Hi Paul and Gene,

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

> To: <tuning@yahoogroups.com>

> Sent: Thursday, January 10, 2002 10:10 AM

> Subject: [tuning] Re: badly tuned remote overtones

>

>

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

>

> > The periodicity-blocks that Gene made from my numerical analysis

> > of Schoenberg's 1911 and 1927 theories are a good start.

>

> Well, given that most of the periodicity blocks imply not 12-tone,

> but rather 7-, 5-, and 2-tone scales, it strikes me that Schoenberg's

> attempted justification for 12-tET, at least as intepreted by you,

> generally fails. No?

I originally said:

> From: monz <joemonz@yahoo.com>

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

> Sent: Tuesday, December 25, 2001 3:44 PM

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

>

>

> Unison-vector matrix:

>

> 1911 _Harmonielehre_ 11-limit system

>

> ( 1 0 0 1 ) = 33:32

> (-2 0 -1 0 ) = 64:63

> ( 4 -1 0 0 ) = 81:80

> ( 2 1 0 -1 ) = 45:44

>

> Determinant = 7

>

> ... <snip> ...

>

> But why do I get a determinant of 7 for the 11-limit system?

> Schoenberg includes Bb and Eb as 7th harmonics in his description,

> which gives a set of 9 distinct pitches. But even when

> I include the 15:14 unison-vector, I still get a determinant

> of -7. And if I use 16:15 instead, then the determinant

> is only 5.

But Paul, you yourself said:

> From: Paul Erlich <paul@stretch-music.com>

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

> Sent: Thursday, July 19, 2001 12:43 PM

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

>

>

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

> >

> > Could anyone out there do some periodicity-block

> > calculations on this theory and say something about that?

>

> It's pretty clear that Schoenberg's theory implies a 12-tone

> periodicity block.

That was quite a while ago ... have you changed your position

on that? I thought that Gene showed clearly that a 12-tone

periodicity-block could be constructed out of Schoenberg's

unison-vectors.

> From: genewardsmith <genewardsmith@juno.com>

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

> Sent: Wednesday, December 26, 2001 12:27 AM

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

>

>

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

>

> > Can someone explain what's going on here, and what candidates

> > may be found for unison-vectors by extending the 11-limit system,

> > in order to define a 12-tone periodicity-block? Thanks.

>

> See if this helps;

>

> We can extend the set {33/32,64/63,81/80,45/44} to an

> 11-limit notation in various ways, for instance

>

> <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5]

>

> where g7 differs from h7 by g7(7)=19.

Gene, how did you come up with 56/55 as a unison-vector?

Why did I get 5 and 7 as matrix determinants for the

scale described by Schoenberg, but you were able to

come up with 12?

> Using this, we find the corresponding block is

>

> (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12)

> (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; the

> Pythagorean scale. We don't need anything new to find a

> 12-note scale; we get

>

> 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8

>

> or variants, the variants coming from the fact that 12

> is even, by using 12 rather than 7 in the denominator.

Can you explain this business about variants in a little

more detail? I understand the general concept, having seen

it in periodicity-blocks I've constructed on my spreadsheet,

but I'd like your take on the particulars for this case.

-monz

_________________________________________________________

Do You Yahoo!?

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

First, I'd like to start this post off with a link to my

"rough draft" of a lattice of the periodicity-block Gene

calculated for Schoenberg's theory:

http://www.ixpres.com/interval/monzo/schoenberg/harm/Genes-pblock.gif

This shows the 12-tone periodicity-block (primarily 3- and 5-limit,

with one 11-limit pitch), and its equivalent p-block cousins at

+/- each of the four unison-vectors.

Now to respond to Paul...

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

> To: <tuning@yahoogroups.com>

> Sent: Friday, January 11, 2002 12:47 PM

> Subject: [tuning] Re: badly tuned remote overtones

>

>

> You seem to be brushing some of the unison vectors you had

> previously reported, and from which Gene derived 7-, 5-, and 2-tone

> periodicity blocks, under the rug.

Ah ... so then this, from Gene:

> From: genewardsmith <genewardsmith@juno.com>

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

> Sent: Wednesday, December 26, 2001 3:25 PM

> Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices

>

> ... This matrix is unimodular, meaning it has determinant +-1.

> If I invert it, I get

>

> [ 7 12 7 -2 5]

> [11 19 11 -3 8]

> [16 28 16 -5 12]

> [20 34 19 -6 14]

> [24 42 24 -7 17]

>

actually *does* specify "7-, 5-, and 2-tone periodicity blocks".

Yes?

> Face it, Monz -- without some careful "fudging", Schoenberg's

> derviation of 12-tET as a scale for 13-limit harmony is not

> the rigorous, unimpeachable bastion of good reasoning that

> you'd like to present it as.

Your point is taken, but please try to understand my objectives

more clearly. I agree with you that "Schoenberg's derviation ...

is not the rigorous, unimpeachable bastion of good reasoning" etc.

I'm simply trying to get a foothold on what was in his mind when

he came up with his radical new ideas for using 12-tET to represent

higher-limit chord identities.

I've seen it written (can't remember where right now) that without

the close personal attachment to Schoenberg that his students had,

it's nearly impossible to understand all the subtleties of his

teaching. I'm just trying to dig into that scenario a bit, and

in a sense to "get closer" to Schoenberg and his mind.

> The contradictions in Schoenberg's arguments were known at least

> as early as Partch's Genesis, and he isn't going to weasel his way

> out of them now :) If 12-tET can do what you and Schoenberg are

> trying to say it can, it can do _anything_, and there would be

> no reason ever to adopt any other tuning system.

Ahh ... well, I think you've put on finger on the crux of the matter.

Schoenberg consciously rejected microtonality and also made a

conscious decision to use the 12-tET tuning as tho it *could* do

"_anything_".

As I've documented again and again, he *did* have a favorable attitude

towards adopting other tuning systems, but was of the opinion that

only in the future would the time be right for that. With us now

living *in* that future, it seems to me that perhaps he was right

after all. Perhaps it's even possible that Schoenberg's actions

in adopting the "new version" of 12-tET ("atonality") helped to

precipitate the current trend towards microtonality and alternative

tunings. ...?

Always curious about these things,

-monz

_________________________________________________________

Do You Yahoo!?

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

I don't recall ever getting a response to this.

Still interested ...

> From: monz <joemonz@yahoo.com>

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

> Sent: Friday, January 11, 2002 2:04 PM

> Subject: [tuning-math] Re: [tuning] Re: badly tuned remote overtones

>

>

>

> First, I'd like to start this post off with a link to my

> "rough draft" of a lattice of the periodicity-block Gene

> calculated for Schoenberg's theory:

>

> http://www.ixpres.com/interval/monzo/schoenberg/harm/Genes-pblock.gif

>

> This shows the 12-tone periodicity-block (primarily 3- and 5-limit,

> with one 11-limit pitch), and its equivalent p-block cousins at

> +/- each of the four unison-vectors.

>

>

> Now to respond to Paul...

>

>

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

> > To: <tuning@yahoogroups.com>

> > Sent: Friday, January 11, 2002 12:47 PM

> > Subject: [tuning] Re: badly tuned remote overtones

> >

> >

> > You seem to be brushing some of the unison vectors you had

> > previously reported, and from which Gene derived 7-, 5-, and 2-tone

> > periodicity blocks, under the rug.

>

>

> Ah ... so then this, from Gene: ...

>

> > From: genewardsmith <genewardsmith@juno.com>

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

> > Sent: Wednesday, December 26, 2001 3:25 PM

> > Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices

> >

> > ... This matrix is unimodular, meaning it has determinant +-1.

