Tenney–Euclidean temperament measures: Difference between revisions
primerr.pdf better mentioned at the top |
m JIP de-abbreviated. Inline math removed (should be used consistently if desired). Reorder the scaling methods by cognitive difficulty |
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\text{TE simple badness} = \text{TE complexity} \times \text{TE error} </math> | \text{TE simple badness} = \text{TE complexity} \times \text{TE error} </math> | ||
TE temperament measures have been extensively studied by [[Graham Breed]] (see [http://x31eq.com/temper/primerr.pdf|''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''), who also proposed [[Cangwu badness]], an important derived measure, which adds a free parameter to TE simple badness that enables one to specify a tradeoff between complexity and error. | TE temperament measures have been extensively studied by [[Graham Breed]] (see [http://x31eq.com/temper/primerr.pdf| ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''), who also proposed [[Cangwu badness]], an important derived measure, which adds a free parameter to TE simple badness that enables one to specify a tradeoff between complexity and error. | ||
== Introduction == | == Introduction == | ||
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=== A Preliminary Note on Scaling Factors === | === A 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> | 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. | ||
norm is used, because these two are equivalent up to a scaling factor, so they will rank temperaments identically. | |||
As a result, it is somewhat common to equivocate between the various choices of scaling factor, and treat the entire thing as "the" Tenney-Euclidean norm, so that we are really only concerned with the results of these metrics up to that equivalence. | As a result, it is somewhat common to equivocate between the various choices of scaling factor, and treat the entire thing as "the" Tenney-Euclidean norm, so that we are really only concerned with the results of these metrics up to that equivalence. | ||
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Because of this, there are different "standards" for scaling that are commonly in use: | Because of this, there are different "standards" for scaling that are commonly in use: | ||
# Taking the simple ''L''<sup>2</sup> norm | |||
# Taking an RMS | # Taking an RMS | ||
# Taking an RMS and also normalizing for the temperament rank | # Taking an RMS and also normalizing for the temperament rank | ||
# Any of the above and also dividing by the norm of the just intonation points (JIP). | |||
# Any of the above and also dividing by the norm of the JIP | |||
Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the | Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the third definition, as do parts of his [http://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | ||
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 Complexity == | == TE Complexity == | ||
Given a [[Wedgies_and_Multivals|wedgie]] M, that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ||M|| is a measure of the | Given a [[Wedgies_and_Multivals|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:Gramian_matrix|Gramian]]. | 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]]. | ||
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||J \wedge M||'_{RMS} = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</math> | ||J \wedge M||'_{RMS} = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</math> | ||
A perhaps simpler way to view this is to start with a mapping matrix | A perhaps simpler way to view this is to start with a mapping matrix V and add an extra row J corresponding to the JIP; we will label this matrix V<sub>J</sub>. Then the simple badness is: | ||
<math>\displaystyle | <math>\displaystyle | ||
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| 6.441×10<sup>-3</sup> : 6.218×10<sup>-3</sup> | | 6.441×10<sup>-3</sup> : 6.218×10<sup>-3</sup> | ||
|} | |} | ||
<references/> | <references /> | ||
[[Category:math]] | [[Category:math]] | ||
[[Category:measure]] | [[Category:measure]] | ||