> 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?

Ahh ... actually Paul ... no.

Now I realize my mistake: 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]

Plugging that into the unison-vector matrix I had already

derived before:

2 3 5 7 11 unison vectors ~cents

[ 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 (without powers of 2) =

[-1 0 0 2]

[-4 0 0 -4] 1

[ 2 0 -12 -4] * --

[ 1 12 0 -2] 12

So it looks to me like Schoenberg's explanation in

_Harmonielehre_ definitely implies a 12-tone periodicity-block.

I'd venture to say that Schoenberg had a good intuitive

grasp of all this, without actually knowing anything about

periodicity-block theory.

-monz

_________________________________________________________

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Hi Gene,

> From: monz <joemonz@yahoo.com>

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

> Sent: Friday, January 11, 2002 1:59 AM

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

>

>

> Now I realize my mistake: 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]

>

>

> Plugging that into the unison-vector matrix I had already

> derived before:

>

> 2 3 5 7 11 unison vectors ~cents

>

> [ 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 (without powers of 2) =

>

> [-1 0 0 2]

> [-4 0 0 -4] 1

> [ 2 0 -12 -4] * --

> [ 1 12 0 -2] 12

>

How does this compare with the other 12-tone periodicity-block

you calculated for Schoenberg? Can you please give a listing

of the pitches inside *this* PB? Thanks.

-monz

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

> How does this compare with the other 12-tone periodicity-block

> you calculated for Schoenberg? Can you please give a listing

> of the pitches inside *this* PB?

The set of pitches inside any non-torsional 12-tone periodicity block

with all the unison vectors tempered out is simply 12-tET.

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

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

> Sent: Friday, January 11, 2002 1:32 PM

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

>

>

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

>

> > How does this compare with the other 12-tone periodicity-block

> > you calculated for Schoenberg? Can you please give a listing

> > of the pitches inside *this* PB?

>

> The set of pitches inside any non-torsional 12-tone periodicity block

> with all the unison vectors tempered out is simply 12-tET.

Right ... but I'm looking for the basic rational implications

in the basic periodicity-block, so how does one find this?

By intentionally leaving out one or more unison-vectors?

Confused,

-monz

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Apologies for wasting so much bandwidth ... but I sent

both of these out this morning with the wrong date, and

they might have been missed.

----- Original Message -----

From: monz <joemonz@yahoo.com>

To: <tuning-math@yahoogroups.com>

Sent: Monday, January 15, 2001 9:38 AM

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

>

> 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

>

>

----- Original Message -----

From: monz <joemonz@yahoo.com>

To: <tuning-math@yahoogroups.com>

Sent: Monday, January 15, 2001 9:59 AM

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

> 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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In addition to still hoping for answers to my previous

questions on this thread, I have some more:

> Message 2577

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

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

> Subject: Re: badly tuned remote overtones

/tuning-math/message/2577

>

>

> Now I realize my mistake: 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]

>

>

> Plugging that into the unison-vector matrix I had already

> derived before:

> <etc.>

Now here, I've also added a row vector for the "8ve",

as Gene had demonstrated:

matrix

2 3 5 7 11 unison vectors ~cents

[ 1 0 0 0 0 ] = 2:1 0

[ 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 0 0 0 0 ]

[ 19 -1 0 0 2 ] 1

[ 28 -4 0 0 -4 ] * --

[ 34 2 0 -12 -4 ] 12

[ 41 1 12 0 -2 ]

OK, so I can see that the first column vector of the

inverse gives the 12-EDO homomorphism:

12-EDO note-name

degree with C = 1/1

h12(2) = 12 C

h12(3) = 19 G

h12(5) = 28 E

h12(7) = 34 Bb

h12(11) = 41 F

And this agrees with the two cases of overtones on "C" and

"G", where Schoenberg equated the "perfect 4th" with the

11th overtone, but not with the case of the "F" fundamental,

where he equated the "augmented 4th" with 11.

But now what do all those other column vectors mean?

They all start with 0 ... what does that mean?

-monz

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In-Reply-To: <004401c19e0a$eba97f20$af48620c@dsl.att.net>

monz wrote:

> inverse

>

> [ 12 0 0 0 0 ]

> [ 19 -1 0 0 2 ] 1

> [ 28 -4 0 0 -4 ] * --

> [ 34 2 0 -12 -4 ] 12

> [ 41 1 12 0 -2 ]

>

>

> OK, so I can see that the first column vector of the

> inverse gives the 12-EDO homomorphism:

...

