Tenney–Euclidean metrics: Difference between revisions

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The '''Tenney-Euclidean metrics''' are [[Wikipedia: Metric (mathematics)|metrics]] defined in Tenney-Euclidean space. These consist of the TE norm, the temperamental norm, and the octave-equivalent TE seminorm.

== ~~The weighting matrix~~ ==

== TE norm ==

Let us define the val weighting matrix W to be the [[wikipedia:diagonal matrix|diagonal matrix]] with values 1, 1/log23, 1/log25 … 1/log2p along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose WaT where the T denotes the transpose. Then the dot product of weighted vals is aW2aT, which makes the Euclidean metric on vals, a measure of complexity, to be || {{val| a2 a3 … a''p''}} ||2 = sqrt (a22 + a32/(log23)2 + … + ap2/(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-2bT, leading to sqrt (b22 + (log23)2b32 + … + (log2''p'')2b''p''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/log23, 1/log25 … 1/log2p along the diagonal. Given a val "a" expressed as a row vector, the corresponding vector in weighted coordinates is aW, with transpose WaT where the T denotes the transpose. Then the dot product of weighted vals is aW2aT, which makes the Euclidean metric on vals, a measure of complexity, to be || {{val| a2 a3 … a''p''}} ||2 = sqrt (a22 + a32/(log23)2 + … + ap2/(log2''p'')2); 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 bW-1, and the dot product is bW-2bT, leading to sqrt (b22 + (log23)2b32 + … + (log2''p'')2b''p''2); 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 ==

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 The '''Tenney-Euclidean metrics''' are [[Wikipedia: Metric (mathematics)|metrics]] defined in Tenney-Euclidean space. These consist of the TE norm, the temperamental norm, and the octave-equivalent TE seminorm.
-== The weighting matrix ==
+== 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 || {{val| a<sub>2</sub> a<sub>3</sub> … a<sub>''p''</sub>}} ||<sub>2</sub> = 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, or TE, norm. Similarly, if b is a monzo, then in weighted coordinates the monzo becomes bW<sup>-1</sup>, and the dot product is bW<sup>-2</sup>b<sup>T</sup>, leading to 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, or TE, complexity.
+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 || {{val| a<sub>2</sub> a<sub>3</sub> … a<sub>''p''</sub>}} ||<sub>2</sub> = 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 bW<sup>-1</sup>, and the dot product is bW<sup>-2</sup>b<sup>T</sup>, leading to 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 ==