Tenney–Euclidean temperament measures: Difference between revisions
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== Introduction == | == Introduction == | ||
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>. | 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>. | ||
=== Preliminary note on scaling factors === | === Preliminary note on scaling factors === | ||
These metrics are mainly used to rank temperaments relative to one another. In that regard, it doesn't matter much if an RMS or an ''L''<sup>2</sup> norm is used, because these two are equivalent up to a scaling factor, so they will rank temperaments identically. | These metrics are mainly used to rank temperaments relative to one another. In that regard, it doesn't matter much if an RMS or an ''L''<sup>2</sup> norm is used, because these two are equivalent up to a scaling factor, so they will rank temperaments identically. | ||
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Note that the above is mainly for comparing temperaments within the same subgroup; when making intra-subgroup comparisons, this can be more complicated. | Note that the above is mainly for comparing temperaments within the same subgroup; when making intra-subgroup comparisons, this can be more complicated. | ||
== TE | == TE complexity == | ||
Given a [[ | Given a [[wedgie]] M, that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ||M|| is a measure of the [[complexity]] of M; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. We may call it '''Tenney-Euclidean complexity''', or '''TE complexity''' since it can be defined in terms of the [[Tenney-Euclidean metrics|Tenney-Euclidean norm]]. | ||
Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [[wikipedia: | Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [[wikipedia: Gramian matrix|Gramian]]. | ||
Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The ''L''<sup>2</sup> norm is one of the standard measures, | Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The ''L''<sup>2</sup> norm is one of the standard measures, | ||
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Graham Breed defines the simple badness slightly differently, again equivalent to a choice of scaling. This is skipped here because, by that definition, it is easier to find TE complexity and TE error first and multiply them together to get the simple badness. | Graham Breed defines the simple badness slightly differently, again equivalent to a choice of scaling. This is skipped here because, by that definition, it is easier to find TE complexity and TE error first and multiply them together to get the simple badness. | ||
=== Reduction to the | === Reduction to the span of a comma === | ||
It is notable that if M is codimension-1, we may view it as representing [[ | It is notable that if M is codimension-1, we may view it as representing [[the dual]] of a single comma. In this situation, the simple badness happens to reduce to the [[Interval span|span]] of the comma, up to a constant multiplicative factor, so that the span of any comma can itself be thought of as measuring the complexity relative to the error of the temperament vanishing that comma. | ||
This relationship also holds if TOP is used rather than TE, as the TOP damage associated with tempering some comma n/d is log(n/d)/( | This relationship also holds if TOP is used rather than TE, as the TOP damage associated with tempering some comma ''n''/''d'' is log(''n''/''d'')/(''nd''), and if we multiply by the complexity ''nd'', we simply get log(''n''/''d'') as our result. | ||
== TE error == | == TE error == | ||