> > If I invert it, I get

> >

> > [ 7 12 7 -2 5]

> > [11 19 11 -3 8]

> > [16 28 16 -5 12]

> > [20 34 19 -6 14]

> > [24 42 24 -7 17]

> >

>

> ... actually *does* specify "7-, 5-, and 2-tone periodicity blocks".

> Yes?

>

>

> > Face it, Monz -- without some careful "fudging", Schoenberg's

> > derviation of 12-tET as a scale for 13-limit harmony is not

> > the rigorous, unimpeachable bastion of good reasoning that

> > you'd like to present it as.

>

>

> Your point is taken, but please try to understand my objectives

> more clearly. I agree with you that "Schoenberg's derviation ...

> is not the rigorous, unimpeachable bastion of good reasoning" etc.

> I'm simply trying to get a foothold on what was in his mind when

> he came up with his radical new ideas for using 12-tET to represent

> higher-limit chord identities.

>

> I've seen it written (can't remember where right now) that without

> the close personal attachment to Schoenberg that his students had,

> it's nearly impossible to understand all the subtleties of his

> teaching. I'm just trying to dig into that scenario a bit, and

> in a sense to "get closer" to Schoenberg and his mind.

>

>

> > The contradictions in Schoenberg's arguments were known at least

> > as early as Partch's Genesis, and he isn't going to weasel his way

> > out of them now :) If 12-tET can do what you and Schoenberg are

> > trying to say it can, it can do _anything_, and there would be

> > no reason ever to adopt any other tuning system.

>

>

> Ahh ... well, I think you've put on finger on the crux of the matter.

>

> Schoenberg consciously rejected microtonality and also made a

> conscious decision to use the 12-tET tuning as tho it *could* do

> "_anything_".

>

> As I've documented again and again, he *did* have a favorable attitude

> towards adopting other tuning systems, but was of the opinion that

> only in the future would the time be right for that. With us now

> living *in* that future, it seems to me that perhaps he was right

> after all. Perhaps it's even possible that Schoenberg's actions

> in adopting the "new version" of 12-tET ("atonality") helped to

> precipitate the current trend towards microtonality and alternative

> tunings. ...?

>

>

> Always curious about these things,

>

> -monz

_________________________________________________________

Do You Yahoo!?

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

I also never got replies on my questions here, and

am still waiting. I'm particularly curious about

how 56/55 was added as a unison-vector. Thanks.

> From: monz <joemonz@yahoo.com>

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

> Sent: Friday, January 11, 2002 1:13 AM

> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

> Hi Paul and Gene,

>

>

>

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

> > To: <tuning@yahoogroups.com>

> > Sent: Thursday, January 10, 2002 10:10 AM

> > Subject: [tuning] Re: badly tuned remote overtones

> >

> >

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

> >

> > > The periodicity-blocks that Gene made from my numerical analysis

> > > of Schoenberg's 1911 and 1927 theories are a good start.

> >

> > Well, given that most of the periodicity blocks imply not 12-tone,

> > but rather 7-, 5-, and 2-tone scales, it strikes me that Schoenberg's

> > attempted justification for 12-tET, at least as intepreted by you,

> > generally fails. No?

>

>

>

> I originally said:

>

>

> > From: monz <joemonz@yahoo.com>

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

> > Sent: Tuesday, December 25, 2001 3:44 PM

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

> >

> >

> > Unison-vector matrix:

> >

> > 1911 _Harmonielehre_ 11-limit system

> >

> > ( 1 0 0 1 ) = 33:32

> > (-2 0 -1 0 ) = 64:63

> > ( 4 -1 0 0 ) = 81:80

> > ( 2 1 0 -1 ) = 45:44

> >

> > Determinant = 7

> >

> > ... <snip> ...

> >

> > But why do I get a determinant of 7 for the 11-limit system?

> > Schoenberg includes Bb and Eb as 7th harmonics in his description,

> > which gives a set of 9 distinct pitches. But even when

> > I include the 15:14 unison-vector, I still get a determinant

> > of -7. And if I use 16:15 instead, then the determinant

> > is only 5.

>

>

>

> But Paul, you yourself said:

>

>

> > From: Paul Erlich <paul@stretch-music.com>

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

> > Sent: Thursday, July 19, 2001 12:43 PM

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

implications

> >

> >

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

> > >

> > > Could anyone out there do some periodicity-block

> > > calculations on this theory and say something about that?

> >

> > It's pretty clear that Schoenberg's theory implies a 12-tone

> > periodicity block.

>

>

> That was quite a while ago ... have you changed your position

> on that? I thought that Gene showed clearly that a 12-tone

> periodicity-block could be constructed out of Schoenberg's

> unison-vectors.

>

>

>

> > From: genewardsmith <genewardsmith@juno.com>

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

> > Sent: Wednesday, December 26, 2001 12:27 AM

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

implications

> >

> >

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

> >

> > > Can someone explain what's going on here, and what candidates

> > > may be found for unison-vectors by extending the 11-limit system,

> > > in order to define a 12-tone periodicity-block? Thanks.

> >

> > See if this helps;

> >

> > We can extend the set {33/32,64/63,81/80,45/44} to an

> > 11-limit notation in various ways, for instance

> >

> > <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5]

> >

> > where g7 differs from h7 by g7(7)=19.

>

>

> Gene, how did you come up with 56/55 as a unison-vector?

> Why did I get 5 and 7 as matrix determinants for the

> scale described by Schoenberg, but you were able to

> come up with 12?

>

>

> > Using this, we find the corresponding block is

> >

> > (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12)

> > (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; the

> > Pythagorean scale. We don't need anything new to find a

> > 12-note scale; we get

> >

> > 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8

> >

> > or variants, the variants coming from the fact that 12

> > is even, by using 12 rather than 7 in the denominator.

>

>

> Can you explain this business about variants in a little

> more detail? I understand the general concept, having seen

> it in periodicity-blocks I've constructed on my spreadsheet,

> but I'd like your take on the particulars for this case.

>

>

>

> -monz

Do You Yahoo!?

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

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

> I also never got replies on my questions here, and

> am still waiting. I'm particularly curious about

> how 56/55 was added as a unison-vector. Thanks.

I added 56/55 to get what I call a notation, and to give me a scale step vector. It was therefore a computational aid which as a byproduct displayed some systems related to Schoenberg's.

> From: genewardsmith <genewardsmith@juno.com>

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

> Sent: Tuesday, January 15, 2002 12:54 PM

> Subject: [tuning-math] Re: badly tuned remote overtones

>

>

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

>

> > I also never got replies on my questions here, and

> > am still waiting. I'm particularly curious about

> > how 56/55 was added as a unison-vector. Thanks.

>

> I added 56/55 to get what I call a notation, and to give

> me a scale step vector. It was therefore a computational

> aid which as a byproduct displayed some systems related

> to Schoenberg's.

OK, right ... I understand all that.

But where did it come from? Is it the result of adding

or subtracting two of the already-existing unison-vectors?

That wouldn't work, would it? ... because all the vectors

in the matrix have to be independent.

-monz

Do You Yahoo!?

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

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

>

> I don't recall ever getting a response to this.

> Still interested ...

>

>

> > From: monz <joemonz@y...>

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

> > Sent: Friday, January 11, 2002 2:04 PM

> > Subject: [tuning-math] Re: [tuning] Re: badly tuned remote

overtones

> >

> >

> >

> > First, I'd like to start this post off with a link to my

> > "rough draft" of a lattice of the periodicity-block Gene

> > calculated for Schoenberg's theory:

> >

> > http://www.ixpres.com/interval/monzo/schoenberg/harm/Genes-

pblock.gif

> >

> > This shows the 12-tone periodicity-block (primarily 3- and 5-

limit,

> > with one 11-limit pitch), and its equivalent p-block cousins at

> > +/- each of the four unison-vectors.

