Tenney–Euclidean temperament measures: Difference between revisions

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.

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.  

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 [https://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well.

 

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>.  

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.  

Graham Breed and [[Gene Ward Smith]] have proposed different RMS norms. Let us denote the RMS norm of ''M'' as ‖''M''‖_RMS. In Graham's paper<ref name="primerr">Graham Breed. [http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''.</ref>, an RMS norm is proposed as

where ''n'' is the number of primes up to the prime limit ''p'', and ''r'' is the rank of the temperament. Thus ''n''^''r'' is the number of permutations of ''n'' things taken ''r'' at a time with repetition, which equals the number of entries of the wedgie in its full tensor form.  

 

: '''Note''': that is the definition used by Graham Breed's temperament finder.  

: '''Note''': that is the definition used by Graham Breed's temperament finder.  

Gene Ward Smith defines the TE error as the ratio ‖''M''_''W'' ∧ ''J''_''W''‖/‖''M''_''W''‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''_''W'' ∧ ''J''_''W''‖/‖''M''_''W''‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:  

Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''M''_''W'' ∧ ''J''_''W''‖_RMS}}. A perhaps simpler way to view this is to start with a mapping matrix ''V''_''W'' and add an extra row ''J''_''W'' corresponding to the just tuning map; we will label this matrix ''Ṽ''_''W''. Then the simple badness is:

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.

The exponent is chosen such that if we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.

Notice the extra factor 1/‖''J''_''U''‖, which is to say we divide it by the norm of the just tuning map. For comparison, Gene's derivation does not have this factor, whereas with Tenney weights, whether this factor is omitted or not has no effects on Graham's derivation since ‖''J''_''W''‖_RMS is unity.  

| 12.435×10⁻³

| 12.882×10⁻³

| 1.554×10⁻³

| 1.610×10⁻³

| 6.218×10⁻³

| 6.441×10⁻³

== Notes ==