mathematical model of torsion-block symmetry?

is there some way to mathematically model
the symmetry in a torsion-block?

see the graphic and its related text in my
Tuning Dictionary definition of "torsion"
-- i've uploaded it to here:
/tuning-math/files/dict/torsion.htm

-monz
"all roads lead to n^0"

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> is there some way to mathematically model
> the symmetry in a torsion-block?
>
> see the graphic and its related text in my
> Tuning Dictionary definition of "torsion"
> -- i've uploaded it to here:
> /tuning-math/files/dict/torsion.htm
>
>
>
> -monz
> "all roads lead to n^0"

i see the green and red lines, but . . . which symmetry exactly are
you referring to?

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, September 30, 2002 2:47 PM
> Subject: [tuning-math] Re: mathematical model of torsion-block symmetry?
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
> > is there some way to mathematically model
> > the symmetry in a torsion-block?
> >
> > see the graphic and its related text in my
> > Tuning Dictionary definition of "torsion"
> > -- i've uploaded it to here:
> > /tuning-math/files/dict/torsion.htm
> >
> >
> >
> > -monz
> > "all roads lead to n^0"
>
> i see the green and red lines, but . . . which symmetry exactly are
> you referring to?

do you see "The thin black line which divides the block in half
diagonally is the torsional interval, 6561:6400 = [-8 2] = (81/80)^2
= (648/625) (2048/2025)^(-1).]" on the diagram?

-monz

--- In tuning-math@y..., "monz" <monz@a...> wrote:
> hi paul,
>
>
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > To: <tuning-math@y...>
> > Sent: Monday, September 30, 2002 2:47 PM
> > Subject: [tuning-math] Re: mathematical model of torsion-block
symmetry?
> >
> >
> > --- In tuning-math@y..., "monz" <monz@a...> wrote:
> > > is there some way to mathematically model
> > > the symmetry in a torsion-block?
> > >
> > > see the graphic and its related text in my
> > > Tuning Dictionary definition of "torsion"
> > > -- i've uploaded it to here:
> > > /tuning-math/files/dict/torsion.htm
> > >
> > >
> > >
> > > -monz
> > > "all roads lead to n^0"
> >
> > i see the green and red lines, but . . . which symmetry exactly
are
> > you referring to?
>
>
> do you see "The thin black line which divides the block in half
> diagonally is the torsional interval, 6561:6400 = [-8 2] = (81/80)^2
> = (648/625) (2048/2025)^(-1).]" on the diagram?
>
>
>
> -monz

sure -- but i don't see any symmetries unique to torsional blocks
associated with reflection about that line or about the center point -
- just the usual symmetry you get when you center a symmetrical shape
about a point of symmetry in the lattice.

however, it seems to be that if you cut the block exactly in half
along the direction of one of its edges, you'd end up with two
instances of a geniune (non-torsional) periodicity block. every note
on the "left" half of the block is connected, with either a green or
a red line, to one and only one note on the "right" half of the
block, and also every note on the "top" half of the block is
connected, with either a green or a red line, to one and only one
note on the "bottom" half of the block. this results from seeing the
black line you referred to as two syntonic commas end-to-end; using
only one of these "half-black-lines" to construct the periodicity
block removes the torsion problem and leaves you with a block half as
big as the original (you have to do a bit of the "wedge-shifting" as
described in my _excursion_ to get these blocks to fit within the
borders of the original torsional block).

making sense?

p.s. are you reading my posts on the tuning list, monzieur? i was
going to post a (fairly serious) critique of your new 12-edo page,
but i'm afraid no one, not even you, would read it, since you haven't
really replied to, or incorporated into your webpages, my last two
lengthy tuning lists posts to you. if i e-mailed this critique to you
privately, would you have a better chance of reading it?

