Tenney–Euclidean temperament measures: Difference between revisions

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.

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, called ''Dirichlet coefficients'', 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.  

We may also note {{nowrap|{{!}}''V''_''W''''V''_''W''{{t}}{{!}} {{=}} {{!}}''VW''²''V''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P''_''W'', which is ''V''_''W''{{t}}(''V''_''W''''V''_''W''{{t}}){{inv}}''V''_''W'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''V''{{t}}(''VW''²''V''{{t}}){{inv}}''V''}}.

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:  

''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap|''G'' {{=}} sin ''θ''}}, where ''θ'' is the angle between ''J''_''W'' and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200&nbsp;sin(''θ''), the TE error in cents.

Some consider the simple badness to be a sort of badness which favors complex temperaments. The '''logflat badness''' is developed to address that. If we define ''B'' to be the simple badness (relative error) of a temperament, and ''C'' to be the complexity, then the logflat badness ''L'' is defined by the formula

In Sintel's Dirichlet coefficients, or Dirichlet badness,  

The different definitions yield different results, but they are related to each other by a factor derived only from the rank and limit. A meaningful comparison of temperaments in the same rank and limit can be provided by picking any one of them.  

Here is a demonstration from [[7-limit]] [[magic]] and [[meantone]], comparing each of the definitions.  

 

{| class="wikitable center-all"

|+ style="font-size: 105%;" | 7-limit magic (left) vs. meantone (right) in TE temperament measures

! {{W|Norm (mathematics) #Euclidean norm|Standard ''L''² norm}}

| 1.799 : 1.350

| 1.074 : 1.382

| 1.610×10⁻³ : 1.554×10⁻³

| 2.937 : 2.204

| 2.631 : 3.384

| 6.441×10⁻³ : 6.218×10⁻³