> >

> >

> > Now to respond to Paul...

> >

> >

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

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

> > > Sent: Friday, January 11, 2002 12:47 PM

> > > Subject: [tuning] Re: badly tuned remote overtones

> > >

> > >

> > > You seem to be brushing some of the unison vectors you had

> > > previously reported, and from which Gene derived 7-, 5-, and 2-

tone

> > > periodicity blocks, under the rug.

> >

> >

> > Ah ... so then this, from Gene: ...

> >

> > > From: genewardsmith <genewardsmith@j...>

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

> > > Sent: Wednesday, December 26, 2001 3:25 PM

> > > Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices

> > >

> > > ... This matrix is unimodular, meaning it has determinant +-1.

> > > If I invert it, I get

> > >

> > > [ 7 12 7 -2 5]

> > > [11 19 11 -3 8]

> > > [16 28 16 -5 12]

> > > [20 34 19 -6 14]

> > > [24 42 24 -7 17]

> > >

> >

> > ... actually *does* specify "7-, 5-, and 2-tone periodicity

blocks".

> > Yes?

I thought he specified those in another post.

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

> I also never got replies on my questions here, and

> am still waiting. I'm particularly curious about

> how 56/55 was added as a unison-vector. Thanks.

>

>

> > From: monz <joemonz@y...>

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

> > Sent: Friday, January 11, 2002 1:13 AM

> > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

> >

> >

> > Hi Paul and Gene,

> >

> >

> >

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

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

> > > Sent: Thursday, January 10, 2002 10:10 AM

> > > Subject: [tuning] Re: badly tuned remote overtones

> > >

> > >

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

> > >

> > > > The periodicity-blocks that Gene made from my numerical

analysis

> > > > of Schoenberg's 1911 and 1927 theories are a good start.

> > >

> > > Well, given that most of the periodicity blocks imply not 12-

tone,

> > > but rather 7-, 5-, and 2-tone scales, it strikes me that

Schoenberg's

> > > attempted justification for 12-tET, at least as intepreted by

you,

> > > generally fails. No?

> >

> >

> >

> > I originally said:

> >

> >

> > > From: monz <joemonz@y...>

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

> > > Sent: Tuesday, December 25, 2001 3:44 PM

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

implications

> > >

> > >

> > > Unison-vector matrix:

> > >

> > > 1911 _Harmonielehre_ 11-limit system

> > >

> > > ( 1 0 0 1 ) = 33:32

> > > (-2 0 -1 0 ) = 64:63

> > > ( 4 -1 0 0 ) = 81:80

> > > ( 2 1 0 -1 ) = 45:44

> > >

> > > Determinant = 7

Well there you go. This shows that Schoenberg's UVs imply a 7-tone

system, not a 12-tone system as you claim.

> > > ... <snip> ...

> > >

> > > But why do I get a determinant of 7 for the 11-limit system?

> > > Schoenberg includes Bb and Eb as 7th harmonics in his

description,

> > > which gives a set of 9 distinct pitches. But even when

> > > I include the 15:14 unison-vector,

In place of which one above?

> > > I still get a determinant

> > > of -7. And if I use 16:15 instead, then the determinant

> > > is only 5.

15:14 and 16:15 are both clearly semitones. Why would you use them as

UVs?

> >

> >

> > > From: genewardsmith <genewardsmith@j...>

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

> > > Sent: Wednesday, December 26, 2001 12:27 AM

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

> implications

> > >

> > >

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

> > >

> > > > Can someone explain what's going on here, and what candidates

> > > > may be found for unison-vectors by extending the 11-limit

system,

> > > > in order to define a 12-tone periodicity-block? Thanks.

> > >

> > > See if this helps;

> > >

> > > We can extend the set {33/32,64/63,81/80,45/44} to an

> > > 11-limit notation in various ways, for instance

> > >

> > > <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5]

> > >

> > > where g7 differs from h7 by g7(7)=19.

> >

> >

> > Gene, how did you come up with 56/55 as a unison-vector?

Because it works to get you 12 as one of the five resulting

cardinalities.

> > Why did I get 5 and 7 as matrix determinants for the

> > scale described by Schoenberg, but you were able to

> > come up with 12?

Only by replacing 33/32 with 56/55.

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

> OK, right ... I understand all that.

>

> But where did it come from? Is it the result of adding

> or subtracting two of the already-existing unison-vectors?

> That wouldn't work, would it? ... because all the vectors

> in the matrix have to be independent.

Any vector that is _not_ the result of adding or subtracting two of

the already-existing unison-vectors would work to create what Gene

calls a "notation" (but is nothing like a musical notation anyone's

ever seen before). In this case, he made a choice (56/55) that makes

the 12-tone system come out nice in this "notation".

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

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

> Sent: Tuesday, January 15, 2002 1:47 PM

> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

> > From: monz <joemonz@y...>

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

> > Sent: Tuesday, December 25, 2001 3:44 PM

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

> >

> >

> > Unison-vector matrix:

> >

> > 1911 _Harmonielehre_ 11-limit system

> >

> > ( 1 0 0 1 ) = 33:32

> > (-2 0 -1 0 ) = 64:63

> > ( 4 -1 0 0 ) = 81:80

> > ( 2 1 0 -1 ) = 45:44

> >

> > Determinant = 7

>

> Well there you go. This shows that Schoenberg's UVs

> imply a 7-tone system, not a 12-tone system as you claim.

Ack! ... Paul, I'm not really *claiming* that Schoenberg's

system implies 12-tone -- I'm just trying to find out more

accurately what Schoenberg had in mind. He himself seems to

have thought that 12-EDO could give a decent representation

of 11-limit harmony. I'm simply trying to reconstruct his

thought-process.

I admit that I've made statements like "it looks to me like

Schoenberg's explanation in _Harmonielehre_ definitely implies

a 12-tone periodicity-block" ... but that was probably premature.

There's still a lot here left for me to understand.

> > ... <snip> ...

> >

> > But why do I get a determinant of 7 for the 11-limit

> > system? Schoenberg includes Bb and Eb as 7th harmonics

> > in his description, which gives a set of 9 distinct pitches.

> > But even when I include the 15:14 unison-vector,

>

> In place of which one above?

In place of the 45:44, which is the UV with which I have

difficulty, because of the inconsistency in Schoenberg's

notation: F = 11/C and C = 11/G, but B (not Bb) = 11/F.

Here are the matrices I used to make that statement:

3 5 7 11 unison vectors ~cents

[ 1 1 -1 0 ] = 15:14 119.4428083

[ 1 0 0 1 ] = 33:32 53.27294323

[-2 0 -1 0 ] = 64:63 27.2640918

[ 4 -1 0 0 ] = 81:80 21.5062896

det = | -7 |

3 5 7 11 unison vectors ~cents

[-1 -1 0 0 ] = 16:15 111.7312853

[ 1 0 0 1 ] = 33:32 53.27294323

[-2 0 -1 0 ] = 64:63 27.2640918

[ 4 -1 0 0 ] = 81:80 21.5062896

det = | 5 |

Leaving in Schoenberg's notational inconsistency means

that we must employ 45:44 in the matrix as well:

3 5 7 11 unison vectors ~cents

[ 2 1 0 -1 ] = 45:44 38.90577323

[ 1 0 0 1 ] = 33:32 53.27294323

[-2 0 -1 0 ] = 64:63 27.2640918

[ 4 -1 0 0 ] = 81:80 21.5062896

And again here, det = | -7 | .

> > I still get a determinant of -7. And if I use 16:15

> > instead, then the determinant is only 5.

