the unison-vector<-->determinant relationship

I just noticed something about the relationship
between unison-vectors and determinants. Let me
know if I've discovered something. Here goes...

I found that

for matrix M = [a b]
[c d]

matrix M' = [(a +/- c) (b +/- d)]
[ c d ]

results in the same determinant.

Has anyone ever noticed this before?
Is it simply a logical result of periodicity-block math?
Is it common knowledge that I missed?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 2:23 AM
> Subject: [tuning-math] the unison-vector<-->determinant relationship
>
>
> I just noticed something about the relationship
> between unison-vectors and determinants. Let me
> know if I've discovered something. Here goes...
>
>
> I found that
>
> for matrix M = [a b]
> [c d]
>
>
> matrix M' = [(a +/- c) (b +/- d)]
> [ c d ]
>
> results in the same determinant.
>
>
>
> Has anyone ever noticed this before?
> Is it simply a logical result of periodicity-block math?
> Is it common knowledge that I missed?

Oops... my bad. That's not quite right.
Change the second part to:

for x = any integer,

matrix M' = [(a+cx) (b+dx)] and [(a-cx) (b-dx)]
[ c d ] [ c d ]

results in the same determinant.

Is there a better way to write that?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

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

This is a basic property of determinants; one proof uses the definition of an nxn determinant in terms of an n-fold wedge product.
If D = v1^v2^v3 ... ^vn, then (v1+x*v2)^v2^v3...^vn =
v1^v2^v3 ... ^vn + x*v2^v2^v3...^vn (distributive law for wedge products). Since v2^v2=0, the second term cancels, and the determinant equals D.

Hi Gene!

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 2:45 AM
> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> This is a basic property of determinants; one proof uses
> the definition of an nxn determinant in terms of an n-fold
> wedge product.
>
> If D = v1^v2^v3 ... ^vn, then (v1+x*v2)^v2^v3...^vn =
> v1^v2^v3 ... ^vn + x*v2^v2^v3...^vn (distributive law
> for wedge products). Since v2^v2=0, the second term cancels,
> and the determinant equals D.

Thanks for that explanation...
unfortunately, it looks like Chinese to me.

If you can at least make it look like Sumerian,
maybe I'll get it... :)

Now that I finally understand a bit about how matrices work,
can you use them to rewrite what you wrote above? I simply
don't comprehend your algebra.

Is there someplace online that explains these "wedge products"?
Perhaps old tuning-math posts?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 8:46 AM
> Subject: Re: [tuning-math] Re: the unison-vector<-->determinant
relationship
>
>
> Is there someplace online that explains these "wedge products"?
> Perhaps old tuning-math posts?

I think I've answered my own question:
http://mathworld.wolfram.com/WedgeProduct.html

But it still looks like Chinese. :(

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

monz wrote:

> > Is there someplace online that explains these "wedge products"?
> > Perhaps old tuning-math posts?

See if you can find the post I made not long after I worked out the
principles. All you need to know is ei^ej = -ej^ei

and again:

> I think I've answered my own question:
> http://mathworld.wolfram.com/WedgeProduct.html
>
> But it still looks like Chinese. :(

Can you work out Python? It's related to Dutch. The code at
<http://x31eq.com/temper.py> includes wedge products. If you
can get it working in an interpreter, you can also try lots of examples.

Graham

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

> But it still looks like Chinese. :(

There are some old posts--until recently I would not have tossed multilinear algebra at people so casually, but there's been a lot of discussion of the wedge product.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 1:38 PM
> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > But it still looks like Chinese. :(
>
> There are some old posts--until recently I would not have
> tossed multilinear algebra at people so casually, but there's
> been a lot of discussion of the wedge product.

Ah... I think I'm getting it. "Multilinear algebra"...

So, for example, on my
"1/6-comma meantone within (4 -1) (19 9) periodicity-block" lattice...

[plug alert]

(which I've just added in miniature with some new text
on my JustMusic software webpage:
<http://www.ixpres.com/interval/monzo/justmusic/introtojm.htm>,)

... you can see that the 1/6-comma meantone vector is has an
off-kilter relationship with 5-limit JI. Two different linear
algebras simultaneous at work, right? (I sure hope so...)

OK, fix what's wrong with this, if anything...

One can find an infinity of closer and closer
fraction-of-a-comma meantone representations of that
5-limit JI periodicity-block, which would be represented
on my lattice here as shiftings of the angle of the vector
representing the meantone. But one could never find one
that would go straight down the middle of the symmetrical
periodicity-block.

