Schoenberg's 1927/34 "Problems of Harmony" theory

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, January 17, 2002 1:13 PM
> Subject: [tuning-math] Re: ERROR IN CARTER'S SCHOENBERG
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > Actually, Partch's examination is based on Schoenberg's
> > 1927/34 theory from "Problems of Harmony", so there's no
> > typo anywhere in that. And as I've shown, that theory
> > results in a 12-tone PB with consistent notation for the
> > 11- and 13-limit ratios.
>
> Where did you show that?

/tuning-math/message/2159
> Message 2159
> From: "monz" <joemonz@y...>
> Date: Tue Dec 25, 2001 6:44 pm
> Subject: lattices of Schoenberg's rational implications
>
>
> ...
>
> 1934 _Problems of Harmony_ 13-limit system
>
> (-2 0 0 -1 1 ) = 104:99
> ( 2 0 -1 0 1 ) = 117:112
> (-2 0 -1 0 0 ) = 64:63
> ( 4 -1 0 0 0 ) = 81:80
> ( 2 1 0 -1 0 ) = 45:44
>
> Determinant = 12

If I include 2:1 , I get

matrix

[ 1 0 0 0 0 ] = 2:1
[ -2 0 0 -1 1 ] = 104:99
[ 2 0 -1 0 1 ] = 117:112
[ -2 0 -1 0 0 ] = 64:63
[ 4 -1 0 0 0 ] = 81:80
[ 2 1 0 -1 0 ] = 45:44

adjoint:

[ 12 0 0 0 0 0 ]
[ 19 -1 1 -1 1 1 ]
[ 28 -4 4 -4 -8 4 ]
[ 34 2 -2 -10 -2 -2 ]
[ 42 -6 6 -6 -6 -6 ]
[ 44 4 8 -8 -4 -4 ]

OK, I see that the first column-vector gives a typical
12-EDO mapping. Interestingly, now the 11th harmonic
is mapped to 42 degrees of 12-EDO -- if "C" is n^0, this
is "F#", the opposite of how Schoenberg mapped it in 1911
(as "F"), and indeed this is exactly how Schoenberg now
notates 11 in "Problems of Harmony". And the "new" 13th
harmonic is mapped to the 44th degree ("Ab"), which again
is how Schoenberg notates it.

As for the other column-vectors:

I can see that all of them map 3 = 1 generator, the "5th",
typical of both meantone and Pythagorean.

Columns 2, 3, 4, and 6 map 5 = 4 generators, also typical
of meantone, and the 5th column maps 5 = -8 generators,
typical of a Pythagorean-based schismic temperament.

Columns 2, 3, 5, and 6 map 7 to -2 generators, the "minor 7th",
the closest approximation in Pythagorean. Column 4 maps
7 = 10 generators, the "augmented 6th", which is a typical
meantone mapping.

Columns 2, 3, and 4 map 11 = 6 generators, the "tritone" or
"augmented 4th", a meantone-like approximation. Columns
5 and 6 map 11 = -6 generators, the Pythagorean "diminished 5th",
again only an approximation.

Columns 2, 5, and 6 map 13 = -4 generators, the "minor 6th",
a meantone-like approximation, and columns 3 and 4 map
13 = 8 generators, the "augmented 5th", a Pythagorean
approximation.

Feedback appreciated.

-monz

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Graham, Gene, Paul,

Can you please verify that what I said here is correct,
or fix and explain if it's not?

