Tenney–Euclidean metrics: Difference between revisions

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__FORCETOC__

=The weighting matrix=

Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || <a2 a3 ... ap| ||_2 = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2); dividing this by sqrt(n), where n = π(p) is the number of primes to p gives the Tenney-Euclidean, or TE, norm. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2); multiplying this by sqrt(n) gives the dual RMS norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.

Let us define the val weighting matrix W to be the [[wikipedia:diagonal matrix|diagonal matrix]] with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || <a2 a3 ... ap| ||_2 = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2); dividing this by sqrt(n), where n = π(p) is the number of primes to p gives the Tenney-Euclidean, or TE, norm. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2); multiplying this by sqrt(n) gives the dual RMS norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.

=Temperamental complexity=

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 __FORCETOC__
 =The weighting matrix=
-Let us define the val weighting matrix W to be the diagonal matrix with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &lt;a2 a3 ... ap| ||_2 = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2); dividing this by sqrt(n), where n = π(p) is the number of primes to p gives the Tenney-Euclidean, or TE, norm. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2); multiplying this by sqrt(n) gives the dual RMS norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.
+Let us define the val weighting matrix W to be the [[wikipedia:diagonal matrix|diagonal matrix]] with values 1, 1/log2(3), 1/log2(5) ... 1/log2(p) along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose Wa* where the * denotes the transpose. Then the dot product of weighted vals is aW^2a*, which makes the Euclidean metric on vals, a measure of complexity, to be || &lt;a2 a3 ... ap| ||_2 = sqrt(a2^2 + a3^2/log2(3)^2 + ... + ap^2/log2(p)^2); dividing this by sqrt(n), where n = π(p) is the number of primes to p gives the Tenney-Euclidean, or TE, norm. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW^(-1), and the dot product is bW^(-2)b*, leading to sqrt(b2^2 + log2(3)^2b3^2 + ... + log2(p)^2bp^2); multiplying this by sqrt(n) gives the dual RMS norm on monzos, a measure of complexity we may call the Tenney-Euclidean, or TE, complexity.
 =Temperamental complexity=