Hey Paul,

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

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

> Sent: Sunday, January 20, 2002 7:48 PM

> Subject: [tuning-math] Re: lattices of Schoenberg's rational implications

>

>

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

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

> >

> > However, I think the only reality for Schoenberg's

> > > system is a tuning where there is ambiguity, as defined by the

> kernel

> > > <33/32, 64/63, 81/80, 225/224>. BTW, is this Minkowski-reduced?

> >

> > Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

>

> Awesome. So this suggests a more compact Fokker parallelepiped

> as "Schoenberg PB" -- here are the results of placing it in different

> positions in the lattice (you should treat the inversions of these as

> implied):

>

> <tables of ratios snipped>

I've always been careful to emphasize that our tuning-theory use

of "lattice" is different from the mathematician's strictly define

uses of the term. I've been searching the web to learn about

Minkowksi reduction, and so now it appears to me that we are

talking about the strict mathematical definition after all, yes?

Please set me straight on this.

Here's an article that you (et al) might find useful:

"Finding a shortest vector in a two-dimensional lattice modulo m"

http://citeseer.nj.nec.com/rote97finding.html

Please, let me know what it means after you've read it. :)

What's the purpose of wanting to find the Minkowski-reduced

version of the PB instead of the actual one defined by

Schoenberg's ratios? How much of a difference is there?

-monz

_________________________________________________________

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In-Reply-To: <001301c1a297$fe1d9ec0$af48620c@dsl.att.net>

monz wrote:

> Here's an article that you (et al) might find useful:

>

> "Finding a shortest vector in a two-dimensional lattice modulo m"

> http://citeseer.nj.nec.com/rote97finding.html

>

> Please, let me know what it means after you've read it. :)

Web searches are, unfortunately, more likely to turn up research articles

than beginners' guides. That may be why my matrix tutorial is so popular.

I do now have a book that covers short vectors. They look very

important, but I still don't understand them (haven't even got to that

chapter).

> What's the purpose of wanting to find the Minkowski-reduced

> version of the PB instead of the actual one defined by

> Schoenberg's ratios? How much of a difference is there?

It means you get the simplest set that define the same temperament. Also

that you have a canonical set to compare with others, although we use

wedgies for that now. If you follow the link I gave before for my unison

vector CGI, you'll see its attempts at Minkowski reduction among the

results. They should all be correct, except for the ones that are wildly

incorrect. That's something I'm still working on. It's something to do

with short vectors.

Graham

> Message 2850

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

> Date: Sun Jan 20, 2002 10:48pm

> Subject: Re: lattices of Schoenberg's rational implications

>

>

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

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

> >

> > > However, I think the only reality for Schoenberg's system

> > > is a tuning where there is ambiguity, as defined by the

> > > kernel <33/32, 64/63, 81/80, 225/224>.

Ah ... so then, Paul, you agreed with me that this PB is

a valid one for p 1-184 of _Harmonielehre_?

> > > BTW, is this Minkowski-reduced?

>

> > Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

>

> Awesome. So this suggests a more compact Fokker parallelepiped

> as "Schoenberg PB" -- here are the results of placing it in different

> positions in the lattice (you should treat the inversions of these as

> implied):

>

>

> 0 1 1

> 84.467 21 20

> 203.91 9 8

> 315.64 6 5

> 386.31 5 4

> 470.78 21 16

> 617.49 10 7

> 701.96 3 2

> 786.42 63 40

> 933.13 12 7

> 968.83 7 4

> 1088.3 15 8

>

>

> 0 1 1

> 119.44 15 14

> 203.91 9 8

> 315.64 6 5

> 386.31 5 4

> 470.78 21 16

> 617.49 10 7

> 701.96 3 2

> 786.42 63 40

> 933.13 12 7

> 968.83 7 4

> 1088.3 15 8

>

>

> 0 1 1

> 119.44 15 14

> 155.14 35 32

> 301.85 25 21

> 386.31 5 4

> 470.78 21 16

> 617.49 10 7

> 701.96 3 2

> 772.63 25 16

> 884.36 5 3

> 968.83 7 4

> 1088.3 15 8

>

>

> 0 1 1

> 84.467 21 20

> 155.14 35 32

> 266.87 7 6

> 386.31 5 4

> 470.78 21 16

> 582.51 7 5

> 701.96 3 2

> 737.65 49 32

> 884.36 5 3

> 968.83 7 4

> 1053.3 147 80

With variant alternate pitches written on the same line

-- and thus with invariant ones on a line by themselves --

these scales are combined into:

