Euclidean Heights and Wonzos

We will need a logarithm base for this, which I will take to be cents, though you may prefer base 2/octaves.

A tuning map, or element of a tuning space, is a real vector
<t1 t2 t3 ... tn| whose entries define where the successive primes are mapped to (in cents.) When applied to a monzo |m1 m2 ... mn> it gives the mapping of the monzo, in cents. We may change coordinates to a weighted tuning map <t1/cents(2) t2/cents(3) ...| in which case we have to change the dual coordinates of the monzo to a weighted monzo ("wonzo"?) by |m1*cents(2) m2*cents(3) ,,,>. If we subtract the JI point <1 1 ... 1| from a weighted tuning map, we get an error
map E; if W is a wonzo then the dot product E.W gives the error of the tuning map in cents.

Define the Euclidean height of a wonzo W, and hence also of the corresponding monzo, to be the oridinary Euclidean norm ||W||. It has values in cents, or in unit-free language its defined in a logarithmic measure of interval size. Define the Euclidean norm of a tuning map or error map to be the ordinary Euclidean norm of the weighted coordinates. Since E.W gives the error, then by Cauchy's inequality we have that

|Error| = |E.W| <= ||E|| ||W||

where of course we may think of the norms as defined on the unweighted coordinates also, by change of coordinates. We may rewrite the above as

|Error|/||W|| <= ||E||

which tells us the relative error (something which is independent of the base of logarithms) is bounded by ||E||.

Euclidean height isn't quite as well-behaved as Tenney height (for one thing, successive integers don't increase monotonically in height) but it isn't too awful, and in any case as I remarked before Euclidean height (for any particular prime limit) is bounded in terms of Tenney height and vice-versa.

On 2 May 2010 18:39, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> Define the Euclidean height of a wonzo W, and hence also of the
> corresponding monzo, to be the oridinary Euclidean norm ||W||.
> It has values in cents, or in unit-free language its defined in a
> logarithmic measure of interval size. Define the Euclidean norm of a
> tuning map or error map to be the ordinary Euclidean norm of the
> weighted coordinates. Since E.W gives the error, then by Cauchy's
> inequality we have that
>
> |Error| = |E.W| <= ||E|| ||W||

If W is a weighted monzo and J is the weighted JI vector (all ones for
Tenney weighting) the TOP-RMS error for the monzo being a unison
vector is

|W.J| / ||W|| ||J||

I can't generalize this formula to more than one unison vector.

For a weighted val-space wedgie Q and JI vector V, the TOP-RMS error is

||Q^V|| / ||Q|| ||V||

You seen to be defining E = Q - V and getting something less useful.

Graham