Tenney–Euclidean temperament measures: Difference between revisions

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== Introduction ==

Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [[wikipedia:Exterior_algebra|wedge product]] of weighted vals or monzos, we may define a norm by means of the usual [[wikipedia:Norm_(mathematics)#Euclidean norm|Euclidean norm]] (aka ''L''2 norm or ℓ2 norm). We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([[wikipedia:Root_mean_square|root mean square]]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||RMS.

Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [[wikipedia:Exterior_algebra|wedge product]] of weighted vals or monzos (where the weighting factors are 1/log2(''p'') for the entry corresponding to ''p''), we may define a norm by means of the usual [[wikipedia:Norm_(mathematics)#Euclidean norm|Euclidean norm]] (aka ''L''2 norm or ℓ2 norm). We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([[wikipedia:Root_mean_square|root mean square]]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||RMS.

=== A Preliminary Note on Scaling Factors ===

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 == Introduction ==
-Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [[wikipedia:Exterior_algebra|wedge product]] of weighted vals or monzos, we may define a norm by means of the usual [[wikipedia:Norm_(mathematics)#Euclidean norm|Euclidean norm]] (aka ''L''<sup>2</sup> norm or ℓ<sub>2</sub> norm). We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([[wikipedia:Root_mean_square|root mean square]]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||<sub>RMS</sub>.
+Given a [[Wedgies_and_Multivals|multival]] or multimonzo which is a [[wikipedia:Exterior_algebra|wedge product]] of weighted vals or monzos (where the weighting factors are 1/log<sub>2</sub>(''p'') for the entry corresponding to ''p''), we may define a norm by means of the usual [[wikipedia:Norm_(mathematics)#Euclidean norm|Euclidean norm]] (aka ''L''<sup>2</sup> norm or ℓ<sub>2</sub> norm). We can rescale this by taking the sum of squares of the entries of the multivector, dividing by the number of entries, and taking the square root. This will give a norm which is the RMS ([[wikipedia:Root_mean_square|root mean square]]) average of the entries of the multivector. The point of this normalization is that measures of corresponding temperaments in different prime limits can be meaningfully compared. If M is a multivector, we denote the RMS norm as ||M||<sub>RMS</sub>.
 === A Preliminary Note on Scaling Factors ===