Fw: copy of tuning-math post I can't send now (Schoenberg)

Graham couldn't access the list, so he sent me this reply.

> From: Graham Breed <graham.breed@isky.co.uk>
> To: <joemonz@yahoo.com>
> Sent: Wednesday, January 16, 2002 10:11 AM
> Subject: copy of tuning-math post I can't send now
>
>
> monz 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.
>
> That's consistent 12-equal spelling, though. The matrix suggests a
> consistent meantone spelling, which is stricter. That is, 7:4 is always a
> minor seventh and never an augmented sixth.
>
> > > 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.
>
> Each column matches up with a row from the original matrix. The
temperament
> shown by the column doesn't temper out the comma shown by the relevant
row.
> So the meantone column is what you get by not tempering out an enharmonic
> diesis.
>
> > > ... 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:
>
> ...
>
> > 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 |
>
> 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. There are some more respectable temperaments that work this
way.
> Like my multiple-29, where 1 and 3 (and 9) come from 1 keyboard, and 5, 7,
> 11 and 13 (and 15) from the other.
>
> The next column is the same idea, but it's 7:1 that's on the second
> keyboard.
>
> The last column is different now. I don't recognize it, but it would mean
> dividing the octave into 3 equal parts, and taking 5:4 as an exact
> third-octave, same as it is in 12-equal. It's the column that doesn't
> temper out the syntonic comma.
>
>
> Graham

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