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Error and complexity Tenney style

🔗Gene Ward Smith <gwsmith@svpal.org>

1/9/2004 8:08:49 PM

We have an error measurement already; a temperament corresonds to a
subspace of the val space, and its error is its minimum distance to
the JIP.

For complexity maybe we want to put a measure on wedgies products.
For example, for a 7-limit wedgie l = [l1 l2 l3 l4 l5 l6] and a monzo
|w1 w2 w3 w4> we can take a wedge product and get a val

<l3*w4+l2*w3+w2*l1, l5*w4+l4*w3-w1*l1,
w4*l6-l4*w2-w1*l2, -w3*l6-l5*w2-w1*l3|

We can use all the same formulas for arbitary points in the Tenney
space. Suppose we wedge with all unit vectors, which is to say,
with w = |w1 w2 w3 w4> such that

||w|| = |w1|+log2(3)|w2|+log2(5)|w3|+log2(7)|w4| = 1

The maximum of the norms of all the corresponding points in the
Tenney space when wedged with all w of norm 1 will give us a norm on
l. We can also start from the val side, and get another norm on wedge
products.