From: "monz" <monz@a...>:
>
>Is there some way to mathematically model
>the symmetry in a torsion-block?
>
>see the graphic and its related text in my
>Tuning Dictionary definition of "torsion"
>-- i've uploaded it to here:
>/tuning-math/files/dict/torsion.htm
>

Well, they are translation symmetries in the quotient group of the full lattice
and the subgroup generated by the unison vectors. The symmetries in the
example are pairs because the element has order 2 in the quotient group,
but there are other elements such as (0,1) with order 6 or (0,2), (1,1) with
order 3. Something like this?

BTW, I think the definition of torsion can be made simpler. You do not need
the condition that some power of the interval is in the unison vector group,
because this is always the case (at least when the periodicity block is finite).
Do I see this correctly?

Hans Straub

hi Hans,

thanks very much for your replies to this, but
i'm afraid some of the math language is over my head.
i defer to Gene, paul, Graham, et al. for comment.

-monz
"all roads lead to n^0"

----- Original Message -----
From: "Hans Straub" <straub@datacomm.ch>
To: <tuning-math@yahoogroups.com>
Sent: Wednesday, October 09, 2002 2:47 PM
Subject: [tuning-math] Re: mathematical model of torsion-block symmetry?

> From: "monz" <monz@a...>:
> >
> >Is there some way to mathematically model
> >the symmetry in a torsion-block?
> >
> >see the graphic and its related text in my
> >Tuning Dictionary definition of "torsion"
> >-- i've uploaded it to here:
> >/tuning-math/files/dict/torsion.htm
> >
>
> Well, they are translation symmetries in the quotient group of the full
lattice
> and the subgroup generated by the unison vectors. The symmetries in the
> example are pairs because the element has order 2 in the quotient group,
> but there are other elements such as (0,1) with order 6 or (0,2), (1,1)
with
> order 3. Something like this?
>
>
> BTW, I think the definition of torsion can be made simpler. You do not
need
> the condition that some power of the interval is in the unison vector
group,
> because this is always the case (at least when the periodicity block is
finite).
> Do I see this correctly?
>
> Hans Straub

>From: "monz" <monz@a...>
>Date: Thu Oct 10, 2002 8:34 am
>Subject: Re: [tuning-math] Re: mathematical model of torsion-block
symmetry?
>
>hi Hans,
>
>thanks very much for your replies to this, but
>i'm afraid some of the math language is over my head.
>i defer to Gene, paul, Graham, et al. for comment.

That would indeed be good - because I often have the impression the things
here are above _my_ head... I still got the impression I might have something
wrong about that torsion thing.

The base lattice (Z^2) is a Z-module (like a vector space but only integers as
coefficients and for scalar multiplication), and so is the quotient (the
elements of which are simply equivalence classes of intervals with respect to
the unison vectors). A periodicity block, BTW, is nothing else but a set of
adjacent representants of the quotient module (one representant for each
equivalence class).

Now, the quotient module being finite, for every element v of it there is an non-
vanishing integer n such that n*v is 0 - in other words: for _every_ interval
there is some power of it that is a combination of unison vectors. This is a
basic property of finite modules - and exactly here is the point where I would
like Gene, Paul, Graham, et al. to explain - for it appears to me that EVERY
periodicity block has torsion...

For a vector v, the smallest non-vanishing integer such that n*v is in the
unison vector submodule is called the order of v. For symmetries in the
quotient module, orders play a prominent role.
In the example on the website (unison vectors (4,-4) and (-4,-2)), the
equivalence class of the diesis vector (0,-3) is an element of order 2, since
2*(0,-3) is in the unison vector submodule. But if you take, e.g., the vector (0,-
2) (interval 32/25), its equivalence class has order 3. If you draw a picture of
this (as is done with the diesis vector in the example), there will appear
triples of connected elements. The same procedure with the vector (0,-1)
(interval (8/5) will partition the periodicity block in hexuples of connected
elements.

