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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