Tenney–Euclidean metrics: Difference between revisions

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The '''Tenney-Euclidean metrics''' are {{w|metric (mathematics)|metrics}} defined in Tenney-Euclidean space. These consist of the TE norm, which measures the [[complexity]] of an [[interval]] in [[just intonation]], the TE temperamental norm, which measures the complexity of an interval ''as mapped by a temperament'', and the octave-equivalent TE seminorms of both.

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For example, for 31et, you get {{val| 31 49/log23 72/log25 87/log27 }}, or roughly {{val| 31 30.92 31.01 30.99 }}. Each of these entries individually tells you how sharp or flat each tuning is relative to the octave. The distance from the origin to this point is ~61.96, which is divided by {{nowrap| sqrt(4) {{=}} 2 }} to receive the norm, 30.98.

Similarly, if '''m''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap| '''m'''''W'' {{=}} ''W''{{inv}}'''m''' }}, and the dot product is {{nowrap| {{subsup|'''m'''|''W''|T}}'''m'''''W'' {{=}} '''m'''{{t}}''W''-2'''m''' }}, leading to {{nowrap| sqrt({{subsup|'''m'''|''W''|T}}'''m'''''W'') {{=}} sqrt({{subsup|''m''|1|2}} + (log23)2{{subsup|''m''|2|2}} + … + (log2''p'')2{{subsup|''m''|''n''|2}}) }}; multiplying this by sqrt(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.

Similarly, if '''m''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap| '''m'''''W'' {{=}} ''W''{{inv}}'''m''' }}, and the dot product is {{nowrap| {{subsup|'''m'''|''W''|T}}'''m'''''W'' {{=}} '''m'''{{t}}''W''−2'''m''' }}, leading to {{nowrap| sqrt({{subsup|'''m'''|''W''|T}}'''m'''''W'') {{=}} sqrt({{subsup|''m''|1|2}} + (log23)2{{subsup|''m''|2|2}} + … + (log2''p'')2{{subsup|''m''|''n''|2}}) }}; multiplying this by sqrt(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.

This is a similar method, but instead of dividing by the logarithm of each prime, you multiply, which can be thought of as scaling the lattice of intervals so that larger primes represent greater distances along their respective axes. Again, this results in a vector that can be treated as a point in Euclidean space. Then, as with vals, the TE norm is the distance from the origin to that point, but this time it is multiplied by the square root of the dimensionality rather than divided.

For example, the TE complexity of 5/3 is the distance from the origin to {{monzo| 0 -1.585 2.322 }}, the scaled version of the monzo {{monzo| 0 -1 1 }}, multiplied by sqrt(3). That value is 2.939.

== TE temperamental norm ==

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+{{inacc}}
 The '''Tenney-Euclidean metrics''' are {{w|metric (mathematics)|metrics}} defined in Tenney-Euclidean space. These consist of the TE norm, which measures the [[complexity]] of an [[interval]] in [[just intonation]], the TE temperamental norm, which measures the complexity of an interval ''as mapped by a temperament'', and the octave-equivalent TE seminorms of both.
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 For example, for 31et, you get {{val| 31 49/log<sub>2</sub>3 72/log<sub>2</sub>5 87/log<sub>2</sub>7 }}, or roughly {{val| 31 30.92 31.01 30.99 }}. Each of these entries individually tells you how sharp or flat each tuning is relative to the octave. The distance from the origin to this point is ~61.96, which is divided by {{nowrap| sqrt(4) {{=}} 2 }} to receive the norm, 30.98.
-Similarly, if '''m''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap| '''m'''<sub>''W''</sub> {{=}} ''W''{{inv}}'''m''' }}, and the dot product is {{nowrap| {{subsup|'''m'''|''W''|T}}'''m'''<sub>''W''</sub> {{=}} '''m'''{{t}}''W''<sup>-2</sup>'''m''' }}, leading to {{nowrap| sqrt({{subsup|'''m'''|''W''|T}}'''m'''<sub>''W''</sub>) {{=}} sqrt({{subsup|''m''|1|2}} + (log<sub>2</sub>3)<sup>2</sup>{{subsup|''m''|2|2}} + … + (log<sub>2</sub>''p'')<sup>2</sup>{{subsup|''m''|''n''|2}}) }}; multiplying this by sqrt(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.
+Similarly, if '''m''' is a monzo, then in weighted coordinates the monzo becomes {{nowrap| '''m'''<sub>''W''</sub> {{=}} ''W''{{inv}}'''m''' }}, and the dot product is {{nowrap| {{subsup|'''m'''|''W''|T}}'''m'''<sub>''W''</sub> {{=}} '''m'''{{t}}''W''<sup>−2</sup>'''m''' }}, leading to {{nowrap| sqrt({{subsup|'''m'''|''W''|T}}'''m'''<sub>''W''</sub>) {{=}} sqrt({{subsup|''m''|1|2}} + (log<sub>2</sub>3)<sup>2</sup>{{subsup|''m''|2|2}} + … + (log<sub>2</sub>''p'')<sup>2</sup>{{subsup|''m''|''n''|2}}) }}; multiplying this by sqrt(''n'') gives the dual RMS norm on monzos which serves as a measure of complexity.
 This is a similar method, but instead of dividing by the logarithm of each prime, you multiply, which can be thought of as scaling the lattice of intervals so that larger primes represent greater distances along their respective axes. Again, this results in a vector that can be treated as a point in Euclidean space. Then, as with vals, the TE norm is the distance from the origin to that point, but this time it is multiplied by the square root of the dimensionality rather than divided.
 For example, the TE complexity of 5/3 is the distance from the origin to {{monzo| 0 -1.585 2.322 }}, the scaled version of the monzo {{monzo| 0 -1 1 }}, multiplied by sqrt(3). That value is 2.939.
 == TE temperamental norm ==