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

_________________________________________________________

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> 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

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--- 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

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> 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

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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...)

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--- 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

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--- 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

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--- 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

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--- 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

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--- 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

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> 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

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--- 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.