maps, uvs

Can someone post a general method for transforming a
list of unison vectors into a map and vice versa?

-Carl

Carl Lumma wrote:

> Can someone post a general method for transforming a
> list of unison vectors into a map and vice versa?

See

<http://x31eq.com/temper.html>

and work out the code. I can see the value in a longer explanation of
that, but I haven't done it and certainly won't at this time of night.

The basic method is to take the wedge product of the vectors, and the
octave-equivalent part is the mapping by generator. Wedge it with some
chromatic unison vector and you get an example ET mapping. You can
combine the two to get either kind of mapping I print out, but you'll have
to check the code to see how.

There's also <http://x31eq.com/vectors.html> that uses matrix
operations instead of wedge products. There, you put the octave at the
top, the chromatic UV second, and the commatic UVs below in a matrix.
Take the adjoint, and the left hand column is your example ET and the next
column is the mapping by generator. They're both mathematically
equivalent to the same things you get from wedge products.

Graham

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Can someone post a general method for transforming a
> list of unison vectors into a map and vice versa?

(1) Wedge the unisons together to get a wedgie

(2) If you have a linear temperament wedgie, wedge this with 2 to
get the map to steps of the generator, and then solve the linear equations to get corresponding octaves for the generator map

(3) If you aren't dealing with a linear temperament, you need to find a basis for the generators--wedging with the elements of this basis will give the map

(4) I've started writing up a paper on the mathematics of temperament,
so I should have this explained in detail. When it's ready I'd like some comments on it!

hi Gene and Carl,

> From: Carl Lumma <carl@lumma.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, March 03, 2002 2:00 PM
> Subject: [tuning-math] maps, uvs
>
>
> Can someone post a general method for transforming a
> list of unison vectors into a map and vice versa?
>
> -Carl

see the second section of:

Tuning Dictionary, "periodicity block"
http://www.ixpres.com/interval/dict/pblock.htm

where i quote Gene's method of finding a notation
which maps to JI pitches. (BTW, there were errors
in this originally ... did i get them all out, Gene?)

to find the mapping to EDOs, put the unison-vectors
in vector form into a matrix, then calculate the
determinant and the inverse. if the inverse is
unimodular (= has a determinant = 1), then it gives
the mapping to EDOs, the cardinality of which (i.e.,
mapping of prime-factor 2) is in the top row. see:

Tuning Dictionary, "matrix"
http://www.ixpres.com/interval/dict/matrix.htm

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, March 03, 2002 4:34 PM
> Subject: Re: [tuning-math] maps, uvs
>
>
> to find the mapping to EDOs, put the unison-vectors
> in vector form into a matrix, then calculate the
> determinant and the inverse. if the inverse is
> unimodular (= has a determinant = 1), then it gives
> the mapping to EDOs, the cardinality of which (i.e.,
> mapping of prime-factor 2) is in the top row. see:
>
> Tuning Dictionary, "matrix"
> http://www.ixpres.com/interval/dict/matrix.htm

oops ... the matrix is unimodular if the determinant
is +1 or -1.

-monz

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In-Reply-To: <002001c1c314$5539d420$af48620c@dsl.att.net>
monz wrote:

> to find the mapping to EDOs, put the unison-vectors
> in vector form into a matrix, then calculate the
> determinant and the inverse. if the inverse is
> unimodular (= has a determinant = 1), then it gives
> the mapping to EDOs, the cardinality of which (i.e.,
> mapping of prime-factor 2) is in the top row. see:
>
> Tuning Dictionary, "matrix"
> http://www.ixpres.com/interval/dict/matrix.htm

Note that mappings to EDOs are covered in
<http://x31eq.com/et.htm>. That's old, but the method still
stands. The generator mapping is the same idea but for octave-equivalent
matrices, and common factors give you the octave division.

Graham