Now, what does this mean?
(please use as much English as possible along with the algebra)

I have similiar questions for the octave-specific versions
of my lattices, which utilize prime-factor 2, and on which
I can also lattice EDOs.

-monz
(so sorrowfully deficient in math...)

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

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

> ... you can see that the 1/6-comma meantone vector is has an
> off-kilter relationship with 5-limit JI. Two different linear
> algebras simultaneous at work, right? (I sure hope so...)

If you look at u^v, then it is linear in u *and* v, hence bilinear. The same is true of a related doodad called the tensor product, and of the dot product, which is a bilinear form. So all this stuff is called "multilinear algebra", except in contexts where something called the Clifford algebra comes to the fore and we have "geometric algebra" instead.

>
> OK, fix what's wrong with this, if anything...
>
> One can find an infinity of closer and closer
> fraction-of-a-comma meantone representations of that
> 5-limit JI periodicity-block, which would be represented
> on my lattice here as shiftings of the angle of the vector
> representing the meantone. But one could never find one
> that would go straight down the middle of the symmetrical
> periodicity-block.
>
> Now, what does this mean?

I'm not following you--why not spell it out, giving the block in question?

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 4:10 PM
> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
>
>
>
> >
> > OK, fix what's wrong with this, if anything...
> >
> > One can find an infinity of closer and closer
> > fraction-of-a-comma meantone representations of that
> > 5-limit JI periodicity-block, which would be represented
> > on my lattice here as shiftings of the angle of the vector
> > representing the meantone. But one could never find one
> > that would go straight down the middle of the symmetrical
> > periodicity-block.
> >
> > Now, what does this mean?
>
> I'm not following you--why not spell it out, giving the block in question?

OK. I'd much rather spell it out with 1/6-comma, since
I've been working more with that.

But I already have a spreadsheet which shows how various
meantones "fit" within a particular periodicity-block.
I posted about it here last week:

my original post:
/tuning-math/message/2102

errata:
/tuning-math/message/2103

Paul's criticism:
/tuning-math/message/2104

And guess what? Oops! It looks like the conclusion I
reached there is that the 7/25-comma meantone *does* indeed
go straight down the middle of the periodicity-block!

Here's my Microsoft Excel spreadsheet latting these meantones:

/tuning-math/files/monz/(6%20-14)%20(4%20-1)%20
PB%20and%207-25cmt.xls

I made all the graphs the same size and put them in
exactly the same place, so if you don't change anything,
you can click on the worksheet tabs at the bottom and
see how the different meantones lie within the periodicity-block.

Hmmm... *if* my lattice *was* wrapped as a cylinder or
torus, would there be any significance to this "middle
of the road" business?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > ... you can see that the 1/6-comma meantone vector is has an
> > off-kilter relationship with 5-limit JI. Two different linear
> > algebras simultaneous at work, right? (I sure hope so...)
>
> If you look at u^v, then it is linear in u *and* v, hence bilinear.
>The same is true of a related doodad called the tensor product,

Ah, so this is related to the tensor product? Is there an ellipsoid
associated with the wedge product??

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 5:36 PM
> Subject: Re: [tuning-math] Re: the unison-vector<-->determinant
relationship
>
>
>
> > From: genewardsmith <genewardsmith@juno.com>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Saturday, December 29, 2001 4:10 PM
> > Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
> >
> >
> >
> > >
> > > OK, fix what's wrong with this, if anything...
> > >
> > > One can find an infinity of closer and closer
> > > fraction-of-a-comma meantone representations of that
> > > 5-limit JI periodicity-block, which would be represented
> > > on my lattice here as shiftings of the angle of the vector
> > > representing the meantone. But one could never find one
> > > that would go straight down the middle of the symmetrical
> > > periodicity-block.
> > >
> > > Now, what does this mean?
> >
> > I'm not following you--why not spell it out, giving the block in
question?
>
>
>
> OK. I'd much rather spell it out with 1/6-comma, since
> I've been working more with that.
>
> But I already have a spreadsheet which shows how various
> meantones "fit" within a particular periodicity-block.
> I posted about it here last week:
>
> my original post:
> /tuning-math/message/2102
>
> errata:
> /tuning-math/message/2103
>
> Paul's criticism:
> /tuning-math/message/2104
>
>
> And guess what? Oops! It looks like the conclusion I
> reached there is that the 7/25-comma meantone *does* indeed
> go straight down the middle of the periodicity-block!
>
> Here's my Microsoft Excel spreadsheet latting these meantones:
>
>
/tuning-math/files/monz/(6%20-14)%20(4%20-1)%20
> PB%20and%207-25cmt.xls
>
> I made all the graphs the same size and put them in
> exactly the same place, so if you don't change anything,
> you can click on the worksheet tabs at the bottom and
> see how the different meantones lie within the periodicity-block.
>
>
> Hmmm... *if* my lattice *was* wrapped as a cylinder or
> torus, would there be any significance to this "middle
> of the road" business?
>
>
> -monz

