Tenney–Euclidean temperament measures: Difference between revisions

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== TE complexity ==

Given a [[wedgie]] ''M'', that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ‖''M''‖ is a measure of the [[complexity]] of ''M''; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave~~. We may call it '''Tenney–Euclidean complexity''', or '''TE complexity''' since it can be defined in terms of the [[Tenney–Euclidean metrics|Tenney–Euclidean norm]]~~.

Given a [[wedgie]] ''M'', that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ‖''M''‖ is a measure of the complexity of ''M''; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave.

Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log23, 1/log25 … 1/log2''p'' along the diagonal. For the prime basis {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }},

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$$ W = \operatorname {diag} (1/\log_2 (Q)) $$

If ''V'' is the mapping matrix of a temperament, then ''VW'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals ('''v'''''w'')''i''.

If ''V'' is the mapping matrix of a temperament, then ''VW'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals (''v''''w'')''i''.

Our first complexity measure of a temperament is given by the ''L''2 norm of the Tenney-weighted wedgie ''M''''W'', which can in turn be obtained from the Tenney-weighted mapping matrix ''V''''W''. This complexity can be easily computed either from the wedgie or from the mapping matrix, using the {{w|Gramian matrix|Gramian}}:

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== TE error ==

We can consider ~~'''~~TE error~~'''~~ to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in TE tuning. In this regard, TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s.

We can consider TE error to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in the tuning where it is minimized. In this regard, TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s.

~~By Graham Breed~~'~~s definition~~<~~ref name="primerr"~~/>, TE error ~~may be accessed via [[Tenney–Euclidean tuning~~|TE tuning map~~]]. If~~ ''T''''W'' is the ~~Tenney-~~weighted tuning map, then the TE error ''G'' can be found by

As with complexity, we may simply define the TE error as the ''L''2 norm of the weighted TE error map. If {{nowrap| ''T''''W'' {{=}} ''TW'' }} is the weighted TE tuning map and {{nowrap| ''J''''W'' {{=}} ''JW'' {{=}} {{val| 1 1 … 1 }} }} is the weighted just tuning map, then the TE error ''E'' is given by

$$

\begin{align}

E &= \norm{T_W - J_W}_2 \\

&= \norm{J_W(V_W^+ V_W - I) }_2 \\

&= \sqrt{J_W(V_W^+ V_W - I)(V_W^+ V_W - I)^\mathsf{T} J_W^\mathsf{T}}

\end{align}

$$

where + denotes the [[pseudoinverse]].

Often, it is desirable to know the average of errors instead of the sum, which corresponds to Graham Breed's definition<ref name="primerr"/>. This error figure, ''G'', can be found by

$$

\begin{align}

G &= \norm{T_W - J_W}_\text{RMS} \\

&= ~~\norm{J_W(V_W^+ V_W - I) }_\text{RMS} \\~~

&= E / \sqrt{n}

&= \sqrt{~~J_W(V_W^+ V_W - I)(V_W^+ V_W - I)^\mathsf{T} J_W^\mathsf{T}/~~n}

\end{align}

$$

If ''T''''W'' is denominated in cents, then ''J''''W'' should be also, so that {{nowrap|''J''''W'' {{=}} {{val| 1200 1200 … 1200 }}}}. Here {{nowrap|''T''''W'' − ''J''''W''}} is the list of weighted errors of each prime harmonic.

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

By Gene Ward Smith~~'s definition,~~ TE error ~~is 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 derives TE error from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''.

From the ratio {{nowrap|(‖''J''''W'' ∧ ''M''‖ / ‖''M''‖)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

From the ratio {{nowrap|(‖''J''''W'' ∧ ''M''''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

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

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

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.

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.

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

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

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

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

''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.

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$$ B = C \cdot E $$

Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''J''''W'' ∧ ''M''''W''‖RMS}}, where {{nowrap|''J''''W'' ~~{{=}} {{val| 1 1 … 1 }}~~}} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that {{nowrap|''a''''i'' {{=}} ''J''''W''·('''v'''''w'')''i''/''n''}} is the mean value of the entries of ('''v'''''w'')''i''. Then note that {{nowrap|''J''''W'' ∧ (('''v'''''w'')1 − ''a''1''J''''W'') ∧ (('''v'''''w'')2 − ''a''2''J''''W'') ∧ … ∧ (('''v'''''w'')''r'' − ''a''''r''''J''''W'') {{=}} ''J''''W'' ∧ ('''v'''''w'')1 ∧ ('''v'''''w'')2 ∧ … ∧ ('''v'''''w'')''r''}}, since wedge products with more than one term ''J''''W'' are zero. The Gram matrix of the vectors ''J''''W'' and {{nowrap|('''v'''''w'')1 − ''a''''i''''J''''W''}} will have ''n'' as the {{nowrap|(1, 1)}} entry, and 0's in the rest of the first row and column. Hence we obtain:

Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''J''''W'' ∧ ''M''''W''‖RMS}}, where {{nowrap|''J''''W'' }} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that {{nowrap|''a''''i'' {{=}} ''J''''W''·(''v''''w'')''i''/''n''}} is the mean value of the entries of (''v''''w'')''i''. Then note that {{nowrap|''J''''W'' ∧ ((''v''''w'')1 − ''a''1''J''''W'') ∧ ((''v''''w'')2 − ''a''2''J''''W'') ∧ … ∧ ((''v''''w'')''r'' − ''a''''r''''J''''W'') {{=}} ''J''''W'' ∧ (''v''''w'')1 ∧ (''v''''w'')2 ∧ … ∧ (''v''''w'')''r''}}, since wedge products with more than one term ''J''''W'' are zero. The Gram matrix of the vectors ''J''''W'' and {{nowrap|(''v''''w'')1 − ''a''''i''''J''''W''}} will have ''n'' as the {{nowrap|(1, 1)}} entry, and 0's in the rest of the first row and column. Hence we obtain:

$$ \norm{ J_W \wedge M_W }'_\text {RMS} = \sqrt{\frac{n}{C(n, r + 1)}} \det((~~\vec{~~v_w})_i \cdot (~~\vec{~~v_w})_j - n a_i a_j) $$

$$ \norm{ J_W \wedge M_W }'_\text {RMS} = \sqrt{\frac{n}{C(n, r + 1)}} \det((v_w)_i \cdot (v_w)_j - n a_i a_j) $$

A perhaps simpler way to view this is to start with a mapping matrix ''V''''W'' and add an extra row ''J''''W'' corresponding to the JIP; we will label this matrix ''V''''J''. Then the simple badness is:

@@ Line 22: / Line 22: @@
 == TE complexity ==
-Given a [[wedgie]] ''M'', that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ‖''M''‖ is a measure of the [[complexity]] of ''M''; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. We may call it '''Tenney–Euclidean complexity''', or '''TE complexity''' since it can be defined in terms of the [[Tenney–Euclidean metrics|Tenney–Euclidean norm]].
+Given a [[wedgie]] ''M'', that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ‖''M''‖ is a measure of the complexity of ''M''; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave.
 Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the prime basis {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }},
@@ Line 28: / Line 28: @@
 $$ W = \operatorname {diag} (1/\log_2 (Q)) $$
-If ''V'' is the mapping matrix of a temperament, then ''V<sub>W</sub>'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals ('''v'''<sub>''w''</sub>)<sub>''i''</sub>.
+If ''V'' is the mapping matrix of a temperament, then ''V<sub>W</sub>'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals (''v''<sub>''w''</sub>)<sub>''i''</sub>.
 Our first complexity measure of a temperament is given by the ''L''<sup>2</sup> norm of the Tenney-weighted wedgie ''M''<sub>''W''</sub>, which can in turn be obtained from the Tenney-weighted mapping matrix ''V''<sub>''W''</sub>. This complexity can be easily computed either from the wedgie or from the mapping matrix, using the {{w|Gramian matrix|Gramian}}:
@@ Line 61: / Line 61: @@
 == TE error ==
-We can consider '''TE error''' to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in TE tuning. In this regard, TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s.
+We can consider TE error to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in the tuning where it is minimized. In this regard, TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s.
-By Graham Breed's definition<ref name="primerr"/>, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If ''T''<sub>''W''</sub> is the Tenney-weighted tuning map, then the TE error ''G'' can be found by
+As with complexity, we may simply define the TE error as the ''L''<sup>2</sup> norm of the weighted TE error map. If {{nowrap| ''T''<sub>''W''</sub> {{=}} ''TW'' }} is the weighted TE tuning map and {{nowrap| ''J''<sub>''W''</sub> {{=}} ''JW'' {{=}} {{val| 1 1 … 1 }} }} is the weighted just tuning map, then the TE error ''E'' is given by
+$$
+\begin{align}
+E &= \norm{T_W - J_W}_2 \\
+&= \norm{J_W(V_W^+ V_W - I) }_2 \\
+&= \sqrt{J_W(V_W^+ V_W - I)(V_W^+ V_W - I)^\mathsf{T} J_W^\mathsf{T}}
+\end{align}
+$$
+where <sup>+</sup> denotes the [[pseudoinverse]].
+Often, it is desirable to know the average of errors instead of the sum, which corresponds to Graham Breed's definition<ref name="primerr"/>. This error figure, ''G'', can be found by
 $$
 \begin{align}
 G &= \norm{T_W - J_W}_\text{RMS} \\
-&= \norm{J_W(V_W^+ V_W - I) }_\text{RMS} \\
+&= E / \sqrt{n}
-&= \sqrt{J_W(V_W^+ V_W - I)(V_W^+ V_W - I)^\mathsf{T} J_W^\mathsf{T}/n}
 \end{align}
 $$
-If ''T''<sub>''W''</sub> is denominated in cents, then ''J''<sub>''W''</sub> should be also, so that {{nowrap|''J''<sub>''W''</sub> {{=}} {{val| 1200 1200 … 1200 }}}}. Here {{nowrap|''T''<sub>''W''</sub> − ''J''<sub>''W''</sub>}} is the list of weighted errors of each prime harmonic.
 : '''Note''': that is the definition used by Graham Breed's temperament finder.
-By Gene Ward Smith's definition, TE error is 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 derives TE error from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''.
-From the ratio {{nowrap|(‖''J''<sub>''W''</sub> ∧ ''M''‖ / ‖''M''‖)<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
+From the ratio {{nowrap|(‖''J''<sub>''W''</sub> ∧ ''M''<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
-$$ \psi = \sqrt{\frac{2(n-r)}{(r+1)(n-1)}} \Psi $$
+$$ \psi = \sqrt{\frac{2(n - r)}{(r + 1)(n - 1)}} \Psi $$
-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.
+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.
-''Ψ'', ''ψ'', and ''G'' error can be related as follows:
+To express ''Ψ'' and ''ψ'' in terms of ''E'':
-$$ G = \sqrt{\frac{n-1}{2n}} \psi = \sqrt{\frac{n-r}{(r+1)n}} \Psi $$
+$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E, \ \psi = \sqrt{\frac{2}{n - 1}} E $$
 ''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.
@@ Line 98: / Line 107: @@
 $$ B = C \cdot E $$
-Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''J''<sub>''W''</sub> ∧ ''M''<sub>''W''</sub>‖<sub>RMS</sub>}}, where {{nowrap|''J''<sub>''W''</sub> {{=}} {{val| 1 1 … 1 }}}} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that {{nowrap|''a''<sub>''i''</sub> {{=}} ''J''<sub>''W''</sub>·('''v'''<sub>''w''</sub>)<sub>''i''</sub>/''n''}} is the mean value of the entries of ('''v'''<sub>''w''</sub>)<sub>''i''</sub>. Then note that {{nowrap|''J''<sub>''W''</sub> ∧ (('''v'''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>1</sub>''J''<sub>''W''</sub>) ∧ (('''v'''<sub>''w''</sub>)<sub>2</sub> − ''a''<sub>2</sub>''J''<sub>''W''</sub>) ∧ … ∧ (('''v'''<sub>''w''</sub>)<sub>''r''</sub> − ''a''<sub>''r''</sub>''J''<sub>''W''</sub>) {{=}} ''J''<sub>''W''</sub> ∧ ('''v'''<sub>''w''</sub>)<sub>1</sub> ∧ ('''v'''<sub>''w''</sub>)<sub>2</sub> ∧ … ∧ ('''v'''<sub>''w''</sub>)<sub>''r''</sub>}}, since wedge products with more than one term ''J''<sub>''W''</sub> are zero. The Gram matrix of the vectors ''J''<sub>''W''</sub> and {{nowrap|('''v'''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>''i''</sub>''J''<sub>''W''</sub>}} will have ''n'' as the {{nowrap|(1, 1)}} entry, and 0's in the rest of the first row and column. Hence we obtain:
+Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''J''<sub>''W''</sub> ∧ ''M''<sub>''W''</sub>‖<sub>RMS</sub>}}, where {{nowrap|''J''<sub>''W''</sub> }} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that {{nowrap|''a''<sub>''i''</sub> {{=}} ''J''<sub>''W''</sub>·(''v''<sub>''w''</sub>)<sub>''i''</sub>/''n''}} is the mean value of the entries of (''v''<sub>''w''</sub>)<sub>''i''</sub>. Then note that {{nowrap|''J''<sub>''W''</sub> ∧ ((''v''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>1</sub>''J''<sub>''W''</sub>) ∧ ((''v''<sub>''w''</sub>)<sub>2</sub> − ''a''<sub>2</sub>''J''<sub>''W''</sub>) ∧ … ∧ ((''v''<sub>''w''</sub>)<sub>''r''</sub> − ''a''<sub>''r''</sub>''J''<sub>''W''</sub>) {{=}} ''J''<sub>''W''</sub> ∧ (''v''<sub>''w''</sub>)<sub>1</sub> ∧ (''v''<sub>''w''</sub>)<sub>2</sub> ∧ … ∧ (''v''<sub>''w''</sub>)<sub>''r''</sub>}}, since wedge products with more than one term ''J''<sub>''W''</sub> are zero. The Gram matrix of the vectors ''J''<sub>''W''</sub> and {{nowrap|(''v''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>''i''</sub>''J''<sub>''W''</sub>}} will have ''n'' as the {{nowrap|(1, 1)}} entry, and 0's in the rest of the first row and column. Hence we obtain:
-$$ \norm{ J_W \wedge M_W }'_\text {RMS} = \sqrt{\frac{n}{C(n, r + 1)}} \det((\vec{v_w})_i \cdot (\vec{v_w})_j - n a_i a_j) $$
+$$ \norm{ J_W \wedge M_W }'_\text {RMS} = \sqrt{\frac{n}{C(n, r + 1)}} \det((v_w)_i \cdot (v_w)_j - n a_i a_j) $$
 A perhaps simpler way to view this is to start with a mapping matrix ''V''<sub>''W''</sub> and add an extra row ''J''<sub>''W''</sub> corresponding to the JIP; we will label this matrix ''V''<sub>''J''</sub>. Then the simple badness is: