Structure metric - Revision history

ArrowHead294: /* Sparsity */ Formatting

2026-01-26T03:41:28Z

Sparsity: Formatting

← Older revision		Revision as of 03:41, 26 January 2026
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	=== Sparsity ===		=== 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(35)) = 4/(t^3 + 2t^2 +1~~), sprea'''d'''~~(Euler(35*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}(357)\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>

	{{todo\|link}}		{{todo\|link}}

ArrowHead294: /* Roundness */ Clean up

2026-01-26T03:31:08Z

Roundness: Clean up

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ArrowHead294: /* Centrality */ Formatting

2026-01-26T03:19:31Z

Centrality: Formatting

← Older revision		Revision as of 03:19, 26 January 2026
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	=== Centrality ===		=== Centrality ===
	The ''eccentricity'' of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in [[John O'Sullivan]]'s scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center. A more refined measure than eccentricity is the ''distance degree'' of a point, which is the sum of the distances from that point to other points; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {1, 3/2}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.		The ''eccentricity'' of a point ''x'' of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in [[John O'Sullivan]]'s scale "Blue," {{nowrap\|{{dash\|1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2}}}}, {{nowrap\|{1, 6/5, 5/4, 3/2}<nowiki/>}} is singled out as the center. A more refined measure than eccentricity is the ''distance degree'' of a point, which is the sum of the distances from that point to other points; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {{nowrap\|{1, 3/2}<nowiki/>}}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.

	=== Roundness ===		=== Roundness ===

ArrowHead294: /* Isometry */ Formatting

2026-01-26T03:16:43Z

Isometry: Formatting

← Older revision		Revision as of 03:16, 26 January 2026
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	== Isometry ==		== Isometry ==
	An ~~[https://en.wikipedia.org/wiki/Isometry~~ isometry] between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance '''d'''(f(a), f(b)) in Y equals '''d'''(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the ~~[https://en.wikipedia.org/wiki/Isometry_group~~ isometry group].		An {{w\|isometry}} between two metric spaces is a distance-preserving mapping; a mapping ''f'' from metric spaces ''X'' and ''Y'' such that the distance '''d'''(''f''(''a''), ''f''(''b'')) in ''Y'' equals '''d'''(''a'', ''b'') in ''X''. If ''f'' is a bijection, then the isometry defines an isometric isomorphism between ''X'' and ''Y''; in this case ''X'' and ''Y'' are said to be isometric. A metric space ''X'' is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the {{w\|isometry group}}.

	In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix ('''d'''(i, j)), where "i" denotes the ith point in some ordering. If S is a permutation matrix on these points, it is an element of the isometry group if and only if ~~S⋅D~~ = ~~D⋅S, where the dot is matrix multiplication~~. In this case, D is permutation-similar to itself by S. An invariant under similarity, and hence permutation similarity in particular, is the characteristic polynomial, as well as related invariants such as the rank, eigenvalues and minimal polynomial. The characteristic polynomial tends to reflect the symmetries of the metric space and the isometry group.		In the case of a finite metric space, the isometry group is defined by a permutation group on the set of points. Any finite metric space is completely characterized by the distance matrix '''d'''(''i'', ''j''), where ''i'' denotes the ith point in some ordering. If '''S''' is a permutation matrix on these points, it is an element of the isometry group if and only if {{nowrap\|'''SD''' {{=}} '''DS'''}}. In this case, '''D''' is permutation-similar to itself by '''S'''. An invariant under similarity, and hence permutation similarity in particular, is the characteristic polynomial, as well as related invariants such as the rank, eigenvalues, and minimal polynomial. The characteristic polynomial tends to reflect the symmetries of the metric space and the isometry group.

	An interesting example of this is given by the ~~[https://en.wikipedia.org/wiki/Hexany~~ hexany], 1~~-15/14-5/4-10/7-3/2-12/7-2. This has distance matrix [[0~~, ~~4, 4, 4, 4, 5], [4, 0, 4, 4, 5, 4], [4, 4, 0, 5, 4, 4], [4, 4, 5, 0, 4, 4], [4, 5, 4, 4, 0, 4], [5, 4, 4, 4, 4, 0]]. If we set f(1) = 1, f(~~15/14~~) = 9/8~~, f(5/4~~) = 6/5~~, f(10/7~~) = 5/4~~, f(3/2~~) = 9/5 and f(~~12/7~~) = 15/8~~, ~~then the distances we get from the new scale 1-9/8-6/5-5/4-9/5-15/8-~~2 ~~are the same as for the hexany; this scale, the [[hexagon\|hexagon]], is isometric to the hexany~~. Also, by mapping the hexany to itself we may find the isometry group, which turns out to be the same 48 element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, (x-21) (x+3)^2 (x+5)^3, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required--both [[27edo\|27edo]] and [[31edo\|31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.		An interesting example of this is given by the {{w\|hexany}}, {{nowrap\|{{dash\|1, 15/14, 5/4, 10/7, 3/2, 12/7, 2}}}}. This has distance matrix"

