Yahoo Tuning Groups Ultimate Backup tuning-math adding/subtracting unison-vectors

Question:

What significance is there in the addition or subtraction
of unison-vectors? In either case, the result is another
unison-vector which may be substituted in the matrix in
place of either of those two, correct? Or does it work
only for subtraction, or only for addition?

Here's the specific example I have in mind:

In Message 2159, I wrote:

> From: "monz" <joemonz@y...>
> Date: Tue Dec 25, 2001 6:44 pm
> Subject: lattices of Schoenberg's rational implications
/tuning-math/message/2159
>
>
> 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

But I see now that adding together the two 11-limit
unison-vectors cancels out prime-factor 11, and we're
left with one of the standard 5-limit "semitones":

3 5 7 11 ratio ~cents

[ 1 0 0 1 ] = 33:32 53.27294323
+ [ 2 1 0 -1 ] = 45:44 38.90577323
----------------
[ 3 1 0 0 ] = 135:128 92.17871646

This is the interval which Rameau called the "mean semitone",
and Ellis called the "larger limma". See my webpage:
http://www.ixpres.com/interval/td/monzo/o483-26new5limitnames.htm

It seems to me that this is perhaps why Schoenberg,
in 1911, gave an inconsistent notation for the 11th harmonic.

If I subtract these two 11-limit unison-vectors, I get:

3 5 7 11 ratio ~cents

[ 1 0 0 1 ] = 33:32 53.27294323
+ [ 2 1 0 -1 ] = 45:44 38.90577323
----------------
[-1 -1 0 2 ] = 121:120 14.36717

What do either of these operations have to do with
Schoenberg's theory?

-monz

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adding/subtracting unison-vectors

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>