Tenney–Euclidean temperament measures: Difference between revisions

Line 34:

$$ \norm{M_W}_2 = \sqrt {\abs{V_W V_W^\mathsf{T}}} $$

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

where {{!}}·{{!}} denotes the determinant, and {{t}} denotes the transpose.

We denote the RMS norm of ''M'' as ‖''M''‖RMS. In Graham ~~Breed~~'s paper<ref name="primerr">[http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''</ref>, an RMS norm is proposed as

Graham Breed and [[Gene Ward Smith]] have proposed different RMS norms. Let us denote the RMS norm of ''M'' as ‖''M''‖RMS. In Graham's paper<ref name="primerr">[http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''</ref>, an RMS norm is proposed as

$$ \norm{M_W}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M}_2}{\sqrt {n^r}} $$

$$ \norm{M_W}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M_W}_2}{\sqrt {n^r}} $$

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. Thus ''n''''r'' is the number of permutations of ''n'' things taken ''r'' at a time with repetition, which equals the number of entries of the wedgie in its full tensor form.

: '''Note''': that is the definition used by Graham Breed's temperament finder.

[[Gene Ward Smith]] has ~~recognized that TE complexity can be interpreted as the RMS norm of the wedgie. That defines~~ another RMS norm,

Gene Ward Smith has defined another RMS norm,

$$ \norm{M_W}_\text{RMS}' = \sqrt {\frac{\abs{V_W V_W^\mathsf{T}}}{C(n, r)}} = \frac {\norm{M}_2}{\sqrt {C(n, r)}} $$

$$ \norm{M_W}_\text{RMS}' = \sqrt {\frac{\abs{V_W V_W^\mathsf{T}}}{C(n, r)}} = \frac {\norm{M_W}_2}{\sqrt {C(n, r)}} $$

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

where {{nowrap|C(''n'', ''r'')}} is the number of combinations of ''n'' things taken ''r'' at a time without repetition, which equals the number of entries of the wedgie in the usual, compressed form.

: '''Note''': that is the definition currently used throughout the wiki, unless stated otherwise.

@@ Line 34: / Line 34: @@
 $$ \norm{M_W}_2 = \sqrt {\abs{V_W V_W^\mathsf{T}}} $$
-where {{!}}''A''{{!}} denotes the determinant of ''A'', and ''A''{{t}} denotes the transpose of ''A''.
+where {{!}}·{{!}} denotes the determinant, and {{t}} denotes the transpose.
-We denote the RMS norm of ''M'' as ‖''M''‖<sub>RMS</sub>. In Graham Breed's paper<ref name="primerr">[http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''</ref>, an RMS norm is proposed as
+Graham Breed and [[Gene Ward Smith]] have proposed different RMS norms. Let us denote the RMS norm of ''M'' as ‖''M''‖<sub>RMS</sub>. In Graham's paper<ref name="primerr">[http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''</ref>, an RMS norm is proposed as
-$$ \norm{M_W}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M}_2}{\sqrt {n^r}} $$
+$$ \norm{M_W}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M_W}_2}{\sqrt {n^r}} $$
-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. Thus ''n''<sup>''r''</sup> is the number of permutations of ''n'' things taken ''r'' at a time with repetition, which equals the number of entries of the wedgie in its full tensor form.
 : '''Note''': that is the definition used by Graham Breed's temperament finder.
-[[Gene Ward Smith]] has recognized that TE complexity can be interpreted as the RMS norm of the wedgie. That defines another RMS norm,
+Gene Ward Smith has defined another RMS norm,
-$$ \norm{M_W}_\text{RMS}' = \sqrt {\frac{\abs{V_W V_W^\mathsf{T}}}{C(n, r)}} = \frac {\norm{M}_2}{\sqrt {C(n, r)}} $$
+$$ \norm{M_W}_\text{RMS}' = \sqrt {\frac{\abs{V_W V_W^\mathsf{T}}}{C(n, r)}} = \frac {\norm{M_W}_2}{\sqrt {C(n, r)}} $$
-where {{nowrap|C(''n'', ''r'')}} is the number of combinations of ''n'' things taken ''r'' at a time, which equals the number of entries of the wedgie.
+where {{nowrap|C(''n'', ''r'')}} is the number of combinations of ''n'' things taken ''r'' at a time without repetition, which equals the number of entries of the wedgie in the usual, compressed form.
 : '''Note''': that is the definition currently used throughout the wiki, unless stated otherwise.