1/1

21/20 15/14

35/32 9/8

7/6 25/21 6/5

5/4

21/16

7/5 10/7

3/2

49/32 25/16 63/40

5/3 12/7

7/4

147/80 15/8

My first question is: this is a 7-limit periodicity-block,

so can you explain how the two 11-limit unison-vectors disappeared?

I've been trying to figure it out but don't see it.

One thing I did notice in connection with this, is that

147/80 is only a little less than 4 cents wider than 11/6,

which is one of the pitches implied in Schoenberg's overtone

diagram (p 23 of _Harmonielehre_) :

vector ratio ~cents

[ -4 1 -1 2 0 ] = 147/80 1053.2931

- [ -1 -1 0 0 1 ] = 11/6 1049.362941

--------------------

[ -3 2 -1 2 -1 ] = 441/440 3.930158439

So I know that 441/440 is tempered out. But I don't see

how to get this as a combination of two of the other

unison-vectors.

Anyway, regarding the 7-limit PB itself:

I could see that all of those pairs and triplets of

alternate pitches are separated by either or both of

the two 7-limit unison-vectors. I made a lattice of

this combination of PBs:

Monzo lattice of 4 variant 12-tone 7-limit periodicity-blocks

calculated by Paul Erlich from Minkowski-reduced form of

my PB for p 1-184 of Schoenberg's _Harmonielehre_ :

/tuning-math/files/monz/mink-red.gif

Dotted lines connect the alternate pitches:

-- the long, somewhat horizontal dotted line represents

the 50:49 = [1 0 2 -2] between the pairs of notes:

21/20 : 15/14 ,

7/6 : 25/21 ,

7/5 : 10/7 ,

49/32 : 25/16 ,

147/80 : 15/8

-- and the short, nearly vertical dotted line represents

the 36:35 [-2 -2 1 1] between the pairs of notes:

35/32 : 9/8 ,

7/6 : 6/5 ,

49/32 : 63/40 ,

5/3 : 12/7 .

These intervals are portrayed graphically in my list of

the scale above.

I was startled by the unusual number of different symmetries

I saw on this lattice.

> Message 2861

> From: graham@m...

> Date: Mon Jan 21, 2002 0:44pm

> Subject: Re: Minkowski reduction (was: ...Schoenberg's rational

implications)

>

>

>

> In-Reply-To: <001301c1a297$fe1d9ec0$af48620c@d...>

> monz wrote:

>

> > What's the purpose of wanting to find the Minkowski-reduced

> > version of the PB instead of the actual one defined by

> > Schoenberg's ratios? How much of a difference is there?

>

> It means you get the simplest set that define the same temperament. Also

> that you have a canonical set to compare with others, although we use

> wedgies for that now. If you follow the link I gave before for my unison

> vector CGI, you'll see its attempts at Minkowski reduction among the

> results. They should all be correct, except for the ones that are wildly

> incorrect. That's something I'm still working on. It's something to do

> with short vectors.

Then, I reasoned that since all of these pitches are separated

by one or two of the unison vectors which define this set of PBs,

the lattice could be further reduced to a 12-tone set, one that

can still "define the same temperament":

Monzo lattice of Monzo's ultimate reduction of Paul Erlich's

4 variant Minkowski-reduced 7-limit PBs for p 1-184 of

Schoenberg's _Harmonielehre_, to one 12-tone PB :

/tuning-math/files/monz/ult-red.gif

Correct?