	Even though the [[Graph-~~theoretic_properties_of_scales~~\|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[~~star\|~~star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[~~nova\|~~Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera Euler(15^n) have the group of the square for both groups, Euler(105^n) gives the group of the cube, and the 5-limit diamond the group of the hexagon.		<math>
			\left[ \begin{matrix}
			0 & 4 & 4 & 4 & 4 & 5 \\
			4 & 0 & 4 & 4 & 5 & 4 \\
			4 & 4 & 0 & 5 & 4 & 4 \\
			4 & 4 & 5 & 0 & 4 & 4 \\
			4 & 5 & 4 & 4 & 0 & 4 \\
			5 & 4 & 4 & 4 & 4 & 0
			\end{matrix} \right]
			</math>

			If we set {{nowrap\|''f''(1) {{=}} 1\|''f''(15/14) {{=}} 9/8\|''f''(5/4) {{=}} 6/5\|''f''(10/7) {{=}} 5/4\|''f''(3/2) {{=}} 9/5}}, and {{nowrap\|''f''(12/7) {{=}} 15/8}}, then the distances we get from the new scale {{nowrap\|{{dash\|1, 9/8, 6/5, 5/4, 9/5, 15/8, 2}}}} are the same as for the hexany; this scale, the [[hexagon]], is isometric to the hexany. Also, by mapping the hexany to itself we may find the isometry group, which turns out to be the same 48-element group of the octahedron as is also derivable from the octahedron of 7-limit interval relationships; however, in this case it has been found entirely from the structure of the interval classes and without reference to harmonic relationships. The characteristic polynomial, {{nowrap\|(''x'' − 21)(''x'' + 3)<sup>2</sup>(''x'' + 5)<sup>3</sup>}}, reflects the high degree of symmetry of the hexany. It should be noted, however, that precise JI tuning is not required—both [[27edo]] and [[31edo]], for example, are well enough in tune to give the same structure of interval classes and hence the same metric space.

			Even though the [[Graph-theoretic properties of scales\|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is {{nowrap\|Cps([2, 3, 5, 7, 11], 2)}}, the {{nowrap\|2{{!}}5}} dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera {{nowrap\|Euler(15<sup>''n'')}} have the group of the square for both groups, {{nowrap\|Euler(105<sup>''n''</sup>)}} gives the group of the cube, and the 5-limit diamond the group of the hexagon.

	== Invariants ==		== Invariants ==

ArrowHead294 at 03:07, 26 January 2026

2026-01-26T03:07:29Z

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ArrowHead294: Section formatting

2026-01-26T03:04:59Z

Section formatting

← Older revision		Revision as of 03:04, 26 January 2026
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	5. {{nowrap\|d(''a'', ''c'') ≤ d(''a'', ''b'') + d(''b'', ''c'')}}		5. {{nowrap\|d(''a'', ''c'') ≤ d(''a'', ''b'') + d(''b'', ''c'')}}