I should have specified... for the benefit of those who
are not able to see my spreadsheet, it forms a 50-tone
periodicity-block from the [3, 5] unison-vectors [6 -14],[-4 1].
Then each spreadsheet lattices a fraction-of-a-comma type
meantone within it: 2/7-, 7/25-, 5/18-, and 3/11-comma.

The 7/25-comma meantone is lattice right down the middle of
the symmetrical periodicity-block.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> The 7/25-comma meantone is lattice right down the middle of
> the symmetrical periodicity-block.

Monz, I can't view your .xls file right now, and I'm wondering what
it means. I've seen quite a few of your lattices for this topic, but
I have to clue as to how to picture what you're describing above.

Can you help?

Have you received my package yet?

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

> Ah, so this is related to the tensor product?

A wedge product can be defined as a quotient of a tensor product; it is an antisymmetrized tensor product, in effect.

Is there an ellipsoid
> associated with the wedge product??

There's a parallepiped associated to it, and you could associate an ellipsoid to that if you wanted to--why do you ask?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > Ah, so this is related to the tensor product?
>
> A wedge product can be defined as a quotient of a tensor product;
it is an antisymmetrized tensor product, in effect.
>
> Is there an ellipsoid
> > associated with the wedge product??
>
> There's a parallepiped associated to it, and you could associate an
ellipsoid to that if you wanted to--why do you ask?

This parallelepiped -- does it exist in the lattice of notes? Perhaps
after some transformation? (I really need to go through the GA Matlab
tutorial you gave me -- haven't had time yet).

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 6:52 PM
> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > The 7/25-comma meantone is lattice right down the middle of
> > the symmetrical periodicity-block.
>
> Monz, I can't view your .xls file right now, and I'm wondering what
> it means. I've seen quite a few of your lattices for this topic, but
> I have to [_sic_: no] clue as to how to picture what you're describing
> above.

OK, let's go back to 1/6-comma meantone, since that's what I've been
mostly working with here.

On my diagram here:
http://www.ixpres.com/interval/monzo/justmusic/introtojm.htm
(you can click on the diagram to open a big version),
you can see that 1/6-comma meantone does not run exactly
down the center of the (19 9),(4 -1) periodicity-block.

There are other fraction-of-a-comma meantones which come
closer to the center, and it seems to me that the one which
*does* run exactly down the middle is 8/49-comma.

Is this derivable from the [19 9],[4 -1] matrix? Is there
any kind of significance to it?

It seems to me that a meantone chain that would run down the
center of a periodicity-block would have the smallest overall
deviation from the most closely implied JI ratios in the
periodicity-block, assuming that the JI lattice is wrapped
into a cylinder. Yes?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

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

> This parallelepiped -- does it exist in the lattice of notes?

It's simply the parallepiped defined by the vectors we are wedging together, so yes. The wedge of two vectors is the directed area of the associated parallogram, of three vectors the directed volume of the associated parallepiped, and so forth.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Saturday, December 29, 2001 6:52 PM
> > Subject: [tuning-math] Re: the unison-vector<-->determinant
relationship
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >
> > > The 7/25-comma meantone is lattice right down the middle of
> > > the symmetrical periodicity-block.
> >
> > Monz, I can't view your .xls file right now, and I'm wondering
what
> > it means. I've seen quite a few of your lattices for this topic,
but
> > I have to [_sic_: no] clue as to how to picture what you're
describing
> > above.
>
>
> OK, let's go back to 1/6-comma meantone, since that's what I've been
> mostly working with here.
>
> On my diagram here:
> http://www.ixpres.com/interval/monzo/justmusic/introtojm.htm
> (you can click on the diagram to open a big version),
> you can see that 1/6-comma meantone does not run exactly
> down the center of the (19 9),(4 -1) periodicity-block.

Right, and if you used a different unison vector besides (19 9) to
define your JI block, you would also see this.

