Dividing through by the common factor of the whole normalised, inverted

matrix does do the trick for my multiple-29 vectors.

I think it's the *commatic* unison vectors that mean you have to do this.

So my unison vector finder needs to be improved (surprise!)

The second column of the normalised octave-specific inverse is always the

same as the first column of the octave-invariant one, with an extra zero.

I was forgetting the extra zero before. It may be this doesn't always

work for really silly unison vectors, but it does for all the examples

I've tried.

The octave-specific column of the octave-specific matrix is important for

getting the right scale-step mapping. This may be what was going wrong

with the multiple-29 before. Whatever, it works now.

I've got a rough and ready Excel spreadsheet showing this at

<http://x31eq.com/vectors.xls>.

You need to install the Analysis ToolPack for the GCD function to work.

Matrix operations work with the standard install.

The Exchange Server at work is currently flaky, and although I have

offline folders I don't seem to be able to get at them offline. So

although I did read an e-mail from Monzo this morning, I can't reply to

it.

I think any commatic unison vector will do to get the generator mapping,

so long as it's orthogonal to the other vectors. One good way of finding

such is to try a 1 in every column until the determinant is non-zero.

I'll try to include these changes in my Python code tonight. Python with

the Numerical extensions is a good way of hacking this stuff, but the

latter had disappeared from the FTP server last I checked, so I don't

know how you'll get hold of them.

The source code to MIDI Relay should include a matrix library for C++.

Graham

--- In tuning-math@y..., graham@m... wrote:

> I think any commatic

You mean chromatic?

> unison vector will do to get the generator mapping,

> so long as it's orthogonal to

You mean linearly independent from?

> the other vectors.

Paul wrote:

> > I think any commatic

>

> You mean chromatic?

Yes.

> > unison vector will do to get the generator mapping,

> > so long as it's orthogonal to

>

> You mean linearly independent from?

I think so, but I didn't take good notes in that lecture.

> > the other vectors.

>

Graham