Tenney–Euclidean temperament measures: Difference between revisions

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</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.

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 Graham Breed's temperament finder.

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

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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~~.

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'', the TE error in cents.

== Examples of each definition ==

@@ Line 108: / Line 108: @@
 </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.
+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 Graham Breed's temperament finder.
 Ψ, ψ and G error can be related as follows:
@@ Line 114: / Line 114: @@
 <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.
+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'', the TE error in cents.
 == Examples of each definition ==