	Suppose ''X'' is the {{w\|indicator function}} (characteristic function) for the set {{nowrap\|S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''}})}}, ''Y'' for the set {{nowrap\|S({{abs\|'''s'''[''b''] − '''s'''[''c'']}}, {{abs\|''b'' − ''c''}})}}, and ''Z'' for the set {{nowrap\|S({{abs\|'''s'''[''a''] − '''s'''[''c'']}}, {{abs\|''a'' - ''c''}})}}, which we may regard as vectors in {{nowrap\|ℝ<sup>'''P'''</sup>}}. Let ''J'' be the '''P'''-dimensional vector {{nowrap\|[1, 1, ..., 1]}} of all 1s. Then what we wish to prove may be rewritten {{nowrap\|'''P''' − ''Z'' · ''J'' ≤ ('''P''' − ''X'' · ''J'') + ('''P''' − ''Y'' · ''J'')}}. This may be rewritten again as {{nowrap\|''Z'' · ''J'' ≥ (''X'' + ''Y'' − ''J'') · ''J''}}. Every index contributing to {{nowrap\|''X'' · ''Y''}} counts as one of ''Z'', and hence {{nowrap\|''Z'' ⋅ ''J'' ≥ ''X'' ⋅ ''Y''}}. The vector {{nowrap\|''X'' + ''Y'' − ''J''}} is 1 at an index where both ''X'' and ''Y'' are 1, is −1 when neither is 1, and 0 otherwise. Hence {{nowrap\|(''X'' + ''Y'' − ''J'') · ''J'' {{=}} ''X'' · ''Y'' − (''J'' − ''X'') · (''J'' − ''Y'')}}, and so is less than or equal to {{nowrap\|''X'' · ''Y''}}, and hence less than or equal to {{nowrap\|''Z'' · ''J''}}.		Suppose '''X''' is the {{w\|indicator function}} (characteristic function) for the set {{nowrap\|S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''}})}}, '''Y''' for the set {{nowrap\|S({{abs\|'''s'''[''b''] − '''s'''[''c'']}}, {{abs\|''b'' − ''c''}})}}, and '''Z''' for the set {{nowrap\|S({{abs\|'''s'''[''a''] − '''s'''[''c'']}}, {{abs\|''a'' - ''c''}})}}, which we may regard as vectors in {{nowrap\|ℝ<sup>'''P'''</sup>}}. Let '''J''' be the '''P'''-dimensional vector {{nowrap\|[1, 1, ..., 1]}} of all 1s. Then what we wish to prove may be rewritten {{nowrap\|'''P''' − '''Z''' · '''J''' ≤ ('''P''' − '''X''' · '''J''') + ('''P''' − '''Y''' · '''J''')}}. This may be rewritten again as {{nowrap\|'''Z''' · '''J''' ≥ ('''X''' + '''Y''' − '''J''') · '''J'''}}. Every index contributing to {{nowrap\|'''X''' · '''Y'''}} counts as one of '''Z''', and hence {{nowrap\|'''Z''' ⋅ '''J''' ≥ '''X''' ⋅ '''Y'''}}. The vector {{nowrap\|'''X''' + '''Y''' − '''J'''}} is 1 at an index where both '''X''' and '''Y''' are 1, is −1 when neither is 1, and 0 otherwise. Hence {{nowrap\|('''X''' + '''Y''' − '''J''') · '''J''' {{=}} '''X''' · '''Y''' − ('''J''' − '''X''') · ('''J''' − '''Y''')}}, and so is less than or equal to {{nowrap\|'''X''' · '''Y'''}}, and hence less than or equal to {{nowrap\|'''Z''' · '''J'''}}.

	These properties mean that the structure metric defines a ''finite metric space''. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.		These properties mean that the structure metric defines a ''finite metric space''. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.

	=Isometry=		== Isometry ==
	An [https://en.wikipedia.org/wiki/Isometry isometry] between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the [https://en.wikipedia.org/wiki/Isometry_group isometry group].		An [https://en.wikipedia.org/wiki/Isometry isometry] between two metric spaces is a distance-preserving mapping; a mapping f from metric spaces X and Y such that the distance d(f(a), f(b)) in Y equals d(a, b) in X. If f is a bijection, then the isometry defines an isometric isomorphism between X and Y; in this case X and Y are said to be isometric. A metric space X is always isometric to itself by the identity map, but it may have nontrivial isometries. The isometries of X with itself define a group, the [https://en.wikipedia.org/wiki/Isometry_group isometry group].

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	Even though the [[Graph-theoretic_properties_of_scales\|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[star\|star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[nova\|Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera Euler(15^n) have the group of the square for both groups, Euler(105^n) gives the group of the cube, and the 5-limit diamond the group of the hexagon.		Even though the [[Graph-theoretic_properties_of_scales\|group of the graph]] is defined entirely in terms of harmonic relationships and the isometry group entirely in terms of interval classes, in the case of the hexany they give the exact same group. Another example of this is Cps([2,3,5,7,11], 2), the 2)5 dekany, where the isometry group and the group of the graph are both 10T13. A more common situation is for the isometry group to be a subgroup of the group of the graph. For instance, [[star\|star]] has a group of order 384 as the group of its graph, and a subgroup of order 4, a Klein 4-group, as its isometry group. [[nova\|Nova]], which is isometric with star and has an isomorphic graph, is similar. On the other hand, scales with a clear geometric symmetry tend to have isomorphic graph groups and isometry groups. For instance, the Euler genera Euler(15^n) have the group of the square for both groups, Euler(105^n) gives the group of the cube, and the 5-limit diamond the group of the hexagon.

