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

>

>

> Well, the definition now is quite incorrect even as a description

> (let alone a definition), as you can see by comparing what you say

> with the example you give.

>

> "Torsion describes a condition where a set of pitch-classes appears

> to create a periodicity-block, when in reality that apparent

> periodicity-block is actually composed of two or more (call it n)

> identical smaller periodicity-blocks, each exactly 1/n as large as

> the apparent one, whose pitch-classes are separated by some interval

> which acts as a unison-vector and which is an integer power of an

> interval included within the actual smaller periodicity-block itself."

>

> You won't find sets of pitch classes separated by such an interval.

ACK!! OK, I don't have time to do anything else now. You guys

hash out how this definition should be corrected, and I'll change

it when I read your follow-up posts later tonight.

-monz

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