Structure metric: Difference between revisions

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=== Sparsity ===

If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the ''average distance'' in the space is S/(2(n-1)(n-2)), and the ''sparsity'' of the space is S/(2(n-1)^2(n-2)). The sparsity is 1 when all points are at the same distance, but otherwise less.

If '''D''' is the distance matrix of a finite metric space of ''n'' points, let ''S'' be the sum of elements of '''D'''. '''S''' can also be described as twice the sum of all the distances in the metric since these are counted twice in '''D'''. Then, the ''average distance'' in the space is {{sfrac|''S''|2(''n'' − 1)(''n'' − 2)}}, and the ''sparsity'' of the space is {{sfrac|''S''|2(''n'' − 1)<sup>2</sup>(''n'' − 2)}}. The sparsity is 1 when all points are at the same distance, but otherwise less.

Tempering will often shrink distances and so increase density. For example, the duodene has a sparsity of 0.3686. Tempering by [[Srutal~~|srutal~~]], where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparsity becomes 0. To give another example, [[~~pentadekany2|~~pentadekany2]], which is Cps([2,3,5,7,9,11], 3), has a sparsity of 0.4796; tempering out 3025/3024 lowers that to 0.4772; tempering further to portent (which tempers out 385/384, 441/440 and 1029/1024 as well as 3025/3024) lowers that to 0.4521, ~~miracle~~ tempering brings it down to 0.4286, and 72et brings that down to 0.4282. If A and B are two metric matricies for the same set of points, then A ''dominates'' B if A - B has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparsity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.

Tempering will often shrink distances and so increase density. For example, the duodene has a sparsity of 0.3686. Tempering by [[Srutal]], where 2048/2025 is tempered out, reduces that to 0.2860, and tempering by meantone gives 0.2364. Tempering both gives 12et, and the sparsity becomes 0. To give another example, [[pentadekany2]], which is {{nowrap|Cps([2, 3, 5, 7, 9, 11], 3)}}, has a sparsity of 0.4796; tempering out 3025/3024 lowers that to 0.4772; tempering further to portent (which tempers out 385/384, 441/440, and 1029/1024 as well as 3025/3024) lowers that to 0.4521, Miracle tempering brings it down to 0.4286, and 72et brings that down to 0.4282. If '''A''' and '''B''' are two metric matricies for the same set of points, then '''A''' ''dominates'' '''B''' if {{nowrap|'''A''' − '''B'''}} has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparsity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.

An invariant related to sparsity is ''spread''. If n is a point, define the spread polynomial of n to be the sum sp(n) = ∑ t^~~'''~~d~~'''~~(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function ~~sprea'''d'''~~(t) = ~~∑1/~~sp(n) over all points n. Spread as a function decreases between 0 and 1, with ~~sprea'''d'''~~(0) = '''P''', the number of notes in the scale and therefore points in the space, and ~~sprea'''d'''~~(1) = 1. We can think of t = 0 as the highest magnification, with each of the points showing clearly, and t = 1 as the lowest, where all points have merged together. In between, at t = 1/2 or (a traditional choice, for some reason) t = exp(-1), we have a sparsity measure. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. This inflection is related to the fact that spread for scales tends to be relatively ~~large--notes~~ tend to be far apart. Especially for larger scales, the tendency of spread to stick close to the maximum value for most values of t in the range 0 to 1 is striking.