-monz

_________________________________________________________

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

> With variant alternate pitches written on the same line

> -- and thus with invariant ones on a line by themselves --

> these scales are combined into:

>

> 1/1

> 21/20 15/14

> 35/32 9/8

> 7/6 25/21 6/5

> 5/4

> 21/16

> 7/5 10/7

> 3/2

> 49/32 25/16 63/40

> 5/3 12/7

> 7/4

> 147/80 15/8

1--21/20--9/8--6/5--5/4--21/16--7/5--3/2--25/16--5/3--7/4--15/8

is one version of this, and has scale steps

(25/24)^2 * (21/20)^3 * (16/15)^4 * (15/14)^3 = 2

Looks awfully familiar...a slight permutation of the order of the steps, and we have

1--21/20--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--7/4--15/8

The very first scale I ever constructed.

> My first question is: this is a 7-limit periodicity-block,

> so can you explain how the two 11-limit unison-vectors disappeared?

The block wasn't that big, and perhaps was a little flat in the 11-direction.

> So I know that 441/440 is tempered out. But I don't see

> how to get this as a combination of two of the other

> unison-vectors.

Maybe it isn't. You should set up the linear equation and solve for it.

Hi Graham,

> From: <graham@microtonal.co.uk>

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

> Sent: Monday, January 21, 2002 9:44 AM

> Subject: [tuning-math] Re: Minkowski reduction (was: ...Schoenberg's

rational implications)

>

>

> > What's the purpose of wanting to find the Minkowski-reduced

> > version of the PB instead of the actual one defined by

> > Schoenberg's ratios? How much of a difference is there?

>

> It means you get the simplest set that define the same temperament. Also

> that you have a canonical set to compare with others, although we use

> wedgies for that now. If you follow the link I gave before for my unison

> vector CGI, you'll see its attempts at Minkowski reduction among the

> results. They should all be correct, except for the ones that are wildly

> incorrect. That's something I'm still working on. It's something to do

> with short vectors.

You're talking about

http://microtonal.co.uk/cgi-bin/uvsurvey.cgi

I put in the four unison-vectors which I had determined from p 1-184

of _Harmonielehre_, that is, <225/224, 81/80, 63/64, 33/32>, and got

these results:

>> [output from Graham's temperament calculator]

>>

>> unison vectors

>>

>> 225:224

>> 81:80

>> 64:63

>>

>> lower limit, got an ET (12, 19, 28, 34)

>>

>> ---------------

>>

>> unison vectors

>>

>> 225:224

>> 81:80

>> 33:32

>>

>> calculated

>> unison vectors

>>

>> 33:32

>> 55:54

>> 77:75

>>

>> 0/1, 1892.5 cent generator

>>

>> basis:

>> (1.0, 1.57708393667)

>>

>> mapping by period and generator:

>> [(1, 0), (0, 1), (-4, 4), (-13, 10), (5, -1)]

>>

>> mapping by steps:

>> [(1, 0), (-1, 1), (-8, 4), (-23, 10), (6, -1)]

>>

>> highest interval width: 11

>> complexity measure: 11 (12 for smallest MOS)

>> highest error: 0.036516 (43.819 cents)

>>

>> -------------

>>

>> unison vectors

>>

>> 225:224

>> 64:63

>> 33:32

>>

>> calculated

>> unison vectors

>>

>> 33:32

>> 64:63

>> 242:225

>>

>> 0/1, 1908.8 cent generator

>>

>> basis:

>> (0.5, 1.59064251985)

>>

>> mapping by period and generator:

>> [(2, 0), (0, 1), (11, -2), (12, -2), (10, -1)]

>>

>> mapping by steps:

>> [(2, 0), (-1, 1), (13, -2), (14, -2), (11, -1)]

>>

>> highest interval width: 4

>> complexity measure: 8 (10 for smallest MOS)

>> highest error: 0.061434 (73.721 cents)

>>

>> ------------

>>

>> unison vectors

>>

>> 81:80

>> 64:63

>> 33:32

>>

>> calculated

>> unison vectors

>>

>> 22:21

>> 33:32

>> 36:35

>>

>> 0/1, 1902.9 cent generator

>>

>> basis:

>> (1.0, 1.58576219547)

>>

>> mapping by period and generator:

>> [(1, 0), (0, 1), (-4, 4), (6, -2), (5, -1)]

>>

>> mapping by steps:

>> [(1, 0), (-1, 1), (-8, 4), (8, -2), (6, -1)]

>>

>> highest interval width: 6

>> complexity measure: 6 (7 for smallest MOS)

>> highest error: 0.066315 (79.577 cents)

So yes, I can see that the last one found both the 36/35

and the 22/21 which Gene found for the Minkowski-reduced

version, to replace two of the UVs I determined.

The second one replaces (225/224 and 81/80) with

(77/75 and 55/54), and the third one replaces 225/224

with 242/225.

So the only difference between your results and Gene's

is that your calculator seems to be fishing around for the

50/49 UV, but it got two of the others.

-monz

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

> I've always been careful to emphasize that our tuning-theory use

> of "lattice" is different from the mathematician's strictly define

> uses of the term.

This is not correct, Monz. There are several mathematical definitions

of "lattice" -- the one we use is most certainly one of these, as

we've discussed numerous times on the tuning list, and applied for

example in crystallographic theory.

> I've been searching the web to learn about

> Minkowksi reduction, and so now it appears to me that we are

> talking about the strict mathematical definition after all, yes?

We have been all along.

> What's the purpose of wanting to find the Minkowski-reduced

> version of the PB instead of the actual one defined by

> Schoenberg's ratios?

There's no purpose, as Schoenberg clearly meant for all the unison

vectors to be tempered out, and thus for 12-tET rather than JI to be

used. Tempering out the original set of unison vectors you posted is

exactly equivalent to tempering out the Minkowski-reduced set.

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

>

> > Message 2850

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

> > Date: Sun Jan 20, 2002 10:48pm

> > Subject: Re: lattices of Schoenberg's rational implications

> >

> >

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

wrote:

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

> > >

> > > > However, I think the only reality for Schoenberg's system

> > > > is a tuning where there is ambiguity, as defined by the

> > > > kernel <33/32, 64/63, 81/80, 225/224>.

>

>

> Ah ... so then, Paul, you agreed with me that this PB is

> a valid one for p 1-184 of _Harmonielehre_?

I don't know . . . I'll give you the benefit of the doubt.

>

>

>

> > > > BTW, is this Minkowski-reduced?

> >

> > > Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

> >

> > Awesome. So this suggests a more compact Fokker parallelepiped

> > as "Schoenberg PB" -- here are the results of placing it in

different

> > positions in the lattice (you should treat the inversions of

these as

> > implied):

> >

> >

> > 0 1 1

> > 84.467 21 20

> > 203.91 9 8

> > 315.64 6 5

> > 386.31 5 4

> > 470.78 21 16

> > 617.49 10 7

> > 701.96 3 2

> > 786.42 63 40

> > 933.13 12 7

> > 968.83 7 4

> > 1088.3 15 8

> >

> >

> > 0 1 1

> > 119.44 15 14

> > 203.91 9 8

> > 315.64 6 5

> > 386.31 5 4

> > 470.78 21 16

> > 617.49 10 7

> > 701.96 3 2

> > 786.42 63 40

> > 933.13 12 7

> > 968.83 7 4

> > 1088.3 15 8

> >

> >

> > 0 1 1

> > 119.44 15 14

> > 155.14 35 32

> > 301.85 25 21

> > 386.31 5 4

> > 470.78 21 16

> > 617.49 10 7

> > 701.96 3 2

> > 772.63 25 16

> > 884.36 5 3

> > 968.83 7 4

> > 1088.3 15 8

> >

> >

> > 0 1 1

> > 84.467 21 20

> > 155.14 35 32

> > 266.87 7 6

> > 386.31 5 4

> > 470.78 21 16

> > 582.51 7 5

> > 701.96 3 2

> > 737.65 49 32

> > 884.36 5 3

> > 968.83 7 4

> > 1053.3 147 80

>

>

>

> With variant alternate pitches written on the same line

> -- and thus with invariant ones on a line by themselves --

> these scales are combined into:

>

> 1/1

> 21/20 15/14

> 35/32 9/8

> 7/6 25/21 6/5

> 5/4

> 21/16

> 7/5 10/7

> 3/2

> 49/32 25/16 63/40

> 5/3 12/7

> 7/4

> 147/80 15/8

>

>

>

> My first question is: this is a 7-limit periodicity-block,

> so can you explain how the two 11-limit unison-vectors disappeared?

They didn't disappear! It's just that in these particular positions,

the parallelepiped all lies within one "power of 11" plane. I'm sure

Gene could produce an example that wouldn't.

> I've been trying to figure it out but don't see it.

>

> One thing I did notice in connection with this, is that

> 147/80 is only a little less than 4 cents wider than 11/6,

> which is one of the pitches implied in Schoenberg's overtone

> diagram (p 23 of _Harmonielehre_) :

>

> vector ratio ~cents

>

> [ -4 1 -1 2 0 ] = 147/80 1053.2931

> - [ -1 -1 0 0 1 ] = 11/6 1049.362941

> --------------------

> [ -3 2 -1 2 -1 ] = 441/440 3.930158439

>

>

> So I know that 441/440 is tempered out.

NO IT ISN'T! I believe it maps to 1 semitone given the set of unison

vectors you've put forward.

> But I don't see

> how to get this as a combination of two of the other

> unison-vectors.

YOU CAN'T!

> Then, I reasoned that since all of these pitches are separated

> by one or two of the unison vectors which define this set of PBs,

> the lattice could be further reduced to a 12-tone set, one that

> can still "define the same temperament":

>

> Monzo lattice of Monzo's ultimate reduction of Paul Erlich's

> 4 variant Minkowski-reduced 7-limit PBs for p 1-184 of

> Schoenberg's _Harmonielehre_, to one 12-tone PB :

>

> /tuning-math/files/monz/ult-red.gif

>

>

> Correct?

What's so "ultimate" about it?

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

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

> Sent: Tuesday, January 22, 2002 4:25 AM

> Subject: [tuning-math] Re: Minkowski reduction (was: ...Schoenberg's

rational implications)

>

>

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

>

> > I've always been careful to emphasize that our tuning-theory use

> > of "lattice" is different from the mathematician's strictly define

> > uses of the term.

>

> This is not correct, Monz. There are several mathematical definitions

> of "lattice" -- the one we use is most certainly one of these, as

> we've discussed numerous times on the tuning list, and applied for

> example in crystallographic theory.

Duh!, of course. My bad! I've mentioned the crystallography bit

myself in my webpage (even in the Dictionary).

> ...

>

> > What's the purpose of wanting to find the Minkowski-reduced

> > version of the PB instead of the actual one defined by

> > Schoenberg's ratios?

>

> There's no purpose, as Schoenberg clearly meant for all the unison

> vectors to be tempered out, and thus for 12-tET rather than JI to be

> used. Tempering out the original set of unison vectors you posted is

> exactly equivalent to tempering out the Minkowski-reduced set.

Thanks, Paul, this helps. Of course, since Schoenberg intended *all*

of the unison-vectors to be tempered out, it doesn't make a difference

which set you look at ... he meant to imply them *all* simultaneously.

-monz

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Get your free @yahoo.com address at http://mail.yahoo.com

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

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

> Sent: Tuesday, January 22, 2002 4:32 AM

> Subject: [tuning-math] Re: Minkowski reduction (was: ...Schoenberg's

rational implications)

>

>

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

> >

> > > Message 2850

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

> > > Date: Sun Jan 20, 2002 10:48pm

> > > Subject: Re: lattices of Schoenberg's rational implications

> > >

> > >

> > > > > [Paul]

> > > > > BTW, is this Minkowski-reduced?

> > >

> > > > [Gene]

> > > > Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

> > >

> > > [Paul]

> > > Awesome. So this suggests a more compact Fokker parallelepiped

> > > as "Schoenberg PB" -- here are the results of placing it in

> > > different positions in the lattice (you should treat the

> > > inversions of these as implied):

> > >

> > > <tables of scales snipped>

> >

> > [monz]

> > My first question is: this is a 7-limit periodicity-block,

> > so can you explain how the two 11-limit unison-vectors disappeared?

>

> [Paul]

> They didn't disappear! It's just that in these particular positions,

> the parallelepiped all lies within one "power of 11" plane. I'm sure

> Gene could produce an example that wouldn't.

OK ... I understood that intuitively ... just couldn't wrap my brain

around the math.

> > [monz]

> > With variant alternate pitches written on the same line

> > -- and thus with invariant ones on a line by themselves --

> > these scales are combined into:

> >

> > 1/1

> > 21/20 15/14

> > 35/32 9/8

> > 7/6 25/21 6/5

> > 5/4

> > 21/16

> > 7/5 10/7

> > 3/2

> > 49/32 25/16 63/40

> > 5/3 12/7

> > 7/4

> > 147/80 15/8

> >

> > ...

> >

> > One thing I did notice in connection with this, is that

> > 147/80 is only a little less than 4 cents wider than 11/6,

> > which is one of the pitches implied in Schoenberg's overtone

> > diagram (p 23 of _Harmonielehre_) :

> >

> > vector ratio ~cents

> >

> > [ -4 1 -1 2 0 ] = 147/80 1053.2931

> > - [ -1 -1 0 0 1 ] = 11/6 1049.362941

> > --------------------

> > [ -3 2 -1 2 -1 ] = 441/440 3.930158439

> >

> >

> > So I know that 441/440 is tempered out.

>

> NO IT ISN'T! I believe it maps to 1 semitone given the set of unison

> vectors you've put forward.

>

> > But I don't see

> > how to get this as a combination of two of the other

> > unison-vectors.

>

> YOU CAN'T!

Oops... my bad. Thanks, Paul. I see it now. If "C" is Schoenberg's

1/1, the 147/80 is mapped to "B" but 11/6 is mapped to "Bb".

This is precisely the note which was misprinted in the diagram in

the English edition ... guess I accepted it for so long that I

got confused.

>

> > Then, I reasoned that since all of these pitches are separated

> > by one or two of the unison vectors which define this set of PBs,

> > the lattice could be further reduced to a 12-tone set, one that

> > can still "define the same temperament":

> >

> > Monzo lattice of Monzo's ultimate reduction of Paul Erlich's

> > 4 variant Minkowski-reduced 7-limit PBs for p 1-184 of

> > Schoenberg's _Harmonielehre_, to one 12-tone PB :

> >

> > /tuning-math/files/monz/ult-red.gif

> >

> >

> > Correct?

>

> What's so "ultimate" about it?

Simply that now it actually finally is reduced to one 12-one set.

The point of my question is: even tho the shape of this particular

PB is different from the 4 identical ones from which it was coalesced,

all the same UVs are in effect, so it's still identical to those 4, yes?

-monz

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

> The point of my question is: even tho the shape of this particular

> PB is different from the 4 identical ones from which it was

coalesced,

> all the same UVs are in effect, so it's still identical to those 4,

yes?

Right . . .

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

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

> > I've always been careful to emphasize that our tuning-theory use

> > of "lattice" is different from the mathematician's strictly define

> > uses of the term.

> This is not correct, Monz. There are several mathematical definitions

> of "lattice" -- the one we use is most certainly one of these, as

> we've discussed numerous times on the tuning list, and applied for

> example in crystallographic theory.

When I use lattice on these lists it always means a discrete subgroup of R^n.