>

> 15:14 and 16:15 are both clearly semitones. Why would

> you use them as UVs?

Well, to give you the same kind of answer which you and Gene

both gave to me and which didn't help me at all ... because

it works to give me 7 or 5 as a cardinality!

Seriously, the reason I chose 15:14 is because I could see

that by equating the 11th harmonic sometimes with the

"perfect 4th" and sometimes with the "augmented 4th",

Schoenberg was implying the tempering-out of a semitone.

When I saw that the determinant of the matrix directly above

(with unison-vectors 33:32, 45:44, 64:63, and 81:80) was -7,

it put up a red flag in my mind. It seemed obvious to me

that Schoenberg's equivalences implied only a 7-tone PB,

i.e., a diatonic scale and not a chromatic one.

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

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

> Sent: Tuesday, January 15, 2002 1:53 PM

> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

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

>

> > OK, right ... I understand all that.

> >

> > But where did it come from? Is it the result of adding

> > or subtracting two of the already-existing unison-vectors?

> > That wouldn't work, would it? ... because all the vectors

> > in the matrix have to be independent.

>

> Any vector that is _not_ the result of adding or subtracting two of

> the already-existing unison-vectors would work to create what Gene

> calls a "notation" (but is nothing like a musical notation anyone's

> ever seen before). In this case, he made a choice (56/55) that makes

> the 12-tone system come out nice in this "notation".

OK, fair enough. But what does it have to do with Schoenberg's

actual theory? I'm having a hard time seeing how 56/55 is implied

by anything Schoenberg actually said.

Just to satisfy my curiosity, I've made a table of the

sum and difference of all pairs of unison-vectors in my

original assessment of Schoenberg's _Harmonielehre_ system,

i.e. the matrix

[ 2 1 0 -1 ] = 45:44

[ 1 0 0 1 ] = 33:32

[-2 0 -1 0 ] = 64:63

[ 4 -1 0 0 ] = 81:80 .

Here are the results:

45/44 and 33/32 :

[-2 2 1 0 -1 ] = 45/44 38.90577323

- [-5 1 0 0 1 ] = 33/32 53.27294323

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

[ 3 1 1 0 -2 ] = 120/121 -14.36717

[-2 2 1 0 -1 ] = 45/44 38.90577323

+ [-5 1 0 0 1 ] = 33/32 53.27294323

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

[-7 3 1 0 0 ] = 135/128 92.17871646

45/44 and 64/63 :

[-2 2 1 0 -1 ] = 45/44 38.90577323

- [ 6 -2 0 -1 0 ] = 64/63 27.2640918

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

[-8 4 1 1 -1 ] = 2835/2816 11.64168143

[-2 2 1 0 -1 ] = 45/44 38.90577323

+ [ 6 -2 0 -1 0 ] = 64/63 27.2640918

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

[ 4 0 1 -1 -1 ] = 80/77 66.16986503

45/44 and 81/80 :

[-2 2 1 0 -1 ] = 45/44 38.90577323

- [-4 4 -1 0 0 ] = 81/80 21.5062896

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

[ 2 -2 2 0 -1 ] = 100/99 17.39948363

[-2 2 1 0 -1 ] = 45/44 38.90577323

+ [-4 4 -1 0 0 ] = 81/80 21.5062896

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

[-6 6 0 0 -1 ] = 729/704 60.41206283

33/32 and 64/63 :

[-5 1 0 0 1 ] = 33/32 53.27294323

- [ 6 -2 0 -1 0 ] = 64/63 27.2640918

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

[-11 3 0 1 1 ] = 2079/2048 26.00885143

[-5 1 0 0 1 ] = 33/32 53.27294323

+ [ 6 -2 0 -1 0 ] = 64/63 27.2640918

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

[ 1 -1 0 -1 1 ] = 22/21 80.53703503

33/32 and 81/80 :

[-5 1 0 0 1 ] = 33/32 53.27294323

- [-4 4 -1 0 0 ] = 81/80 21.5062896

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

[-1 -3 1 0 1 ] = 55/54 31.76665363

[-5 1 0 0 1 ] = 33/32 53.27294323

+ [-4 4 -1 0 0 ] = 81/80 21.5062896

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

[-9 5 -1 0 1 ] = 2673/2560 74.77923283

64/63 and 81/80 :

[ 6 -2 0 -1 0 ] = 64/63 27.2640918

- [-4 4 -1 0 0 ] = 81/80 21.5062896

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

[10 -6 1 -1 0 ] = 5120/5103 5.757802203

[ 6 -2 0 -1 0 ] = 64/63 27.2640918

+ [-4 4 -1 0 0 ] = 81/80 21.5062896

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

[ 2 2 -1 -1 0 ] = 36/35 48.7703814

Does anyone have comments on this?

-monz

Do You Yahoo!?

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

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

> > Any vector that is _not_ the result of adding or subtracting two

of

> > the already-existing unison-vectors would work to create what

Gene

> > calls a "notation" (but is nothing like a musical notation

anyone's

> > ever seen before). In this case, he made a choice (56/55) that

makes

> > the 12-tone system come out nice in this "notation".

>

>

> OK, fair enough. But what does it have to do with Schoenberg's

> actual theory? I'm having a hard time seeing how 56/55 is implied

> by anything Schoenberg actually said.

It isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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

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

> Sent: Tuesday, January 15, 2002 3:00 PM

> Subject: [tuning-math] Re: badly tuned remote overtones

>

>

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

>

> > > Any vector that is _not_ the result of adding or

> > > subtracting two of the already-existing unison-vectors

> > > would work to create what Gene calls a "notation"

> > > (but is nothing like a musical notation anyone's

> > > ever seen before). In this case, he made a choice

> > > (56/55) that makes the 12-tone system come out nice

> > > in this "notation".

> >

> >

> > OK, fair enough. But what does it have to do with

> > Schoenberg's actual theory? I'm having a hard time

> > seeing how 56/55 is implied by anything Schoenberg

> > actually said.

>

> It isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Ah ... I just tried something else, and now there's a tiny

flicker of light beaming thru the fog ...

So in other words, Gene simply found a unison-vector that made

it easy for him to construct a 12-tone PB using some of the

unison-vectors supplied by Schoenberg, yes?

Well, the other day I did find my own 12-tone PB for Schoenberg,

but making use of that again now (with a new twist), I found

something very interesting. I wrote:

/tuning-math/message/2577

> Message 2577

> From: "monz" <joemonz@y...>

> Date: Fri Jan 11, 2002 4:59 am

> Subject: Re: badly tuned remote overtones

>

>

> ... I had failed to take into consideration the 5-limit

> enharmonicity required by Schoenberg. To construct a

> periodicity-block according to his descriptions, one would

> have to temper out one of the "enharmonic equivalents".

>

> We may choose 2048:2025 =

>

> [2]

> [3] * [11 -4 -2]

> [5]

>

> <etc.>

I think one would *have* to include a 5-limit "enharmonic

unison-vector" here, since Schoenberg explicitly equated A#=Bb,

C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent

stuff.

Well, look what happens if one includes 2048:2025 along with

*all* the unison-vectors explicitly stated by Schoenberg, in

other words, the same matrix Gene used except that it has

2048:2025 instead of 56:55 :

matrix

2 3 5 7 11 unison vectors ~cents

[ -2 2 1 0 -1 ] = 45:44 38.90577323

[ 11 -4 -2 0 0 ] = 2048:2025 19.55256881

[ -5 1 0 0 1 ] = 33:32 53.27294323

[ 6 -2 0 -1 0 ] = 64:63 27.2640918

[ -4 4 -1 0 0 ] = 81:80 21.5062896

inverse

[ 12 7 12 0 -2 ]

[ 19 11 19 0 -3 ]

[ 28 16 28 0 -5 ]

[ 34 20 34 -1 -6 ]

[ 41 24 42 0 -7 ]

determinant = | 1 |

So here, one can see that there are two possible

mappings to 12-EDO, and that the only difference between

them is the mapping of 11: h12(11)=41 but g12(11)=42.