> There are other fraction-of-a-comma meantones which come
> closer to the center, and it seems to me that the one which
> *does* run exactly down the middle is 8/49-comma.
>
> Is this derivable from the [19 9],[4 -1] matrix?

You should find that the interval corresponding to (19 9), AS IT
APPEARS in 8/49-comma meantone, is a very tiny interval.

> Is there
> any kind of significance to it?

In my opinion, no.
>
> It seems to me that a meantone chain that would run down the
> center of a periodicity-block would have the smallest overall
> deviation from the most closely implied JI ratios in the
> periodicity-block, assuming that the JI lattice is wrapped
> into a cylinder. Yes?

Assuming the lattice is not wrapped into a cylinder, it might make
some infinitesimal difference. If the lattice is wrapped around a
cylinder, as it should be if you're talking about an actual meantone
tuning, then you're implying an infinite number of JI ratios no
matter what meantone tuning you're using.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > This parallelepiped -- does it exist in the lattice of notes?
>
> It's simply the parallepiped defined by the vectors we are wedging
>together, so yes. The wedge of two vectors is the directed area of
>the associated parallogram, of three vectors the directed volume of
>the associated parallepiped, and so forth.

So the wegde product is simply the boundary of the Fokker periodicity
block??

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 7:48 PM
> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > There are other fraction-of-a-comma meantones which come
> > closer to the center, and it seems to me that the one which
> > *does* run exactly down the middle is 8/49-comma.
> >
> > Is this derivable from the [19 9],[4 -1] matrix?
>
> You should find that the interval corresponding to (19 9), AS IT
> APPEARS in 8/49-comma meantone, is a very tiny interval.

Ah... so then 8/49-comma meantone does *not* run *exactly*
down the middle. How could one calculate the meantone which
*does* run exactly down the middle?

> > Is there any kind of significance to it?
>
> In my opinion, no.
> >
> > It seems to me that a meantone chain that would run down the
> > center of a periodicity-block would have the smallest overall
> > deviation from the most closely implied JI ratios in the
> > periodicity-block, assuming that the JI lattice is wrapped
> > into a cylinder. Yes?
>
> Assuming the lattice is not wrapped into a cylinder, it might make
> some infinitesimal difference. If the lattice is wrapped around a
> cylinder, as it should be if you're talking about an actual meantone
> tuning, then you're implying an infinite number of JI ratios no
> matter what meantone tuning you're using.

Right, I understand that. But since any given meantone interval
can only be closest to *one* particular 5-limit JI ratio, which
should fall within the periodicity-block, the vector of the meantone
chain will *still* imply a unique periodicity-block, will it not?
(assuming that the periodicity-block is replicated a comma away
as one travels around the cylinder)

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Saturday, December 29, 2001 7:48 PM
> > Subject: [tuning-math] Re: the unison-vector<-->determinant
relationship
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > There are other fraction-of-a-comma meantones which come
> > > closer to the center, and it seems to me that the one which
> > > *does* run exactly down the middle is 8/49-comma.
> > >
> > > Is this derivable from the [19 9],[4 -1] matrix?
> >
> > You should find that the interval corresponding to (19 9), AS IT
> > APPEARS in 8/49-comma meantone, is a very tiny interval.
>
>
> Ah... so then 8/49-comma meantone does *not* run *exactly*
> down the middle. How could one calculate the meantone which
> *does* run exactly down the middle?

It's 55-tET.
>
> Right, I understand that. But since any given meantone interval
> can only be closest to *one* particular 5-limit JI ratio,

While functioning as an infinite number.

> which
> should fall within the periodicity-block, the vector of the meantone
> chain will *still* imply a unique periodicity-block, will it not?

I don't get it. What's the vector of the meantone chain? Is (19 9) an
example?

> (assuming that the periodicity-block is replicated a comma away
> as one travels around the cylinder)

Assuming to the answer to the last question it "yes", then I'd
say, "no, one would get a "strip" rather than a periodicity block",
but then of course I'd be ignoring your "since any given meantone
interval can only be closest to *one* particular 5-limit JI ratio".
If I take that part seriously, I have two comments:

(a) You are NOT, with your current method, mapping identical meantone
intervals to identical JI ratios, and

