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

http://groups.yahoo.com/group/tuning-math/message/20

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > [Paul:]
> > > So shall we call our integer limit 12?
> >
> > [monz:]
> > Sure! Guess what?... that ties this in nicely with
> > Schoenberg's alleged integer-limit of 12 in his
> > _Harmonielehre_ (the explanation disparaged by Partch).
>
>
> Umm . . . I thought that explanation used a _prime-limit_
> of 13, not an _integer-limit_ of 12. In particular, Partch
> showed that Schoenberg's two derivations of the note C# --
> as the 11th harmonic of G and as the 13th harmonic of F --
> hence as 33/32 and 13/12 -- differed by virtually an entire
> semitone (i.e., Schoenberg assumed a "unison vector" of 143:128).


Damn, Paul! Duh again!

I had signed off for the night, and just realized this error
and came back to the computer to correct it, and you've
already explained it sufficiently!

Here's the full scoop:



The incorrect part of my statement was the mention of
Partch's analysis.

The Schoenberg work Partch cites is a lecture given in
1934 called in the English translation in _Style and Idea_
"Problems of Harmony".

I was correct in saying that an alleged 12-integer-limit
would connect our optimization with Schoenberg's in his
_Harmonielehre_ of 1911. That's precisely how he explains
the origin of the diatonic scale, plus the first couple
of chromatic alterations which suggest the 12-EDO paradigm
he hints at in a couple of sections of the 1911 edition.

(In the more commonly found 1922 edition he expands quite
a bit at these points and presents fully 12-EDO outlines.)

He obviously decided on a prime-limit of 13 some time later.


I'm interested now in whether Schoenberg thought of his
1934 analysis as a prime-limit or an odd-limit, because
my hazy immediate recollection suggests the latter.

I'll take a closer look at the Schoenberg article to see
if it's possible to determine this, and also make sure that
my dates are accurate.

But for sure, the 12-integer-limit is in _Harmonielehre_.
FTR, Schoenberg actually wrote it during the summer of
1910. It was published in 1911.


Hmmm... 1910 was the same summer Mahler composed his 10th
Symphony, probably reflecting a good deal of the influence
I believe Schoenberg was having on Mahler, who supported
Schoenberg (financially and otherwise) for years past the
point when he could no longer understand Schoenberg's work.

Mahler wrote to Schoenberg in 1909 that "I have the score of
your [Schoenberg's 2nd] Quartet with me here [in New York]
and study it from time to time, but it's difficult for me."

I believe that Mahler's work shows the influence of Schoenberg
as early as the _7th Symphony_, 1905.

So research into this kind of tuning paradigm may have
some bearing on my attempts to experimentally retune
Mahler's work.

Interesting.



-monz
http://www.monz.org
"All roads lead to n^0"