Making sense? I hope I got this right...
-
Hans Straub
Hans Straub
http://home.datacomm.ch/straub/mamuth

--- In tuning-math@y..., "hs" <straub@d...> wrote:

> The base lattice (Z^2) is a Z-module (like a vector space but only integers as
> coefficients and for scalar multiplication), and so is the quotient (the
> elements of which are simply equivalence classes of intervals with respect to
> the unison vectors). A periodicity block, BTW, is nothing else but a set of
> adjacent representants of the quotient module (one representant for each
> equivalence class).

I've mentioned this before, but readers used to the "abelian group" terminology should keep in mind that abelian group and Z-module mean the same thing.

> Now, the quotient module being finite...

Whups--you are sticking "2" into the mix when you conclude this.
The math is more straightforward if you treat 2 as just another prime number.

>
> I've mentioned this before, but readers used to the "abelian group" terminology should keep in mind that abelian group and Z-module mean the same thing.
>

I like the Z-module approach because it emphasizes the vector properties -
but if people are more used to abelian groups, I can use that, of course.

> > Now, the quotient module being finite...
>
> Whups--you are sticking "2" into the mix when you conclude this.
> The math is more straightforward if you treat 2 as just another prime number.
>

2 is just another prime number, sure - but where exactly do you think I
confuse something? So far, I see no flaw in my reasoning... It still seems to
me that the "torsion" definition in the tuning dictionary describes a rather
trivial property that every finite periodicity block has (after all, a "torsion
group", as decribed on mathworld.wolfram.com, is nothing but a finite group).
If this were so, I would suggest you remove this term from the dictionary.

Regards,

Hans Straub

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

> BTW, I think the definition of torsion can be made simpler. You do
not need
> the condition that some power of the interval is in the unison
vector group,
> because this is always the case (at least when the periodicity
block is finite).
> Do I see this correctly?
>
> Hans Straub

i'm not sure. torsion is a pathological condition, especially from
the point of view of generating temperaments. it's certainly not true
that all finite periodicity blocks exhibit torsion. so what exactly
are you saying?

hans, it looks like you're tripping up over this 2 thing, which may
have confused graham in the past as well. 2 most certainly does make
a difference, and watching it closely allows one to detect the
(relatively few) torsional periodicity blocks from among the many non-
torsional ones which are well-behaved. the example on monz's page has
24 notes but the group really only has 12 elements (per octave,
anyway). if you temper out the unison vectors you get 12-equal, not
24-equal. hence it's "pathological" in terms of the old chalmers bit
about justifying ETs in terms of fokker periodicity blocks with the
same number of notes.

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:
> >
> > I've mentioned this before, but readers used to the "abelian
group" terminology should keep in mind that abelian group and Z-
module mean the same thing.
> >
>
> I like the Z-module approach because it emphasizes the vector
properties -
> but if people are more used to abelian groups, I can use that, of
course.
>
> > > Now, the quotient module being finite...
> >
> > Whups--you are sticking "2" into the mix when you conclude this.
> > The math is more straightforward if you treat 2 as just another
prime number.
> >
>
> 2 is just another prime number, sure - but where exactly do you
think I
> confuse something? So far, I see no flaw in my reasoning... It
still seems to
> me that the "torsion" definition in the tuning dictionary describes
a rather
> trivial property that every finite periodicity block has (after
all, a "torsion
> group", as decribed on mathworld.wolfram.com, is nothing but a
finite group).
> If this were so, I would suggest you remove this term from the
dictionary.
>
> Regards,
>
> Hans Straub

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:
> >
> > I've mentioned this before, but readers used to the "abelian group" terminology should keep in mind that abelian group and Z-module mean the same thing.
> >
>
> I like the Z-module approach because it emphasizes the vector properties -
> but if people are more used to abelian groups, I can use that, of course.

I suspect you like it because you are Swiss--Z-module is Continental, whereas an American mathematician will normally say abelian group.

