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Tenney–Euclidean temperament measures: Difference between revisions

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== Preliminaries ==
== Preliminaries ==
There have been several minor variations in the definition of TE temperament measures, which differ from each other only in their choice of multiplicative scaling factor. The reason these differences come up is because we are adopting different averaging methods for the entries of a multivector.
There have been several minor variations in the definition of TE temperament measures, which differ from each other only in their choice of multiplicative scaling factor. These differences come up because we are adopting different averaging methods for the entries of a multivector. To start with, consider a norm defined by means of the usual {{w|norm (mathematics) #Euclidean norm|Euclidean norm}}, a.k.a. ''L''<sup>2</sup> or ℓ<sub>2</sub> norm. The result of this is a kind of a sum of all the entries. We can rescale this in several ways, for example by taking a {{w|root mean square}} (RMS) average of the entries, by normalizing for the temperament's rank, and/or by normalizing up to the norm of the [[just tuning map]].  


To start with, we may define a norm by means of the usual {{w|norm (mathematics) #Euclidean norm|Euclidean norm}}, a.k.a. ''L''<sup>2</sup> norm or ℓ<sub>2</sub> norm. The result of this is a kind of a sum of all the entries. We can rescale this in several ways, for example by taking a {{w|root mean square}} (RMS) average of the entries.  
''These metrics are meant to be used to rank temperaments within the same [[rank]] and [[subgroup]]''. It does not matter much which scheme is used, because 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.  


Here are the different standards for scaling that are commonly in use:
[[Graham Breed]]'s original definitions<ref name="primerr">Graham Breed. [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''.</ref> are based on an RMS norm normalized for the temperament's rank, as are parts of his [https://x31eq.com/temper/ Temperament Finder], although other scaling and normalization methods are sometimes used as well.
# Taking the simple ''L''<sup>2</sup> norm
# Taking an RMS
# 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]]).  


As these metrics are mainly used to rank temperaments within the same [[rank]] and [[just intonation subgroup]], it does not matter much which scheme is used, because they 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.
More recently, it is proposed that the metrics should be normalized to allow us to meaningfully compare temperaments across subgroups and even ranks. [[Sintel]]'s scheme in 2023 is the first attempt at this goal<ref name="sintel">Sintel. [https://github.com/Sin-tel/temper/blob/c0d5c36e3c189f64860f4aea288ff3ff3bc34982/lib_temper/temper.py "Collection of functions for dealing with regular temperaments"], Temperament Calculator.</ref>, though still with notable caveats.  


Graham Breed's original definitions<ref name="primerr">Graham Breed. [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''.</ref> tend to use the third definition, as do parts of his [https://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well.
Each of these will be discussed in detail below.
 
It is also possible to normalize the metrics to allow us to meaningfully compare temperaments across subgroups and even ranks. [[Sintel]]'s scheme in 2023 is the first attempt at this goal<ref name="sintel">Sintel. [https://github.com/Sin-tel/temper/blob/c0d5c36e3c189f64860f4aea288ff3ff3bc34982/lib_temper/temper.py "Collection of functions for dealing with regular temperaments"], Temperament Calculator.</ref>.  


== TE complexity ==
== TE complexity ==
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