Yahoo Tuning Groups Ultimate Backup 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

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🔗monz <joemonz@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

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🔗monz <joemonz@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

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

🔗monz <joemonz@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

_________________________________________________________
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🔗genewardsmith <genewardsmith@juno.com>

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.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.
>
>
> > 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.

🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

> 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

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🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

> 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

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🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

> 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

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🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.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)?

🔗monz <joemonz@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:
> >
> > > 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

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🔗monz <joemonz@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

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🔗paulerlich <paul@stretch-music.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?

🔗monz <joemonz@yahoo.com>

> 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

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