(b) if you really meant "pitches" rather than "intervals", I'd argue
that the mappings you are producing involve a rather arbitrary rule,
and don't reflect the musical properties of the meantone tunings. The
only case in which they would is if you specifically knew you were
not going to use any of the consonances that "wrap" around the block,
AND you were interested in using a simultaneous JI tuning with the
meantone that would minimize the _pitch_ differences between the two -
- a very contrived scenario.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 8:07 PM
> Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > From: paulerlich <paul@s...>
> > > To: <tuning-math@y...>
> > > Sent: Saturday, December 29, 2001 7:48 PM
> > > Subject: [tuning-math] Re: the unison-vector<-->determinant
> relationship
> > >
> > >
> > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > There are other fraction-of-a-comma meantones which come
> > > > closer to the center, and it seems to me that the one which
> > > > *does* run exactly down the middle is 8/49-comma.
> > > >
> > > > Is this derivable from the [19 9],[4 -1] matrix?
> > >
> > > You should find that the interval corresponding to (19 9), AS IT
> > > APPEARS in 8/49-comma meantone, is a very tiny interval.
> >
> >
> > Ah... so then 8/49-comma meantone does *not* run *exactly*
> > down the middle. How could one calculate the meantone which
> > *does* run exactly down the middle?
>
> It's 55-tET.

Not if the periodicity-block is a parallelogram. 10/57-comma meantone
is much closer to 55-EDO than 1/6-comma meantone, yet it is further
away from the center of this periodicity-block.

I suppose that if some of the ratios in the periodicity-block were
transposed, so that the block had a different shape, then 10/57 *would*
run down the center ... but not for this one.

> >
> > Right, I understand that. But since any given meantone interval
> > can only be closest to *one* particular 5-limit JI ratio,
>
> While functioning as an infinite number.
>
> > which
> > should fall within the periodicity-block, the vector of the meantone
> > chain will *still* imply a unique periodicity-block, will it not?
>
> I don't get it. What's the vector of the meantone chain? Is (19 9) an
> example?

Well... the only way I can explain it is that on my lattice,
the meantone chain which runs right down the center of this
periodicity-block *does* follow the same angle on the lattice
as (19 9).

>
> > (assuming that the periodicity-block is replicated a comma away
> > as one travels around the cylinder)
>
> Assuming to the answer to the last question it "yes", then I'd
> say, "no, one would get a "strip" rather than a periodicity block",
> but then of course I'd be ignoring your "since any given meantone
> interval can only be closest to *one* particular 5-limit JI ratio".
> If I take that part seriously, I have two comments:
>
> (a) You are NOT, with your current method, mapping identical meantone
> intervals to identical JI ratios, and

Not really sure what you mean by this.

> (b) if you really meant "pitches" rather than "intervals", I'd argue
> that the mappings you are producing involve a rather arbitrary rule,
> and don't reflect the musical properties of the meantone tunings. The
> only case in which they would is if you specifically knew you were
> not going to use any of the consonances that "wrap" around the block,
> AND you were interested in using a simultaneous JI tuning with the
> meantone that would minimize the _pitch_ differences between the two -
> - a very contrived scenario.

OK, now I'm getting more confused again.

Again -- the reader is supposed to *imagine* that my mappings
wrap cylindrically. So I don't understand why you're pointing
out cases where the consonances that wrap are not used.

Also, I'm not thinking specifically of pitches. I asked this
before but didn't get an answer that was clear -- what's the
real difference? If I assume that my flat lattice is supposed
to wrap cylindrically, then why does it matter whether I'm
considering pitches or intervals? Isn't it the same?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, December 29, 2001 8:25 PM
> Subject: Re: [tuning-math] Re: the unison-vector<-->determinant
relationship
>
>
>
> > From: paulerlich <paul@stretch-music.com>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Saturday, December 29, 2001 8:07 PM
> > Subject: [tuning-math] Re: the unison-vector<-->determinant relationship
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > From: paulerlich <paul@s...>
> > > > To: <tuning-math@y...>
> > > > Sent: Saturday, December 29, 2001 7:48 PM
> > > > Subject: [tuning-math] Re: the unison-vector<-->determinant
> > relationship
> > > >
> > > >
> > > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > > >
> > > > > There are other fraction-of-a-comma meantones which come
> > > > > closer to the center, and it seems to me that the one which
> > > > > *does* run exactly down the middle is 8/49-comma.
> > > > >
> > > > > Is this derivable from the [19 9],[4 -1] matrix?
> > > >
> > > > You should find that the interval corresponding to (19 9), AS IT
> > > > APPEARS in 8/49-comma meantone, is a very tiny interval.
> > >
> > >
> > > Ah... so then 8/49-comma meantone does *not* run *exactly*
> > > down the middle. How could one calculate the meantone which
> > > *does* run exactly down the middle?
> >
> > It's 55-tET.
>
>
> Not if the periodicity-block is a parallelogram. 10/57-comma meantone
> is much closer to 55-EDO than 1/6-comma meantone, yet it is further
> away from the center of this periodicity-block.
>
> I suppose that if some of the ratios in the periodicity-block were
> transposed, so that the block had a different shape, then 10/57 *would*
> run down the center ... but not for this one.
>
>
> > >
> > > Right, I understand that. But since any given meantone interval
> > > can only be closest to *one* particular 5-limit JI ratio,
> >
> > While functioning as an infinite number.
> >
> > > which
> > > should fall within the periodicity-block, the vector of the meantone
> > > chain will *still* imply a unique periodicity-block, will it not?
> >
> > I don't get it. What's the vector of the meantone chain? Is (19 9) an
> > example?
>
>
> Well... the only way I can explain it is that on my lattice,
> the meantone chain which runs right down the center of this
> periodicity-block *does* follow the same angle on the lattice
> as (19 9).