> But now what do all those other column vectors mean?

> They all start with 0 ... what does that mean?

Starting with 0 means they'll do as octave-equivalent generator mappings.

The first is some kind of meantone because it starts with -1 and -4, so a

third is four fifths. Then the 2 means that 7:4 approximates as a minor

seventh, rather than an augmented sixth. That sounds right for

Schoenberg's assumptions. Then the 1 means that 11:8 approximates the

same as 4:3. You could check Arnie's spelling against these, to see if he

really did need enharmonic equivalence.

The other columns will be other temperaments. Looks like two that have a

pair of 12-equal scales a comma apart to get better 7:4 or 11:8

approximations. And than a temperament that divides the octave into two

equal parts. Could be diaschismic. Yes, that's right, an 11-limit

Paultone, or whatever that's calling itself these days.

Graham

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

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

> Sent: Wednesday, January 16, 2002 5:57 AM

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

>

>

> In-Reply-To: <004401c19e0a$eba97f20$af48620c@dsl.att.net>

> monz wrote:

>

> > inverse

> >

> > [ 12 0 0 0 0 ]

> > [ 19 -1 0 0 2 ] 1

> > [ 28 -4 0 0 -4 ] * --

> > [ 34 2 0 -12 -4 ] 12

> > [ 41 1 12 0 -2 ]

> >

> >

> > OK, so I can see that the first column vector of the

> > inverse gives the 12-EDO homomorphism:

>

> ...

>

> > But now what do all those other column vectors mean?

> > They all start with 0 ... what does that mean?

>

> Starting with 0 means they'll do as octave-equivalent

> generator mappings. The first is some kind of meantone

> because it starts with -1 and -4, so a third is four fifths. ...

I noticed that myself ... thanks, Graham!

> ... Then the 2 means that 7:4 approximates as a minor

> seventh, rather than an augmented sixth. That sounds right

> for Schoenberg's assumptions. Then the 1 means that

> 11:8 approximates the same as 4:3. You could check Arnie's

> spelling against these, to see if he really did need

> enharmonic equivalence.

OK, I noticed these also, since I figured out that this column

was some sort of meantone, but didn't really think about the

"minor 7th vs. augmented 6th" question ... I simply saw that

it gave the numbers of a typical linear mapping but with the

signs reversed. Thanks for that!

If you see my last post (ERROR IN CARTER'S SCHOENBERG) you'll

see that Schoenberg did indeed consistently spell the note

which represents the 11th harmonic the same as that which

represents 4:3. This was big news to me, because the English

translation has a misprint in the diagram, and I've based years

of research on that error, until today.

> The other columns will be other temperaments. ...

OK, I assumed that too, after seeing that the first two

columns both gave temperaments. Thanks for confirmation.

> ... Looks like two that have a pair of 12-equal scales a

> comma apart to get better 7:4 or 11:8 approximations. And

> than a temperament that divides the octave into two equal

> parts. Could be diaschismic. Yes, that's right, an 11-limit

> Paultone, or whatever that's calling itself these days.

Now *those* are things I'd never figure out. I'd be

interested in knowing more. But please base analysis on

what I now think is the most appropriate unison-vector matrix:

matrix

2 3 5 7 11 unison-vectors ~cents

[ 1 0 0 0 0 ] = 2:1 0

[ 7 0 -3 0 0 ] = 128:125 41.05885841

[ -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

fractional inverse

[ 12 0 0 0 0 ]

[ 19 -1 0 0 3 ]

[ 28 -4 0 0 0 ]

[ 34 2 0 -12 -6 ]

[ 41 1 12 0 -3 ]

determinant = | 12 |

And when I wrote after this:

> The only cardinality given by this matrix is 12,

I was wrong about that, correct? As Graham shows,

the other columns with all those zeros do indeed

express temperament mappings.

-monz

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

> If you see my last post (ERROR IN CARTER'S SCHOENBERG) you'll

> see that Schoenberg did indeed consistently spell the note

> which represents the 11th harmonic the same as that which

> represents 4:3. This was big news to me, because the English

> translation has a misprint in the diagram, and I've based years

> of research on that error, until today.

I told you that you were the man to sort Schoenberg out. :)

> I was wrong about that, correct? As Graham shows,

> the other columns with all those zeros do indeed

> express temperament mappings.

They give what I call "vals", but these are ones which define your PB.