Tenney–Euclidean temperament measures: Difference between revisions

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$$ \text{TE simple badness} = \text{TE complexity} \times \text{TE error} $$

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. ~~Each~~ of ~~these variations will be discussed below~~.

== 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.

~~== Note on scaling factors ==~~

To start with, we may define a norm by means of the usual {{w|norm (mathematics) #Euclidean norm|Euclidean norm}}, a.k.a. ''L''2 norm or ℓ2 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.

Given a [[wedgies and multivals|multival]] or multimonzo which is a {{w|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 {{w|norm (mathematics) #Euclidean norm|Euclidean norm}} ~~(aka~~ ''L''2 norm or ℓ2 norm). We can rescale this several ways, for example by taking a {{w|root mean square}} (RMS) average of the entries of the multivector. These metrics are mainly used to rank temperaments relative to one another. In that regard, it does not matter much if an RMS or an ''L''2 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.

~~Because of this, there~~ are different "standards" for scaling that are commonly in use:

Here are the different standards for scaling that are commonly in use:

# Taking the simple ''L''2 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.

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.

~~An important point of this normalization~~ is to allow us to meaningfully compare ~~measures of corresponding~~ temperaments ~~in different [[just intonation subgroup]]s~~. ~~However, none of them has been quite successful at this goal until~~ [[Sintel]] ~~developed a~~ scheme in 2023.

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.

== TE complexity ==

@@ Line 4: / Line 4: @@
 $$ \text{TE simple badness} = \text{TE complexity} \times \text{TE error} $$
-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. Each of these variations will be discussed below.
+== 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.
-== Note on scaling factors ==
+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.
-Given a [[wedgies and multivals|multival]] or multimonzo which is a {{w|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 {{w|norm (mathematics) #Euclidean norm|Euclidean norm}} (aka ''L''<sup>2</sup> norm or ℓ<sub>2</sub> norm). We can rescale this several ways, for example by taking a {{w|root mean square}} (RMS) average of the entries of the multivector. These metrics are mainly used to rank temperaments relative to one another. In that regard, it does not 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. 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.
-Because of this, there are different "standards" for scaling that are commonly in use:
+Here are the different standards for scaling that are commonly in use:
 # 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.
 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.
-An important point of this normalization is to allow us to meaningfully compare measures of corresponding temperaments in different [[just intonation subgroup]]s. However, none of them has been quite successful at this goal until [[Sintel]] developed a scheme in 2023.
+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.
 == TE complexity ==