Tenney–Euclidean temperament measures: Difference between revisions

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The '''TE simple badness''' of M, which we may also call the '''relative error''' of M, may be considered error relativized to the complexity of the temperament. It is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step.

Gene Ward Smith defines the simple badness of M as ||J∧M||RMS, where J = {{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 a''i'' = J·v''i''/''n'' is the mean value of the entries of v''i''. Then note that J∧(v1 - a1J)∧(v2 - a2J)∧…∧(v''r'' - a''r''J) = J∧v1∧v2∧...∧v''r'', since wedge products with more than one term J are zero. The Gram matrix of the vectors J and v1 - a''i''J will have ''n'' as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:

Gene Ward Smith defines the simple badness of M as ||J∧M||RMS, where J = {{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 a''i'' = J·v''i''/''n'' is the mean value of the entries of v''i''. Then note that J∧(v1 - a1J)∧(v2 - a2J)∧…∧(v''r'' - a''r''J) = J∧v1∧v2∧...∧v''r'', since wedge products with more than one term J are zero. The Gram matrix of the vectors J and v1 - a''i''J will have ''n'' as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:

<math>\displaystyle

\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</math>

\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} \operatorname {det}([v_i \cdot v_j - na_ia_j])</math>

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

<math>\displaystyle

\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} det(V_J V_J^T)</math>

\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} \operatorname {det}(V_J V_J^T)</math>

So that we can basically view the simple badness as the TE complexity of the "pseudo-temperament" formed by adding the JIP to the mapping matrix as if it were another val.

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<math>\displaystyle

S(A)C(A)^{r/(n-r)}

S(A)C(A)^{r/(n - r)} \\

= \lVert J \wedge M \rVert \lVert M \rVert^{r/(n - r)}

</math>

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 The '''TE simple badness''' of M, which we may also call the '''relative error''' of M, may be considered error relativized to the complexity of the temperament. It is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step.
-Gene Ward Smith defines the simple badness of M as ||J∧M||<sub>RMS</sub>, where J = {{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 a<sub>''i''</sub> = J·v<sub>''i''</sub>/''n'' is the mean value of the entries of v<sub>''i''</sub>. Then note that J∧(v<sub>1</sub> - a<sub>1</sub>J)∧(v<sub>2</sub> - a<sub>2</sub>J)∧…∧(v<sub>''r''</sub> - a<sub>''r''</sub>J) = J∧v<sub>1</sub>∧v<sub>2</sub>∧...∧v<sub>''r''</sub>, since wedge products with more than one term J are zero. The Gram matrix of the vectors J and v<sub>1</sub> - a<sub>''i''</sub>J will have ''n'' as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:
+Gene Ward Smith defines the simple badness of M as ||J∧M||<sub>RMS</sub>, where J = {{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 a<sub>''i''</sub> = J·v<sub>''i''</sub>/''n'' is the mean value of the entries of v<sub>''i''</sub>. Then note that J∧(v<sub>1</sub> - a<sub>1</sub>J)∧(v<sub>2</sub> - a<sub>2</sub>J)∧…∧(v<sub>''r''</sub> - a<sub>''r''</sub>J) = J∧v<sub>1</sub>∧v<sub>2</sub>∧...∧v<sub>''r''</sub>, since wedge products with more than one term J are zero. The Gram matrix of the vectors J and v<sub>1</sub> - a<sub>''i''</sub>J will have ''n'' as the (1,1) entry, and 0s in the rest of the first row and column. Hence we obtain:
 <math>\displaystyle
-\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} det([v_i \cdot v_j - na_ia_j])</math>
+\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} \operatorname {det}([v_i \cdot v_j - na_ia_j])</math>
 A perhaps simpler way to view this is to start with a mapping matrix V and add an extra row J corresponding to the JIP; we will label this matrix V<sub>J</sub>. Then the simple badness is:
 <math>\displaystyle
-\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} det(V_J V_J^T)</math>
+\lVert J \wedge M \rVert'_\text {RMS} = \sqrt{\frac{n}{C(n,r+1)}} \operatorname {det}(V_J V_J^T)</math>
 So that we can basically view the simple badness as the TE complexity of the "pseudo-temperament" formed by adding the JIP to the mapping matrix as if it were another val.
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 <math>\displaystyle
-S(A)C(A)^{r/(n-r)}
+S(A)C(A)^{r/(n - r)} \\
+= \lVert J \wedge M \rVert \lVert M \rVert^{r/(n - r)}
 </math>