Tenney–Euclidean temperament measures: Difference between revisions

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The '''Tenney–Euclidean temperament measures''' ('''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness. These are evaluations of a temperament's [[complexity]], [[error]], and [[badness]], respectively. 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. Nonetheless, the following relationship always holds:

\text{TE simple badness} = \text{TE complexity} \times \text{TE error} </math>

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.

Let us denote a weighted mapping matrix, whose rows are the weighted vals ''v_i'', as ''V''. The ''L''² norm is one of the standard complexity measures:

<math>\displaystyle

\norm{ M }_2 = \sqrt {\abs{VV^\mathsf{T}}}</math>

where {{!}}''A''{{!}} denotes the determinant of ''A'', and ''V''{{t}} denotes the transpose of ''V''.  

\norm{ M }_\text{RMS} = \sqrt {\abs{\frac {VV^\mathsf{T}}{n}}} = \frac {\norm{ M }_2}{\sqrt {n^r}}</math>

\norm{ M }_\text{RMS}' = \sqrt {\frac{\abs{VV^\mathsf{T}}}{\binom{n}{r}}} = \frac {\norm{ M }_2}{\sqrt {\binom{n}{r}}}</math>

where ({{subsup||''r''|''n''}}) is the number of combinations of ''n'' things taken ''r'' at a time, which equals the number of entries of the wedgie.  

If ''W'' is a [[Wikipedia: Diagonal matrix|diagonal matrix]] with 1, 1/log₂3, …, 1/log₂''p'' along the diagonal and A is the matrix corresponding to V with unweighted vals as rows, then {{nowrap|''V'' {{=}} ''AW''}} and {{nowrap|{{!}}''VV''{{t}}{{!}} {{=}} {{!}}''AW''&#x2009;²''A''{{t}}{{!}}}}. This may be related to the [[Tenney–Euclidean metrics|TE tuning projection matrix]] ''P'', which is ''V''{{t}}(''VV''{{t}}){{inv}}''V'', and the corresponding matrix for unweighted monzos {{nowrap|''P'' {{=}} ''A''{{t}}(''AW''&#x2009;²''A''{{t}}){{inv}}''A''}}.

The '''TE simple badness''' of ''M'', which we may also call the '''relative error''' of ''M'', may be considered error relativized to the complexity of the temperament. It is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step.  

Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''J'' ∧ ''M''‖_RMS}}, where {{nowrap|''J'' {{=}} {{val| 1 1 … 1 }}}} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that {{nowrap|''a''_''i'' {{=}} ''J''·''v''_''i''/''n''}} is the mean value of the entries of ''v''_''i''. Then note that {{nowrap|''J'' ∧ (''v''₁ − ''a''₁''J'') ∧ (''v''₂ − ''a''₂''J'') ∧ … ∧ (''v''_''r'' − ''a''_''r''''J'') {{=}} ''J'' ∧ ''v''₁ ∧ ''v''₂ ∧ … ∧ ''v''_''r''}}, since wedge products with more than one term ''J'' are zero. The Gram matrix of the vectors ''J'' and {{nowrap|''v''₁ − ''a''_''i''''J''}} will have ''n'' as the (1,&nbsp;1) entry, and 0's in the rest of the first row and column. Hence we obtain:

\norm{ J \wedge M }'_\text {RMS} = \sqrt{\frac{n}{\binom{n}{r+1}}} \abs{v_i \cdot v_j - na_ia_j}</math>

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_J. Then the simple badness is:

\norm{ J \wedge M }'_\text {RMS} = \sqrt{\frac{n}{\binom{n}{r+1}}} \abs{V_J V_J^\mathsf{T}}</math>

This relationship also holds if TOP is used rather than TE, as the TOP damage associated with tempering some comma ''n''/''d'' is log(''n''/''d'')/(''nd''), and if we multiply by the complexity ''nd'', we simply get log(''n''/''d'') as our result.

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

= \norm{ J \wedge M } \norm{ M }^{r/(n - r)}

By Graham Breed's definition, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If ''T'' is the tuning map, then the TE error ''G'' can be found by

G &= \norm{ T - J }_\text{RMS} \\

&= \norm{ J(V^+V - I) }_\text{RMS} \\

&= \sqrt{J(V^+V - I)(V^+V - I)^\mathsf{T}J^\mathsf{T}/n}

If ''T'' is denominated in cents, then ''J'' should be also, so that {{nowrap|''J'' {{=}} {{val| 1200 1200 … 1200 }}}}. Here {{nowrap|''T'' − ''J''}} is the list of weighted mistunings of each prime harmonics.

By Gene Ward Smith's definition, the TE error is derived from the relationship of TE simple badness and TE complexity. We denote this definition of TE error Ψ.  

From the ratio {{nowrap|(‖''J'' ∧ ''M''‖ / ‖''M''‖)²}} we obtain {{nowrap|{{sfrac|({{subsup||''r'' + 1|''n''}})|''n''({{subsup||''r''|''n''}})}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank one with this for rank ''r'', the ''n'' cancels, and we get {{nowrap|{{sfrac|''n'' − 1|2}} · {{sfrac|''r'' + 1|''n'' − ''r''}} {{=}} {{sfrac|(''r'' + 1)(''n'' − 1)|2(''n'' − ''r'')}}}}. It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank-''r'' temperament then

\psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi</math>

is an '''adjusted error''' which makes the error of a rank ''r'' temperament correspond to the errors of the edo vals which support it; so that requiring the edo val error to be less than {{nowrap|(1 + ε)ψ}} for any positive ε results in an infinite set of vals supporting the temperament.  

Ψ, ψ, and ''G'' error can be related as follows:  

<math>\displaystyle G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi</math>

''G'' and ψ error both have the advantage that higher rank temperament error corresponds directly to rank one error, but the RMS normalization has the further advantage that in the rank one case, {{nowrap|''G'' {{=}} sin ''θ''}}, where ''θ'' is the angle between ''J'' 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.