An invariant related to sparsity is ''spread''. If ''n'' is a point, define the spread polynomial of ''n'' to be the sum <math>\operatorname{sp}(n) = \sum t^{\mathbf{d}(n, i)}</math> over all points ''i'', where ''t'' is an indeterminate. Then the spread is the rational function <math>\operatorname{spr}(n) = \sum \frac{1}{\operatorname{sp}(n)}</math> over all points ''n''. Spread as a function decreases between 0 and 1, with {{nowrap|spr(0) {{=}} '''P'''}}, the number of notes in the scale and therefore points in the space, and {{nowrap|spr(1) {{=}} 1}}. We can think of {{nowrap|''t'' {{=}} 0}} as the highest magnification, with each of the points showing clearly, and {{nowrap|''t'' {{=}} 1|| as the lowest, where all points have merged together. In between, at {{nowrap|''t'' {{=}} 1/2}} or (a traditional choice, for some reason) {{nowrap|''t'' {{=}} exp(−1)}}, we have a sparsity measure. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. This inflection is related to the fact that spread for scales tends to be relatively large—notes tend to be far apart. Especially for larger scales, the tendency of spread to stick close to the maximum value for most values of ''t'' in the range 0 to 1 is striking.

In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance~~, sprea'''d'''~~(Euler(3*5)) = 4/(t^3 + 2t^2 +1~~), sprea'''d'''~~(Euler(3*5*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1~~), sprea'''d'''~~(hexany) = 6/(t^6 + 4t^4 + 1~~), sprea'''d'''~~(dekany) = 10/(3t^9 + 6t^7 + 1~~), sprea'''d'''~~(pentadekany) = 16/(6t^14 + 8t^11 + 1~~), sprea'''d'''~~(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).

In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance:

<math>

\begin{align}

\operatorname{spr}\left(\operatorname{Euler}(3*5)\right) &= \frac{4}{t^3 + 2t^2 + 1} \\

\operatorname{spr}\left(\operatorname{Euler}(3*5*7)\right) &= \frac{8}{t^7 + 3t^6 + 3t^4 + 1} \\

\operatorname{spr}(\text{hexany}) &= \frac{6}{t^6 + 4t^4 + 1} \\

\operatorname{spr}(\text{dekany}) &= \frac{10}{3t^9 + 6t^7 + 1} \\

\operatorname{spr}(\text{pentadekany}) &= \frac{16}{6t^{14} + 8t^{11} + 1} \\

\operatorname{spr}(\text{eikosany}) &= \frac{20}{t^{19} + 9t^{18} + 9t^{14} + 1}

\end{align}

</math>

@@ Line 130: / Line 130: @@
 === Sparsity ===
-If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the ''average distance'' in the space is S/(2(n-1)(n-2)), and the ''sparsity'' of the space is S/(2(n-1)^2(n-2)). The sparsity is 1 when all points are at the same distance, but otherwise less.
+If '''D''' is the distance matrix of a finite metric space of ''n'' points, let ''S'' be the sum of elements of '''D'''. '''S''' can also be described as twice the sum of all the distances in the metric since these are counted twice in '''D'''. Then, the ''average distance'' in the space is {{sfrac|''S''|2(''n'' − 1)(''n'' − 2)}}, and the ''sparsity'' of the space is {{sfrac|''S''|2(''n'' − 1)<sup>2</sup>(''n'' − 2)}}. The sparsity is 1 when all points are at the same distance, but otherwise less.
-Tempering will often shrink distances and so increase density. For example, the duodene has a sparsity of 0.3686. Tempering by [[Srutal|srutal]], where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparsity becomes 0. To give another example, [[pentadekany2|pentadekany2]], which is Cps([2,3,5,7,9,11], 3), has a sparsity of 0.4796; tempering out 3025/3024 lowers that to 0.4772; tempering further to portent (which tempers out 385/384, 441/440 and 1029/1024 as well as 3025/3024) lowers that to 0.4521, miracle tempering brings it down to 0.4286, and 72et brings that down to 0.4282. If A and B are two metric matricies for the same set of points, then A ''dominates'' B if A - B has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparsity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.
+Tempering will often shrink distances and so increase density. For example, the duodene has a sparsity of 0.3686. Tempering by [[Srutal]], where 2048/2025 is tempered out, reduces that to 0.2860, and tempering by meantone gives 0.2364. Tempering both gives 12et, and the sparsity becomes 0. To give another example, [[pentadekany2]], which is {{nowrap|Cps([2, 3, 5, 7, 9, 11], 3)}}, has a sparsity of 0.4796; tempering out 3025/3024 lowers that to 0.4772; tempering further to portent (which tempers out 385/384, 441/440, and 1029/1024 as well as 3025/3024) lowers that to 0.4521, Miracle tempering brings it down to 0.4286, and 72et brings that down to 0.4282. If '''A''' and '''B''' are two metric matricies for the same set of points, then '''A''' ''dominates'' '''B''' if {{nowrap|'''A''' − '''B'''}} has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparsity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.
-An invariant related to sparsity is ''spread''. If n is a point, define the spread polynomial of n to be the sum sp(n) = ∑ t^'''d'''(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function sprea'''d'''(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with sprea'''d'''(0) = '''P''', the number of notes in the scale and therefore points in the space, and sprea'''d'''(1) = 1. We can think of t = 0 as the highest magnification, with each of the points showing clearly, and t = 1 as the lowest, where all points have merged together. In between, at t = 1/2 or (a traditional choice, for some reason) t = exp(-1), we have a sparsity measure. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. This inflection is related to the fact that spread for scales tends to be relatively large--notes tend to be far apart. Especially for larger scales, the tendency of spread to stick close to the maximum value for most values of t in the range 0 to 1 is striking.
+An invariant related to sparsity is ''spread''. If ''n'' is a point, define the spread polynomial of ''n'' to be the sum <math>\operatorname{sp}(n) = \sum t^{\mathbf{d}(n, i)}</math> over all points ''i'', where ''t'' is an indeterminate. Then the spread is the rational function <math>\operatorname{spr}(n) = \sum \frac{1}{\operatorname{sp}(n)}</math> over all points ''n''. Spread as a function decreases between 0 and 1, with {{nowrap|spr(0) {{=}} '''P'''}}, the number of notes in the scale and therefore points in the space, and {{nowrap|spr(1) {{=}} 1}}. We can think of {{nowrap|''t'' {{=}} 0}} as the highest magnification, with each of the points showing clearly, and {{nowrap|''t'' {{=}} 1|| as the lowest, where all points have merged together. In between, at {{nowrap|''t'' {{=}} 1/2}} or (a traditional choice, for some reason) {{nowrap|''t'' {{=}} exp(−1)}}, we have a sparsity measure. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. This inflection is related to the fact that spread for scales tends to be relatively large—notes tend to be far apart. Especially for larger scales, the tendency of spread to stick close to the maximum value for most values of ''t'' in the range 0 to 1 is striking.
-In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance, sprea'''d'''(Euler(3*5)) = 4/(t^3 + 2t^2 +1), sprea'''d'''(Euler(3*5*7))  = 8/(t^7 + 3t^6 + 3t^4 + 1), sprea'''d'''(hexany) = 6/(t^6 + 4t^4 + 1), sprea'''d'''(dekany) = 10/(3t^9 + 6t^7 + 1), sprea'''d'''(pentadekany) = 16/(6t^14 + 8t^11 + 1), sprea'''d'''(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).
+In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance:
+<math>
+\begin{align}
+\operatorname{spr}\left(\operatorname{Euler}(3*5)\right) &= \frac{4}{t^3 + 2t^2 + 1} \\
+\operatorname{spr}\left(\operatorname{Euler}(3*5*7)\right) &= \frac{8}{t^7 + 3t^6 + 3t^4 + 1} \\
+\operatorname{spr}(\text{hexany}) &= \frac{6}{t^6 + 4t^4 + 1} \\
+\operatorname{spr}(\text{dekany}) &= \frac{10}{3t^9 + 6t^7 + 1} \\
+\operatorname{spr}(\text{pentadekany}) &= \frac{16}{6t^{14} + 8t^{11} + 1} \\
+\operatorname{spr}(\text{eikosany}) &= \frac{20}{t^{19} + 9t^{18} + 9t^{14} + 1}
+\end{align}
+</math>
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