Question on name of mathematical operation

Diamonds are constructed by multiplying a harmonic series by a subharmonic one and in general Wilson's cross sets are generated by multiplying two sets of pitches to create a new rectangular array of notes. I have also seen pitch sets generated by transposing a harmonic series as ratios by the ratios of another.

I presume these operations are analogous to creating matrices as the tensor product of two vectors, but are they? If not is there a name for these operations?

Similarly, is there a name for the operation in which a new matrix is created by multiplying two others element by element (like addition is done)?

--John

Hi John,

On Tue, Jun 25, 2013 at 11:24 AM, John H. Chalmers <JHCHALMERS@ucsd.edu> wrote:
>
> Diamonds are constructed by multiplying a harmonic series by a
> subharmonic one and in general Wilson's cross sets are generated by
> multiplying two sets of pitches to create a new rectangular array of
> notes. I have also seen pitch sets generated by transposing a harmonic
> series as ratios by the ratios of another.
>
> I presume these operations are analogous to creating matrices as the
> tensor product of two vectors, but are they? If not is there a name for
> these operations?

If you formulate your harmonic series as a row vector H of integers,
like [1 2 3 4 5], and your subharmonic series as a column vector S of
rationals like [1/1;1/2;1/3;1/4;1/5], then the outer product S*H gives
you the matrix

[1/1 2/1 3/1 4/1 5/1]
[1/2 2/2 3/2 4/2 5/2]
[1/3 2/3 3/3 4/3 5/3]
[1/4 2/4 3/4 4/4 5/4]
[1/5 2/5 3/5 4/5 5/5]

which is the tonality diamond you want.

If we want to get more abstract and treat H and S as vectors in the
same vector space, then we can instead look at the tensor product
S(x)H to get the same result as a tensor with two covariant indices.

Whether this is sufficient or depends on which tonality diamonds you
want to treat as equivalent to one another: for instance, if instead
you take H(x)S, you get the tensor

[1/1 1/2 1/3 1/4 1/5]
[2/1 2/2 2/3 2/4 2/5]
[3/1 3/2 3/3 3/4 3/5]
[4/1 4/2 4/3 4/4 4/5]
[5/1 5/2 5/3 5/4 5/5]

which is the same tensor but with the indices reversed. If we were to
treat it as a matrix, it'd be the transpose of the original matrix. Do
we consider this to be the same tonality diamond as the original? If
so, then we declare a(x)b = b(x)a, giving us the symmetric algebra
rather than the tensor algebra.

If we want to further declare that the permutations of this matrix are
equivalent to the original, things get even hairier. There are lots of
related algebraic structures that give you the same result without
these problems though.

One that I find particularly elegant is: start with the set of
strictly positive rationals Q+, and look at the free commutative
idempotent monoid on it, aka the set of finite-dimensional sets with
elements taken in Q, where the binary operation is set union U. Then,
define an additional binary operation * as follows:

1) if A and B are singletons {a} and {b} respectively, A*B = {a*b}
2) * distributes over U

So {a, b, c} * {d, e} = {a*d, b*d, c*d, a*e, b*e, c*e}.

The resulting structure forms a semiring, which I suggested might be a
useful way of thinking about scales and chords:
/tuning-math/message/21251. I guess
you can add "cross sets" to the list of the things this structure can
do as well.

> Similarly, is there a name for the operation in which a new matrix is
> created by multiplying two others element by element (like addition is
> done)?

That would be called the "Hadamard product":

http://en.wikipedia.org/wiki/Matrix_multiplication#Hadamard_product

Mike