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