This is *precisely* the inconsistency in Schoenberg's

1911 notation of 11 which I've mentioned many times!

So now, does that mean that *this* periodicity-block

is the prime candidate for the one Schoenberg probably

had in mind? Are there further difficulties, because of

the inconsistent mapping of 11?

Still curious,

-monz

Do You Yahoo!?

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

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

>

> I think one would *have* to include a 5-limit "enharmonic

> unison-vector" here, since Schoenberg explicitly equated A#=Bb,

> C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent

> stuff.

Did he do this explicitly within any of the 'constructions of unison

vectors' you gleaned from him?

And anyway, why not 128:125? Seems simpler . . .

> So now, does that mean that *this* periodicity-block

> is the prime candidate for the one Schoenberg probably

> had in mind?

What do you mean, *this* periodicity block? How many notes does it

contain? You only reported a determinany of 1, but that wasn't for a

PB, that was for a "notation" with one "extra" unison vector relative

to what would be needed for a PB.

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

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

> Sent: Tuesday, January 15, 2002 3:42 PM

> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

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

> >

> > I think one would *have* to include a 5-limit "enharmonic

> > unison-vector" here, since Schoenberg explicitly equated A#=Bb,

> > C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent

> > stuff.

>

> Did he do this explicitly within any of the 'constructions

> of unison vectors' you gleaned from him?

Well, not specifically *this* interval. But according to his

notational usage, *any* of the 5-limit enharmonicities should apply.

> And anyway, why not 128:125? Seems simpler . . .

OK, Paul, I tried 128:125 in place of 2048:2025, and the

inverse I get is:

[ 12 7 12 0 -9 ]

[ 19 11 19 0 -14 ]

[ 28 16 28 0 -21 ]

[ 34 20 34 -1 -26 ]

[ 41 24 42 0 -31 ]

So you're right ... this still shows the inconsistent

mapping to 11 in h12(11)=41, g12(11)=42. Naturally, since

I only replace one row of the UV-matrix, there's only

one column of the inverse that's different (see? ... I really

*am* learning this stuff!), and that's the last column.

So, I asked before ... what do these other columns mean?

What's -h9 showing us that's different from the -h2 in

my other matrix?

[ -2 ]

[ -3 ]

[ -5 ]

[ -6 ]

[ -7 ]

And what about that h0 column? What does that mean?

I also tried plugging the skhisma into the "5-limit

enharmonicity" row of the matrix (other than 81:80, that is).

This time, the last column of the inverse reads:

[ -5 ]

[ -8 ]

[ -11 ]

[ -14 ]

[ -17 ]

and all other columns are the same as the two I derived

before, but with the signs reversed. If I use the complement

of the skhisma, the other four rows are the same as before

and the last one is as above but all positive.

If I assume what is probably the most basic case, and

plug the Pythagorean comma into that row, again I get reversed

signs for the first four columns, and a last column of:

[ -12 ]

[ -19 ]

[ -27 ]

[ -34 ]

[ -41 ]

which is identical to the first column except for the

mapping of 5, so that (if we use "f") we may say that

-f12(-5)=-27 whereas -h12(-5) and -g12(-5) both = 28 .

So, now it seems that I've found the inconsistency in

Schoenberg's mapping of 5 as well.

>> Schoenberg 1911, _Harmonielehre_, p

>> (Carter English translation, p 24)

>>

>> The two tones _E_ and _B_ appear in the first octave

>> [of his illustration -- i.e., the second "8ve" of the

>> overtone series], but _E_ is challenged by _Eb_, B by _Bb_.

>> ... The second octave resolved the question in favor of

>> _E_ and _B_.

I'm not interested right now in arguing the logic of

Schoenberg's statement. He invokes the "decreasing audibility"

of the higher overtones in this very paragraph, and then

ignores that, in his estimation of which overtones may

be perceived more clearly than others.

But the point I'm making is that Schoenberg was fudging

the notation of both 5 and 11 in his description, and *this*

matrix seems to disclose all of that.

> > So now, does that mean that *this* periodicity-block

> > is the prime candidate for the one Schoenberg probably

> > had in mind?

>

> What do you mean, *this* periodicity block? How many notes does it

> contain? You only reported a determinany of 1, but that wasn't for a

> PB, that was for a "notation" with one "extra" unison vector relative

> to what would be needed for a PB.

Yikes! All too true. So then, what relevance does my construction

have, if any at all? It seems to me to show the mechanics of

Schoenberg's notational inconsistency.

Can you or someone else clear up this business about the difference

between PBs and "notations"? Is it that prime-factor 2 is left out

of PB calculations (assuming "8ve"-equivalency) but must be included

for "notation"?

Paul, I understand the criticisms you've written about what

I'm trying to do here, but I still think the effort is worthwhile.

Certainly, Schoenberg's "pantonal" [= atonal] style assumed

that any of the 12-EDO pitches could be used equally well as

the center of its own tonal universe. It would be informative

to see how each of those universes may be modeled conceptually,

and how they relate to each other.

-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: Tuesday, January 15, 2002 4:18 PM

> Subject: Re: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

> If I assume what is probably the most basic case, and

> plug the Pythagorean comma into that row, again I get reversed

> signs for the first four columns, and a last column of:

>

> [ -12 ]

> [ -19 ]

> [ -27 ]

> [ -34 ]

> [ -41 ]

>

> which is identical to the first column except for the

> mapping of 5, so that (if we use "f") we may say that

> -f12(-5)=-27 whereas -h12(-5) and -g12(-5) both = 28 .

Oops ... my bad. The last number is missing the minus sign.

So it's " -h12(-5) and -g12(-5) both = -28 ".

Gene, is there any reason to notate it like this?

Wouldn't it be equivalent to reverse all signs and say

" f12(5)=27 whereas h12(5) and g12(5) both = 28 "?

-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: Tuesday, January 15, 2002 3:42 PM

> > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

> >

> >

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

> > >

> > > I think one would *have* to include a 5-limit "enharmonic

> > > unison-vector" here, since Schoenberg explicitly equated A#=Bb,

> > > C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent

> > > stuff.

> >

> > Did he do this explicitly within any of the 'constructions

> > of unison vectors' you gleaned from him?

>

>

> Well, not specifically *this* interval. But according to his

> notational usage, *any* of the 5-limit enharmonicities should apply.

Right, but . . . did he apply any of them explicitly within any of

the 'constructions of unison vectors' you gleaned from him?

Otherwise, you're just "assuming the answer".

>

> > And anyway, why not 128:125? Seems simpler . . .

>

>

> OK, Paul, I tried 128:125 in place of 2048:2025, and the

> inverse I get is:

>

> [ 12 7 12 0 -9 ]

> [ 19 11 19 0 -14 ]

> [ 28 16 28 0 -21 ]

> [ 34 20 34 -1 -26 ]

> [ 41 24 42 0 -31 ]

>

>

> So you're right ... this still shows the inconsistent

> mapping to 11 in h12(11)=41, g12(11)=42. Naturally, since

> I only replace one row of the UV-matrix, there's only

> one column of the inverse that's different (see? ... I really

> *am* learning this stuff!), and that's the last column.

>

> So, I asked before ... what do these other columns mean?

> What's -h9 showing us that's different from the -h2 in

> my other matrix?

>

> [ -2 ]

> [ -3 ]

> [ -5 ]

> [ -6 ]

> [ -7 ]

>

> And what about that h0 column? What does that mean?

I'll leave these questions to Gene . . .

>

>

> I also tried plugging the skhisma into the "5-limit

> enharmonicity" row of the matrix (other than 81:80, that is).

> This time, the last column of the inverse reads:

>

> [ -5 ]

> [ -8 ]

> [ -11 ]

> [ -14 ]

> [ -17 ]

>

> and all other columns are the same as the two I derived

> before, but with the signs reversed. If I use the complement

> of the skhisma, the other four rows are the same as before

> and the last one is as above but all positive.

>

>

> If I assume what is probably the most basic case, and

> plug the Pythagorean comma into that row,

??? Why is that the most basic case?

> So, now it seems that I've found the inconsistency in

> Schoenberg's mapping of 5 as well.

Only if you assume the Pythagorean comma, right?

>

>

> > > So now, does that mean that *this* periodicity-block

> > > is the prime candidate for the one Schoenberg probably

> > > had in mind?

> >

> > What do you mean, *this* periodicity block? How many notes does

it

> > contain? You only reported a determinany of 1, but that wasn't

for a

> > PB, that was for a "notation" with one "extra" unison vector

relative

> > to what would be needed for a PB.

>

>

> Yikes! All too true. So then, what relevance does my construction

> have, if any at all? It seems to me to show the mechanics of

> Schoenberg's notational inconsistency.

Right.

> Can you or someone else clear up this business about the difference

> between PBs and "notations"? Is it that prime-factor 2 is left out

> of PB calculations (assuming "8ve"-equivalency) but must be included

> for "notation"?

You need to include the prime-factor 2 for PB calculations too, if

you're to weed out cases of torsion. As for "notation", I suggest you

ask Gene.

> Paul, I understand the criticisms you've written about what

> I'm trying to do here, but I still think the effort is worthwhile.

>

> Certainly, Schoenberg's "pantonal" [= atonal] style assumed

> that any of the 12-EDO pitches could be used equally well as

> the center of its own tonal universe. It would be informative

> to see how each of those universes may be modeled conceptually,

> and how they relate to each other.

Any of the PBs that give you a determinant of 12, if all the unison

vectors are tempered out, implies 12-tET. Geometrically, this will be

modeled by a torus or hyper-torus . . . can you make out the

inflatable torus model in the photocopy of the Hall article I sent

you (sorry the photocopy didn't come out so good -- check your

library for a better version)?

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

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

> Sent: Tuesday, January 15, 2002 5:09 PM

> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

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

> >

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

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

> > > Sent: Tuesday, January 15, 2002 3:42 PM

> > > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

> > >

> > >

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

> > > >

> > > > I think one would *have* to include a 5-limit "enharmonic

> > > > unison-vector" here, since Schoenberg explicitly equated A#=Bb,

> > > > C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent

> > > > stuff.

> > >

> > > Did he do this explicitly within any of the 'constructions

> > > of unison vectors' you gleaned from him?

> >

> >

> > Well, not specifically *this* interval. But according to his

> > notational usage, *any* of the 5-limit enharmonicities should apply.

>

> Right, but . . . did he apply any of them explicitly within any of

> the 'constructions of unison vectors' you gleaned from him?

> Otherwise, you're just "assuming the answer".

Well, on p 176 of _Harmonielehre_ (p 155 of the Carter translation),

Schoenberg illustrates the "Circle of 5ths", and explicitly notates

the equivalences Cb=B, Gb=F#, and Db=C# for the major keys, and

ab=g#, eb=g#, and bb=a# for the minor keys that are +5, +6,

and +7 "5ths" (respectively) from the origin C-major/a-minor.

This is in Chapter 9, "Modulation", and all thru his discussion

of modulation Schoenberg assumes the enharmonic equivalence of

keys like this. This is before he ever gets into anything about

the enharmonic equivalence of implied 5-limit harmonies.

He never explicitly says that a "5th" is always to be

interpreted as a 3:2, but his explanation of the basic tones

of the scale as overtones of the subdominant, dominant, and

tonic implies that he's thinking of "5ths" as either 3-limit

intervals or meantone generators, or both. And in the

"Table of the Circle of Fifths for C Major (a minor)", on

the following page, he lists the keys next to each other going

in opposite directions around the circle, so that one column

has only the "sharp" keys and the other only the "flat" keys.

To me, this implies even more strongly a conception based on

a generating "5th", whether it's Pythagorean or meantone.

And again, the enharmonic equivalences are explicitly stated.

If meantone is assumed instead of Pythagorean, then the

enharmonic equivalence illustrated in Schoenberg's

"Circle of 5ths" is the diesis 128:125 = [7 0 -3].

So I can see why you might consider *this* to be the

"missing link" unison-vector.

So, to answer your question directly: Schoenberg equates

pairs of enharmonically-equivalent pitches, but no, he

never explicitly mentions whether those pairs of pitches

are derived via Pythagorean or meantone tuning. So yes,

I'm assuming certain intervals as unison-vectors, based

as much as possible on the pitch-relationships explicitly

detailed by Schoenberg. But I would venture to say, based

on what he wrote in _Harmonielehre_, that he expected *both*

the Pythagorean comma *and* the diesis (and all of their

combinations) to be tempered out.

> >

> > > And anyway, why not 128:125? Seems simpler . . .

> >

> >

> > OK, Paul, I tried 128:125 in place of 2048:2025, and the

> > inverse I get is:

> >

> > [ 12 7 12 0 -9 ]

> > [ 19 11 19 0 -14 ]

> > [ 28 16 28 0 -21 ]

> > [ 34 20 34 -1 -26 ]

> > [ 41 24 42 0 -31 ]

> >

> >

> > So you're right ... this still shows the inconsistent

> > mapping to 11 in h12(11)=41, g12(11)=42. Naturally, since

> > I only replace one row of the UV-matrix, there's only

> > one column of the inverse that's different (see? ... I really

> > *am* learning this stuff!), and that's the last column.

> >

> > ...

> >

> > If I assume what is probably the most basic case, and

> > plug the Pythagorean comma into that row,

>

> ??? Why is that the most basic case?

This is my thinking: Schoenberg's theory certainly assumes

all the "traditional" enharmonic equivalence of the 12-EDO scale

-- by "traditional", I mean all of the enharmonic equivalences

that may arise in the 3- and 5-limits.

To my mind, the 3-limit (linear, 1-D) is both historically and

conceptually more basic than 5-limit (planar, 2-D). The

notational difference between a "sharp" and what later became

its enharmonically equivalent "flat", ocurred first in Pythagorean

tuning. And so, along this line of reasoning, the Pythagorean

comma is historically and conceptually a more basic enharmonicity

than any of the 5-limit examples. However, as implied above, I

will also grant the possibility that Schoenberg may have intended

the diesis as a unison-vector, and will examine that case below

as well.

Also, I understand Gene's "notation" a little better now.

So, taking this particular matrix as an example,

2 3 5 7 11 unison-vector ~cents

[ -2 2 1 0 -1 ] = 45:44 38.90577323

[-19 12 0 0 0 ] = 531441:524288 23.46001038

[ -5 1 0 0 1 ] = 33:32 53.27294323

[ 6 -2 0 -1 0 ] = 64:63 27.2640918

[ -4 4 -1 0 0 ] = 81:80 21.5062896

inverse

[ 12 -7 12 0 12 ]

[ 19 -11 19 0 19 ]

[ 28 -16 28 0 27 ]

[ 34 -20 34 -1 34 ]

[ 41 -24 42 0 41 ]

So, for an example of how the unison-vector maps to

a homomorphism, the matrix describing the mapping

of 45:44 to h12 is:

[ 12 ] [ -2 2 1 0 -1 ]

[ 19 ]

[ 28 ]

[ 34 ]

[ 41 ]

which translates into

(12*-2)+(19*2)+(28*1)+(34*0)+(41*-1)

= -24 + 38 + 28 + 0 + -41

= 1

So when I look at how all the unison-vectors map to

the homomorphisms, I get:

homomorphism

h12 -h7 h12 h0 h12 unison-vector

[ 1 0 0 0 0 ] 45:44

[ 0 1 0 0 0 ] 531441:524288

[ 0 0 1 0 0 ] 33:32

[ 0 0 0 1 0 ] 64:63

[ 0 0 0 0 1 ] 81:80

So now let me try to get this straight. This matrix is

telling us that one of three mappings to 12-EDO may be

chosen, in which we distinguish either 45:44, 33:32, or

81:80 as pairs of distinct notes. Correct?

Plugging 128:125 into the 2nd row instead of the Pythagorean

comma, a look at the mapping of unison-vectors gives us

exactly the same matrix as above. (But of course, this

time the last column of the inverse gives a 9-EDO rather

than 12-EDO mapping, so that there are two 12-EDO mappings

this time rather than three.)

So assuming 128:125 to be a unison-vector, we still may choose

between either of two 12-EDO mappings, in which we distinguish

either 45:44 or 33:32. With 128:125 as a unison-vector, along

with the others we use here, 12-EDO *always* tempers out the

syntonic comma 81:80. With the Pythagorean Comma as a UV instead,

12-EDO may or may not temper out the syntonic comma, depending

on which homomorphism is chosen.

>

> > So, now it seems that I've found the inconsistency in

> > Schoenberg's mapping of 5 as well.

>

> Only if you assume the Pythagorean comma, right?

Right -- that's the only example I've found so far

which changes the mapping of 5.

Gene's PB (using 56:55 as a UV) found an inconsistent

mapping to 7, and he and I have both found several which

map 11 inconsistently.

> You need to include the prime-factor 2 for PB calculations

> too, if you're to weed out cases of torsion.

Ahh! ... now *that's* a useful little tidbit!! Thanks!

> Any of the PBs that give you a determinant of 12, if all the unison

> vectors are tempered out, implies 12-tET. Geometrically, this will be

> modeled by a torus or hyper-torus . . . can you make out the

> inflatable torus model in the photocopy of the Hall article I sent

> you (sorry the photocopy didn't come out so good -- check your

> library for a better version)?

The picture in the Hall article is pretty hard to make out ...

but there's an identical diagram of a "Chicken-wire Torus"

in "Parsimonious Graphs: A Study in Parsimony, Contextual

Transformations, and Modes of Limited Transposition" by Jack

Douthett and Peter Steinbach, in _Journal of Music Theory_ 42:2

(Fall 1998), on p 248. So I understand how it works and can

see it to some degree on this diagram. Of course, the actual

*physical* model Hall used is preferable ... where can I get one?!

-monz

Do You Yahoo!?

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 15, 2002 5:09 PM

> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

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

> > > > I think one would *have* to include a 5-limit "enharmonic

> > > > unison-vector" here, since Schoenberg explicitly equated A#=Bb,

> > > > C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent

> > > > stuff.

> > >

> > > Did he do this explicitly within any of the 'constructions

> > > of unison vectors' you gleaned from him?

> >

> >

> > Well, not specifically *this* interval. But according to his

> > notational usage, *any* of the 5-limit enharmonicities should apply.

>

> Right, but . . . did he apply any of them explicitly within any of

> the 'constructions of unison vectors' you gleaned from him?

> Otherwise, you're just "assuming the answer".

I wanted to attempt a more rigorous answer to this.

Back in July, when I started all this about Schoenberg, I wrote:

/tuning-math/messages/516?expand=1

> Message 516

> From: monz <joemonz@y...>

> Date: Wed Jul 18, 2001 5:16am

> Subject: lattices of Schoenberg's rational implications

>

>

> ...

>

> Schoenberg then extends the diagram to include the

> following overtones:

>

> fundamental partials

>

> F 2...12, 16

> C 2...11

> G 2...12

Here, I will further adapt Schoenberg's diagram to make

his explanation as clear as possible, by adding the

partial-numbers and the fundamentals, which are the two

factors which when multiplied together give the relative

frequency-number of each note. The fundamentals are

F = 4, C = 6, G = 9.

d = 12*9 = 108

c = 11*9 = 99

b = 10*9 = 90

a = 9*9 = 81

g = 8*9 = 72

f = 11*6 = 66

f = 16*4 = 64

(f = 7*9 = 63)

e = 10*6 = 60

d = 9*6 = 54 d = 6*9 = 54

c = 12*4 = 48 c = 8*6 = 48

b = 5*9 = 45

b = 11*4 = 44

(bb= 7*6 = 42)

a = 10*4 = 40

g = 9*4 = 36 g = 6*6 = 36 g = 4*9 = 36

f = 8*4 = 32

e = 5*6 = 30

(eb= 7*4 = 28)

d = 3*9 = 27

c = 6*4 = 24 c = 4*6 = 24

a = 5*4 = 20

g = 3*6 = 18 g = 2*9 = 18

f = 4*4 = 16

c = 3*4 = 12 c = 2*6 = 12

f = 2*4 = 8

> (eb) (bb)

> c d e f g a b c d e f g a b c d

> [44] [64]

> (28) (42) [66]

> 24 27 30 32 36 40 45 48 54 60 63 72 81 90 99 108

>

>

> ...

>

> The partial-numbers are also given for the resulting scale

> at the bottom of the diagram, showing that 7th/F (= eb-28)

> is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker

> than 5th/G (= b-45).

>

> Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C

> (= f-66) are all weaker still, thus I have included them in

> square brackets. These overtones are not even mentioned by

> Schoenberg.

These are all the unison-vectors implied by Schoenberg's diagram:

E 5*6=30 : Eb 4*7=28 = 15:14

B 11*4=44 : Bb 7*6=42 = 22:21

B 5*9=45 : B 11*4=44 = 45:44

B 5*9=45 : Bb 7*6=42 = 15:14

F 16*4=64 : F 7*9=63 = 64:63

F 11*6=66 : F 16*4=64 = 33:32

F 11*6=66 : F 7*9=63 = 22:21

A 9*9=81 :(A 20*4=80) = 81:80

C 11*9=99 :(C 24*4=96) = 33:32

(The high "A" and "C" in parentheses are not explicitly indicated

by Schoenberg, but may be inferred from his theory.)

So the only 5-limit unison-vector indicated here is the 81:80

syntonic comma, and even that is only inferred but not stated.

Its applicability to his theory, as well as that of other

5-limit UVs, must be inferred from a careful study of other

explanations in _Harmonielehre_, as I indicated in my last post.

The 15:14 arises only in connected with the notes Schoenberg

himself placed in parentheses. The other UVs are explicitly

indicated by Schoenberg.

-monz

Do You Yahoo!?

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

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

> > Right, but . . . did he apply any of them explicitly within any

of

> > the 'constructions of unison vectors' you gleaned from him?

> > Otherwise, you're just "assuming the answer".

>

>

> Well, on p 176 of _Harmonielehre_ (p 155 of the Carter translation),

> Schoenberg illustrates the "Circle of 5ths", and explicitly notates

> the equivalences Cb=B, Gb=F#, and Db=C# for the major keys, and

> ab=g#, eb=g#, and bb=a# for the minor keys that are +5, +6,

> and +7 "5ths" (respectively) from the origin C-major/a-minor.

