Tenney–Euclidean metrics: Difference between revisions
→TE norm: need to point out monzo is defined as a row vector in this article |
Rewrite the first section cuz in the next section the monzos are col vectors |
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== TE norm == | == TE norm == | ||
Let us define the val weighting matrix W to be the [[wikipedia:diagonal matrix|diagonal matrix]] with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>p along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa<sup>T</sup> where the <sup>T</sup> denotes the transpose. Then the dot product of weighted vals is aW<sup>2</sup>a<sup>T</sup>, which makes the Euclidean metric on vals, a measure of complexity, to be || | Let us define the val weighting matrix W to be the [[wikipedia:diagonal matrix|diagonal matrix]] with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>p along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is v = aW, with transpose v<sup>T</sup> = Wa<sup>T</sup> where the <sup>T</sup> denotes the transpose. Then the dot product of weighted vals is vv<sup>T</sup> = aW<sup>2</sup>a<sup>T</sup>, which makes the Euclidean metric on vals, a measure of complexity, to be ||v||<sub>2</sub> = sqrt (vv<sup>T</sup>) = sqrt (a<sub>2</sub><sup>2</sup> + a<sub>3</sub><sup>2</sup>/(log<sub>2</sub>3)<sup>2</sup> + … + a<sub>p</sub><sup>2</sup>/(log<sub>2</sub>''p'')<sup>2</sup>); dividing this by sqrt (''n''), where ''n'' = π(''p'') is the number of primes to ''p'' gives the '''Tenney-Euclidean complexity''', or '''TE complexity'''. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes m = W<sup>-1</sup>b, and the dot product is m<sup>T</sup>m = b<sup>T</sup>W<sup>-2</sup>b, leading to sqrt (m<sup>T</sup>m) = sqrt (b<sub>2</sub><sup>2</sup> + (log<sub>2</sub>3)<sup>2</sup>b<sub>3</sub><sup>2</sup> + … + (log<sub>2</sub>''p'')<sup>2</sup>b<sub>''p''</sub><sup>2</sup>); multiplying this by sqrt (''n'') gives the dual RMS norm on monzos, a measure of complexity we may call the '''Tenney-Euclidean norm''', or '''TE norm'''. | ||
== Temperamental complexity == | == Temperamental complexity == | ||