And I felt that I should point out that since many periodicity-blocks
could be found which have the same determinant, there are thus many
different meantones which "go with" a particular determinant, depending
on the actual unison-vectors employed in the definition of the
periodicity-block.

For this particular (19 9),(4 -1) block, 8/49-comma meantone seems
*by eye* to be the meantone which splits the periodicity-block exactly
in half, and thus has the lowest average deviation from the entire
set of JI ratios within the block. Again I ask: how can this be
derived mathematically?

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

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

> > > > > There are other fraction-of-a-comma meantones which come
> > > > > closer to the center, and it seems to me that the one which
> > > > > *does* run exactly down the middle is 8/49-comma.
> > > > >
> > > > > Is this derivable from the [19 9],[4 -1] matrix?
> > > >
> > > > You should find that the interval corresponding to (19 9), AS
IT
> > > > APPEARS in 8/49-comma meantone, is a very tiny interval.
> > >
> > >
> > > Ah... so then 8/49-comma meantone does *not* run *exactly*
> > > down the middle. How could one calculate the meantone which
> > > *does* run exactly down the middle?
> >
> > It's 55-tET.
>
>
> Not if the periodicity-block is a parallelogram. 10/57-comma
meantone
> is much closer to 55-EDO than 1/6-comma meantone, yet it is further
> away from the center of this periodicity-block.

Hmm . . . the line you want is the vector (19 9). So any generator
3^a/b * 5^c/d that is a solution to the equation

a/b * 19 + c/d * 9 = 0

would work. This gives

a/b*19 = -c/d*9

Does this help?

> > (a) You are NOT, with your current method, mapping identical
meantone
> > intervals to identical JI ratios, and
>
>
> Not really sure what you mean by this.

For example, if you look at the fifth D-A, in some of your diagrams
this is mapped to 40:27, while C-G is mapped to 3:2, and yet the two
intervals are tuned _identically_ in any of the meantones in question.

> > (b) if you really meant "pitches" rather than "intervals", I'd
argue
> > that the mappings you are producing involve a rather arbitrary
rule,
> > and don't reflect the musical properties of the meantone tunings.
The
> > only case in which they would is if you specifically knew you
were
> > not going to use any of the consonances that "wrap" around the
block,
> > AND you were interested in using a simultaneous JI tuning with
the
> > meantone that would minimize the _pitch_ differences between the
two -
> > - a very contrived scenario.
>
>
> OK, now I'm getting more confused again.
>
> Again -- the reader is supposed to *imagine* that my mappings
> wrap cylindrically.

If they are so wrapped, all the diagrams for the different meantones
will look _exactly the same_. So what's the point of all the
different shapes?

> So I don't understand why you're pointing
> out cases where the consonances that wrap are not used.

Because in those cases, you're mapping a JI tuning which will
function similarly to the meantone in question.

> Also, I'm not thinking specifically of pitches. I asked this
> before but didn't get an answer that was clear -- what's the
> real difference?

See my remark about D-A and C-G above. You did the same thing when
you mapped Partch's 43 to 72-tET. You concerned yourself with
approximating the pitches, while approximating the consonant
intervals leads to a more musically relevant result.

> If I assume that my flat lattice is supposed
> to wrap cylindrically, then why does it matter whether I'm
> considering pitches or intervals?

Well, again, in that case all your diagrams would look identical, but
they don't, so it looks like you're implying something different.