OK, but isn't this separate from the 'constructions of unison

vectors' in which Schoenberg tries to arrive at a 13-limit

justification of 12-tET? I mean, this is traditional harmony, which

Schoenberg loves to explain, but is trying to break away from in his

own music, no?

>

> To my mind, the 3-limit (linear, 1-D) is both historically and

> conceptually more basic than 5-limit (planar, 2-D). The

> notational difference between a "sharp" and what later became

> its enharmonically equivalent "flat", ocurred first in Pythagorean

> tuning. And so, along this line of reasoning, the Pythagorean

> comma is historically and conceptually a more basic enharmonicity

> than any of the 5-limit examples. However, as implied above, I

> will also grant the possibility that Schoenberg may have intended

> the diesis as a unison-vector, and will examine that case below

> as well.

Schoenberg sees the diatonic scale not as an essentially 3-limit

entity, as I do, but as an essentially 5-limit entity. So why would

the chromatic scale fall back to 3-limit in his thinking? Doesn't

seem to make sense.

>

>

> Also, I understand Gene's "notation" a little better now.

> So, taking this particular matrix as an example,

>

>

> 2 3 5 7 11 unison-vector ~cents

>

> [ -2 2 1 0 -1 ] = 45:44 38.90577323

> [-19 12 0 0 0 ] = 531441:524288 23.46001038

> [ -5 1 0 0 1 ] = 33:32 53.27294323

> [ 6 -2 0 -1 0 ] = 64:63 27.2640918

> [ -4 4 -1 0 0 ] = 81:80 21.5062896

>

>

> inverse

>

> [ 12 -7 12 0 12 ]

> [ 19 -11 19 0 19 ]

> [ 28 -16 28 0 27 ]

> [ 34 -20 34 -1 34 ]

> [ 41 -24 42 0 41 ]

>

Adjoint?

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

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

> Sent: Wednesday, January 16, 2002 4:00 PM

> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones

>

>

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

>

> > > Right, but . . . did he apply any of them explicitly within

> > > any of the 'constructions of unison vectors' you gleaned

> > > from him? Otherwise, you're just "assuming the answer".

> >

> >

> > Well, on p 176 of _Harmonielehre_ (p 155 of the Carter translation),

> > Schoenberg illustrates the "Circle of 5ths", and explicitly notates

> > the equivalences Cb=B, Gb=F#, and Db=C# for the major keys, and

> > ab=g#, eb=g#, and bb=a# for the minor keys that are +5, +6,

> > and +7 "5ths" (respectively) from the origin C-major/a-minor.

>

> OK, but isn't this separate from the 'constructions of unison

> vectors' in which Schoenberg tries to arrive at a 13-limit

> justification of 12-tET?

Careful, Paul ... the 13-limit justification of 12-tET only

came in 1927/34, in the lecture/article "Problems of Harmony".

Here, I'm specifically interested in Schoenberg's 1911

theory (really 1910, published a year later), and this only

goes up to 11-limit and does not claim to generate the

entire 12-tET scale. The _Harmonielehre_ illustration only

results in a 9-tone scale (C, D, Eb, E, F, G, A, Bb, B) if

one considers the unison-vectors to be tempered out.

What I've been trying to do is to gather as much data as

possible on Schoenberg's conception of the 12-tET scale at

the time he wrote _Harmonielehre_, to construct periodicity-blocks

which "explain" the finity of his tonal universe at that time.

As I've already noted, the 13-limit unison-vectors in his

1927/34 explanation clearly delineate a 12-tone PB. But the

earlier version from _Harmonielehre_ has been harder to glean.

The misprint in the diagram in the Carter translation certainly

didn't help me with this! Now I'm *really* glad that I finally

went thru the effort to obtain a copy of the original German

edition!

> I mean, this is traditional harmony, which Schoenberg loves

> to explain, but is trying to break away from in his own music, no?

Yes, you're right about that. But the whole of _Harmonielehre_

is permeated with the quest for "the truth", and that starts

right with these diagrams I've been examining. Schoenberg felt

that he should begin by tearing apart musical sounds themselves,

to study just what it is that we hear. And he finds that the

overtones that are already present in every harmonic timbre

seem not only to agree quite well with the construction of

chords in traditional tonality, but also to offer a paradigm

for the construction of the more unusual chords he wanted to

put into his own music of the time.

Aside from the overtone and circle-of-5th diagrams, Schoenberg

offers very little else in the way of graphical assistance

other than the copious examples in musical staff notation.

And he continually stressed, not only in this book but for the

rest of his life, that his style was an *evolution* out of what

he had learned from Bach, Mozart, Beethoven, Brahms, Wagner, and

Mahler. So his explanations of traditional harmony are entirely

relevant to his own pantonal/atonal work, from his perpective.

> > To my mind, the 3-limit (linear, 1-D) is both historically and

> > conceptually more basic than 5-limit (planar, 2-D). The

> > notational difference between a "sharp" and what later became

> > its enharmonically equivalent "flat", ocurred first in Pythagorean

> > tuning. And so, along this line of reasoning, the Pythagorean

> > comma is historically and conceptually a more basic enharmonicity

> > than any of the 5-limit examples. However, as implied above, I

> > will also grant the possibility that Schoenberg may have intended

> > the diesis as a unison-vector, and will examine that case below

> > as well.

>

> Schoenberg sees the diatonic scale not as an essentially 3-limit

> entity, as I do, but as an essentially 5-limit entity. So why would

> the chromatic scale fall back to 3-limit in his thinking? Doesn't

> seem to make sense.

Wow, I have to concede that you're right about that, Paul!

Very good. This is analagous to the case of Ben Johnston.

His "basic scale" is exactly the same 7-tone 5-limit JI

diatonic scale Schoenberg illustrates in his first diagram,

which goes up to the 6th harmonic on F, C, and G.

Schoenberg's second diagram (on the very next page in

_Harmonielehre_) is the one which goes up to the 12th

harmonics (and the isolated 16th in one case) to illustrate

how E and B "won" over Eb and Bb in the diatonic scale.

So yes, I can see that it's much more likely that a

5-limit unison-vector would come into play, than a 3-limit

one. In fact, my guess is that Schoenberg most likely

thought of "normal" pitch-relationships as having a

basis in some kind of meantone/12-EDO hybrid, which

after all is what the notation "spells" ... unless one

is assuming Pythagorean tuning.

>

> >

> >

> > Also, I understand Gene's "notation" a little better now.

> > So, taking this particular matrix as an example,

> >

> >

> > 2 3 5 7 11 unison-vector ~cents

> >

> > [ -2 2 1 0 -1 ] = 45:44 38.90577323

> > [-19 12 0 0 0 ] = 531441:524288 23.46001038

> > [ -5 1 0 0 1 ] = 33:32 53.27294323

> > [ 6 -2 0 -1 0 ] = 64:63 27.2640918

> > [ -4 4 -1 0 0 ] = 81:80 21.5062896

> >

> >

> > inverse

> >

> > [ 12 -7 12 0 12 ]

> > [ 19 -11 19 0 19 ]

> > [ 28 -16 28 0 27 ]

> > [ 34 -20 34 -1 34 ]

> > [ 41 -24 42 0 41 ]

> >

> Adjoint?

Well ... *this* is the adjoint of this matrix:

[ -12 7 -12 0 -12 ]

[ -19 11 -19 0 -19 ]

[ -28 16 -28 0 -27 ]

[ -34 20 -34 1 -34 ]

[ -41 24 -42 0 -41 ]

and since the determinant is | -1 |, the inverse is as

I gave it.

-monz

. But since the

determinant is

Do You Yahoo!?

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