> > > Now, the quotient module being finite...
> >
> > Whups--you are sticking "2" into the mix when you conclude this.
> > The math is more straightforward if you treat 2 as just another prime number.
> >
>
> 2 is just another prime number, sure - but where exactly do you think I
> confuse something?

Monzo's example was the block defined by 2048/2025 and 648/625; if we mod out the free group on three generators {2,3,5} by the subgroup defined by the above, we produce a mapping onto Z x Z/2Z. This has a nontrivial torsion part, so the block is a torsion block.

Using wedge products, which in the 5-limit we can identify with the cross-product, we have 2048/2025 ^ 648/625 = [11 -4 -2] ^ [3 4 -4] =
[24 38 56] = 2 * [12 19 28], showing the 2-torsion. For this to work, the vectors need to be defined using the 2; Monzo unfortunately left this off and the page should be changed.

>
> Monzo's example was the block defined by 2048/2025 and 648/625; if we mod out the free group on three generators {2,3,5} by the subgroup defined by the above, we produce a mapping onto Z x Z/2Z. This has a nontrivial torsion part, so the block is a torsion block.
>
> Using wedge products, which in the 5-limit we can identify with the cross-product, we have 2048/2025 ^ 648/625 = [11 -4 -2] ^ [3 4 -4] =
> [24 38 56] = 2 * [12 19 28], showing the 2-torsion. For this to work, the vectors need to be defined using the 2; Monzo unfortunately left this off and the page should be changed.
>

Ah, now it is starting to look clearer. So I did have something wrong :-(
Tricky thing that without the 2, there is a Z/12Z component, but with the 2
there is none... BTW, are you sure the quotient is Z x Z/2Z? It appears to me
it should be Z x Z x Z/12Z. What would be the basis, then?

And another question concerning this: the periodicity blocks I have seen
displayed so far all seemed to be drawn without the 2 (one of the reasons for
my mistake above). Somehow you must use octave idenitifcation - or am I
missing something again?

Regards,
Hans Straub

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:
>
> And another question concerning this: the periodicity blocks I have
seen
> displayed so far all seemed to be drawn without the 2 (one of the
reasons for
> my mistake above). Somehow you must use octave idenitifcation - or
am I
> missing something again?
>
> Regards,
> Hans Straub

it's true that you must use octave identification. most musicians
think of pitch in "pitch-class" terms, which means "modulo" the
octave.

all of the BP periodicity blocks that have been displayed use 3,
instead of 2, as the interval of equivalence. some musicians claim
they can "hear" equivalence this way.

as far as we know, though, octave-equivalence is universal among the
world's musical cultures.

projecting down to a 2-less subspace does tweak the various distance
metrics, though. that is why i like to use a triangular lattice,
instead of a rectangular one, when dealing with these 2-less
subspaces. it represents the "average" or "effective" distance
between pitch classes much better that way.

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

>BTW, are you sure the quotient is Z x Z/2Z? It appears to me
> it should be Z x Z x Z/12Z. What would be the basis, then?

Here's one way to explain it:

Define the homomorphic mappings, or "vals", h12 = [12,19,28],
h4 = [4,6,9] and g = [1,2,3]. Then we may define a mapping from the postive rationals into themselves by

H(q) = (16/15)^h12(q) (2025/2048)^h4(q) (625/648)^g(q)

You may verify that q/H(q) is always some power of 81/80; so that
any positive rational number q may be written as

q = H(q) (81/80)^n

If we mod out the kernel generated by 2048/2025 and 648/625, H(q) is
sent to h12(q), so that it maps to Z. Since (648/625)/(2048/2025) =
(81/80)^2, even exponoents n are sent to the identity, and odd exponents to the torsion part; this part of the mapping is the Z/2Z part. In other words, the mapping is

q --> [h(12,q), n mod 2]

If you like, you can think of the even n as being played on the blue piano, and the odd n as being played on the red piano, with any tuning difference between pianos being up to you.