Tenney–Euclidean temperament measures: Difference between revisions

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

Gene Ward Smith defines TE error as the ratio ‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''.

Gene Ward Smith defines the TE error as the ratio ‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:

From the ratio {{nowrap|(‖''M''''W'' ∧ ''J''''W''‖/‖''M''''W''‖)2}} we obtain {{nowrap|{{sfrac|''C''(''n'', ''r'' + 1)|''n''⋅''C''(''n'', ''r'')}} {{=}} {{~~sfrac|''n'' − ''r''|''n''(''~~r'' + 1~~)}}}~~}~~. If we take the ratio of this for rank 1 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

$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$

~~$$ \psi = \sqrt{\frac~~{~~2(n - r)}~~{(~~r + 1)(~~n - 1)}} ~~\Psi $$~~

Also, if we set the rank ''r'' to 1, we get {{nowrap| (''n'' − 1)/2 }}. It follows that dividing TE error by this value gives a constant of proportionality such that

~~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.~~

$$ \psi = \sqrt{\frac{2}{n - 1}} E $$

~~To express~~ ''Ψ'' ~~and~~ ''ψ'' ~~in terms~~ of ''E'':

gives another error, called the ''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.

~~$$ \Psi = \sqrt{\frac{r + 1~~}~~{n - r~~}~~} E, \ \psi = \sqrt{\frac{2}{n - 1}} E $$~~

''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap|''G'' {{=}} sin ''θ''}}, where ''θ'' is the angle between ''J''''W'' and the val in question. Multiplying by 1200 to obtain a result in cents leads to 1200 sin(''θ''), the TE error in cents.

@@ Line 86: / Line 86: @@
 : '''Note''': that is the definition used by Graham Breed's temperament finder.
-Gene Ward Smith defines TE error as the ratio ‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''.
+Gene Ward Smith defines the TE error as the ratio ‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows:
-From the ratio {{nowrap|(‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖)<sup>2</sup>}} we obtain {{nowrap|{{sfrac|''C''(''n'', ''r'' + 1)|''n''⋅''C''(''n'', ''r'')}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank 1 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
+$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$
-$$ \psi = \sqrt{\frac{2(n - r)}{(r + 1)(n - 1)}} \Psi $$
+Also, if we set the rank ''r'' to 1, we get {{nowrap| (''n'' − 1)/2 }}. It follows that dividing TE error by this value gives a constant of proportionality such that
-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.
+$$ \psi = \sqrt{\frac{2}{n - 1}} E $$
-To express ''Ψ'' and ''ψ'' in terms of ''E'':
+gives another error, called the ''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.
-$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E, \ \psi = \sqrt{\frac{2}{n - 1}} E $$
 ''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap|''G'' {{=}} sin ''θ''}}, where ''θ'' is the angle between ''J''<sub>''W''</sub> 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.