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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:  
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:  


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<math>\displaystyle
<math>\displaystyle
\lVert M \rVert_2 = \sqrt {\left|VV^\mathsf{T}\right|}</math>
\norm{ M }_2 = \sqrt {\abs{VV^\mathsf{T}}}</math>


where {{!}}''A''{{!}} denotes the determinant of ''A'', and ''V''{{t}} denotes the transpose of ''V''.  
where {{!}}''A''{{!}} denotes the determinant of ''A'', and ''V''{{t}} denotes the transpose of ''V''.  
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<math>\displaystyle
<math>\displaystyle
\lVert M \rVert_\text{RMS} = \sqrt {\left|\frac {VV^\mathsf{T}}{n}\right|} = \frac {\lVert M \rVert_2}{\sqrt {n^r}}</math>
\norm{ M }_\text{RMS} = \sqrt {\abs{\frac {VV^\mathsf{T}}{n}}} = \frac {\norm{ M }_2}{\sqrt {n^r}}</math>


where ''n'' is the number of primes up to the prime limit ''p'', and ''r'' is the rank of the temperament, which equals the number of vals wedged together to compute the wedgie.  
where ''n'' is the number of primes up to the prime limit ''p'', and ''r'' is the rank of the temperament, which equals the number of vals wedged together to compute the wedgie.  
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<math>\displaystyle
<math>\displaystyle
\lVert M \rVert_\text{RMS}' = \sqrt {\frac{\left|VV^\mathsf{T}\right|}{\binom{n}{r}}} = \frac {\lVert M \rVert_2}{\sqrt {\binom{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.  
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.  
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<math>\displaystyle
<math>\displaystyle
\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{\binom{n}{r+1}}} \left|v_i \cdot v_j - na_ia_j\right|</math>
\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<sub>J</sub>. Then the simple badness is:
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<sub>J</sub>. Then the simple badness is:


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


So that we can basically view the simple badness as the TE complexity of the "pseudo-temperament" formed by adding the JIP to the mapping matrix as if it were another val.
So that we can basically view the simple badness as the TE complexity of the "pseudo-temperament" formed by adding the JIP to the mapping matrix as if it were another val.
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<math>\displaystyle
<math>\displaystyle
S(A)C(A)^{r/(n - r)} \\
S(A)C(A)^{r/(n - r)} \\
= \lVert J \wedge M \rVert \lVert M \rVert^{r/(n - r)}
= \norm{ J \wedge M } \norm{ M }^{r/(n - r)}
</math>
</math>


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<math>\displaystyle
<math>\displaystyle
\begin{align}
\begin{align}
G &= \lVert T - J \rVert_\text{RMS} \\
G &= \norm{ T - J }_\text{RMS} \\
&= \lVert J(V^+V - I) \rVert_\text{RMS} \\
&= \norm{ J(V^+V - I) }_\text{RMS} \\
&= \sqrt{J(V^+V - I)(V^+V - I)^\mathsf{T}J^\mathsf{T}/n}
&= \sqrt{J(V^+V - I)(V^+V - I)^\mathsf{T}J^\mathsf{T}/n}
\end{align}
\end{align}
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