Reflection of a Temperament is the Temperament of a Reflection

Let us say we have an n-tuple of eigenvalues S such that S^2 = (1, 1, …, 1, 1). Let us also construct a matrix Q by augmenting n linearly-independant column monzos (our eigenvectors).

Then our Reflection matrix A (a linear transform) is the following:
A = Q*diag(S)*Q^-1

Note that A is an involutory matrix (i.e. it is its own inverse).

Theorem:
The reflection of a temperament is the same as tempering a reflection.

Proof:
A = A^-1 (by definition of an involutory matrix)
A*m = A^-1*m
T*(A*m) = (T*A^-1)*m

Where m is any nx1 monzo, and T is a (n-1)xn mapping matrix.

Ryan Avella

--- In tuning-math@yahoogroups.com, "Ryan Avella" <domeofatonement@...> wrote:

There are other interesting transformations aside from involutions. You might look here:
http://xenharmonic.wikispaces.com/Using+Scala+to+transform+just+intonation
here:
http://xenharmonic.wikispaces.com/The+Seven+Limit+Symmetrical+Lattices
here:
http://xenharmonic.wikispaces.com/Composing+with+tablets
or here:
http://xenharmonic.wikispaces.com/Graph-theoretic+properties+of+scales