Tenney–Euclidean temperament measures: Difference between revisions

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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 (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament.

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

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

<math>\displaystyle

G = \lVert T - J \rVert_\text{RMS} = \~~sqrt~~{\~~frac~~{(T - J) ~~\cdot~~ (T - J)^\mathsf{T}}{n}}</math>

\begin{align}

G &= \lVert T - J \rVert_\text{RMS} \\

&= \lVert J(V^+V - I) \rVert_\text{RMS} \\

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

\end{align}

</math>

where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = {{val|1200 1200 … 1200}}. Here T - J is the list of weighted mistunings of each prime harmonics. Note: this is the definition used by the temperament finder.

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<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, 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 sin θ, TE error as it appears on the temperament finder pages.

== Examples of each definition ==

@@ Line 91: / Line 91: @@
 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 (1 + ε)ψ for any positive ε results in an infinite set of vals supporting the temperament.
-By Graham Breed's definition, TE error may be accessed directly via [[Tenney-Euclidean Tuning|TE tuning map]]. If T is the tuning map, then the TE error G can be found by
+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
 <math>\displaystyle
-G = \lVert T - J \rVert_\text{RMS} = \sqrt{\frac{(T - J) \cdot (T - J)^\mathsf{T}}{n}}</math>
+\begin{align}
+G &= \lVert T - J \rVert_\text{RMS} \\
+&= \lVert J(V^+V - I) \rVert_\text{RMS} \\
+&= \sqrt{J(V^+V - I)(V^+V - I)^\mathsf{T}J^\mathsf{T}/n}
+\end{align}
+</math>
 where the dot represents the ordinary dot product. If T is denominated in cents, then J should be also, so that J = {{val|1200 1200 … 1200}}. Here T - J is the list of weighted mistunings of each prime harmonics. Note: this is the definition used by the temperament finder.
@@ Line 102: / Line 107: @@
 <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,  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 sin θ, TE error as it appears on the temperament finder pages.
 == Examples of each definition ==