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, January 18, 2002 2:52 AM
> Subject: [tuning-math] Schoenberg's 1927/34 "Problems of Harmony" theory
>
>
> ...
>
> > [Schoenberg] 1934 _Problems of Harmony_ 13-limit system
>
> ...
>
> matrix
>
> [ 1 0 0 0 0 ] = 2:1
> [ -2 0 0 -1 1 ] = 104:99
> [ 2 0 -1 0 1 ] = 117:112
> [ -2 0 -1 0 0 ] = 64:63
> [ 4 -1 0 0 0 ] = 81:80
> [ 2 1 0 -1 0 ] = 45:44
>
>
> adjoint:
>
> [ 12 0 0 0 0 0 ]
> [ 19 -1 1 -1 1 1 ]
> [ 28 -4 4 -4 -8 4 ]
> [ 34 2 -2 -10 -2 -2 ]
> [ 42 -6 6 -6 -6 -6 ]
> [ 44 4 8 -8 -4 -4 ]
>
>
> OK, I see that the first column-vector gives a typical
> 12-EDO mapping. Interestingly, now the 11th harmonic
> is mapped to 42 degrees of 12-EDO -- if "C" is n^0, this
> is "F#", the opposite of how Schoenberg mapped it in 1911
> (as "F"), and indeed this is exactly how Schoenberg now
> notates 11 in "Problems of Harmony". And the "new" 13th
> harmonic is mapped to the 44th degree ("Ab"), which again
> is how Schoenberg notates it.
>
>
> As for the other column-vectors:
>
> I can see that all of them map 3 = 1 generator, the "5th",
> typical of both meantone and Pythagorean.
>
> Columns 2, 3, 4, and 6 map 5 = 4 generators, also typical
> of meantone, and the 5th column maps 5 = -8 generators,
> typical of a Pythagorean-based schismic temperament.
>
> Columns 2, 3, 5, and 6 map 7 to -2 generators, the "minor 7th",
> the closest approximation in Pythagorean. Column 4 maps
> 7 = 10 generators, the "augmented 6th", which is a typical
> meantone mapping.
>
> Columns 2, 3, and 4 map 11 = 6 generators, the "tritone" or
> "augmented 4th", a meantone-like approximation. Columns
> 5 and 6 map 11 = -6 generators, the Pythagorean "diminished 5th",
> again only an approximation.
>
> Columns 2, 5, and 6 map 13 = -4 generators, the "minor 6th",
> a meantone-like approximation, and columns 3 and 4 map
> 13 = 8 generators, the "augmented 5th", a Pythagorean
> approximation.

----------

I appreciated Graham's explanation of this ...

> > adjoint
> >
> > [ 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 |
>
> See the third column has all zeros except for a 12 right at the bottom.
> That means, trivially, it has a greatest common divisor of 12. (We don't
> count zeros in the gcd.) As a rule of thumb, the GCD of a column tells
you
> how many *equal* steps the equivalence interval is being divided into. In
> this case, we're dividing the octave into 12 equal steps.
>
> Dividing the octave this way is the same as defining a new period to be a
> fraction of the original equivalence interval. Divide the whole column
> through by the GCD, and you get the mapping within the period. In this
> case, that gives [0 0 0 0 1]. The first zero tells you that the octave
> takes the same value as it does in 12-equal. Not much of a surprise. The
> next four zeros tell you that 3:1, 5:1 and 7:1 are also taken from
12-equal.
> And the 1 in the last column tells you that 11:1 is one generator step
away
> from it's value in 12-equal. For 11:1 to be just, you'd have a 51 cent
> generator.
>
> This is a fairly trivial example, and not much use as a temperament. But
> you could realise it by having two keyboards tuned a quartertone apart.
To
> play an 11-limit otonality, you'd have C-E-G-Bb-D on one keyboard, and F
on
> the other.

... and would appreciate some insight into how it would apply
to my description of the 1927/34 13-limit system shown at the top
of this post.

Also, Graham's explanation here is of the matrix which I believe
is a good candidate for modeling Schoenberg's 1911 theory. Can
someone please expand on what Graham says and show what it might
have to do with Schoenberg's adoption of 12-EDO as his preferred tuning?

I've shown that Schoenberg rejected microtonality on practical grounds
(lack of instruments, impact on his own financial situation, etc.),
but his acceptance of 12-EDO eventually became so strong that,
in addition to the known numerological motives behind many of his
choices, my hunch is that there's probably a strong aesthetic motive
as well, and an explanation like this would help to formulate that.

Just trying to make sure that I understand this stuff.
Thanks.

-monz

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