Super easy way to calculate generators of a temperament

1) Take mapping matrix

[1 1 0]
[0 1 4]

2) augmented with [1 0;0 1]

[1 1 0 1 0]
[0 1 4 0 1]

3) take null space using rational or integer arithmetic

4 1 -1
-4 -1 0
1 0 0
0 1 0
0 0 1

The second last column tells you that 1 of the 4th column (which is |1
0>), plus some magic configuration of the other columns, maps to the
null vector. The third column tells you the same thing with the 5th
column (which is |0 1>).

The magic configuration for the 4th column happens to be |1 -1 0>, and
for the 5th column it's |-1 0 0>, which is 2/3 and 1/2 respectively.
However, this is what you'd have to have to -cancel out- the |1 0> and
|0 1> columns, respectively, and get the null vector. So you know the
generators are actually |-1 1 0> and |1 0 0>.

You may have to do some "elementary column operations" on the null
space matrix in order to get it in that form. I believe you can also
take the transpose, rotate the matrix 180 degrees, and put it in
Hermite or rref form.

This is a bit abstract but makes sense if you consider matrix
multiplication on the right by a column vector as giving you a
weighted sum of the column vectors of the matrix on the left, where
the weights are the coefficients of the column vector.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> This is a bit abstract

"Abstract" isn't the word. When you say "nth column", you don't tell us nth column of what.

Fixed

1) Take mapping matrix

[1 1 0]
[0 1 4]

2) augmented with [1 0;0 1]

[1 1 0 1 0]
[0 1 4 0 1]

3) take null space using rational or integer arithmetic

[4 1 -1]
[-4 -1 0]
[1 0 0]
[0 1 0]
[0 0 1]

The second last column of the above null space matrix tells you that 1 *
the 4th column of the mapping matrix (which is |1 0>), plus some magic
configuration of the other columns in the mapping matrix, maps to the
null vector. The third column of the above null space matrix tells you the
same thing with the 5th column of the mapping matrix (which is |0 1>).

-Mike

On May 14, 2012, at 1:00 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> This is a bit abstract

"Abstract" isn't the word. When you say "nth column", you don't tell us nth
column of what.