	=Invariants=		== Invariants ==
	A metric invariant is a property of a metric space which is preserved under isometry. The metric invariants of the structure metric define properites of the scale from which it derives.		A metric invariant is a property of a metric space which is preserved under isometry. The metric invariants of the structure metric define properites of the scale from which it derives.

	==Centrality==		=== Centrality ===
	The ''eccentricity'' of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in [[John O'Sullivan]]'s scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center. A more refined measure than eccentricity is the ''distance degree'' of a point, which is the sum of the distances from that point to other points; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {1, 3/2}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.		The ''eccentricity'' of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in [[John O'Sullivan]]'s scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center. A more refined measure than eccentricity is the ''distance degree'' of a point, which is the sum of the distances from that point to other points; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {1, 3/2}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.

	==Roundness==		=== Roundness ===
	The [https://en.wikipedia.org/wiki/Gromov_product Gromov product] is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.		The [https://en.wikipedia.org/wiki/Gromov_product Gromov product] is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = (d(x, y) + d(x, z) - d(y, z))/2. Assuming x is 1, this becomes (y, z) = (d(1, y) + d(1, z) - d(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.

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	p = ∞ otonal and utonal tetrad; this implies the space is ultrametric		p = ∞ otonal and utonal tetrad; this implies the space is ultrametric

	==Sparsity==		=== 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 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.

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	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, spread(Euler(35)) = 4/(t^3 + 2t^2 +1), spread(Euler(35*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(pentadekany) = 16/(6t^14 + 8t^11 + 1), spread(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, spread(Euler(35)) = 4/(t^3 + 2t^2 +1), spread(Euler(35*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(pentadekany) = 16/(6t^14 + 8t^11 + 1), spread(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).

			{{todo\|link}}

	[[Category:Math]]		[[Category:Math]]
	[[Category:Terms]]		[[Category:Terms]]
	~~{{todo\|link}}~~

ArrowHead294: /* Properties */

2026-01-26T03:02:46Z

Properties

← Older revision		Revision as of 03:02, 26 January 2026
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	The ''structure metric'' is a {{w\|metric (mathematics)\|distance function}} on the notes of a [[periodic scale]] within a single period, which give to it the property of being a {{w\|metric space\|finite metric space}}. If '''s''' is a periodic scale with quasiperiod '''P''', and if ''c'' is an interval {{nowrap\|'''s'''[''i'' + ''j''] − '''s'''[''i'']}} with {{nowrap\|0 ≤ ''i'' < '''P'''}}, then we may define the specific interval set S(''c'', ''j'') to be {{nowrap\|{i {{!}} '''s'''[''i'' + ''j''] − '''s'''[''i''] {{=}} ''c''}<nowiki/>}} with {{nowrap\|0 ≤ ''i'' < '''P'''}}, that is, indicies for the set of intervals with specific, chromatic size ''c'' and generic, scalar interval ''j''. If #S(''c'', ''j'') is the cardinality of S(''c'', ''j''), then we set d('''s'''[''a''], '''s'''[''b'']), which we will abbreviate as d(''a'', ''b''), to be {{nowrap\|'''P''' − #S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''}})}}.		The ''structure metric'' is a {{w\|metric (mathematics)\|distance function}} on the notes of a [[periodic scale]] within a single period, which give to it the property of being a {{w\|metric space\|finite metric space}}. If '''s''' is a periodic scale with quasiperiod '''P''', and if ''c'' is an interval {{nowrap\|'''s'''[''i'' + ''j''] − '''s'''[''i'']}} with {{nowrap\|0 ≤ ''i'' < '''P'''}}, then we may define the specific interval set S(''c'', ''j'') to be {{nowrap\|{i {{!}} '''s'''[''i'' + ''j''] − '''s'''[''i''] {{=}} ''c''}<nowiki/>}} with {{nowrap\|0 ≤ ''i'' < '''P'''}}, that is, indicies for the set of intervals with specific, chromatic size ''c'' and generic, scalar interval ''j''. If #S(''c'', ''j'') is the cardinality of S(''c'', ''j''), then we set d('''s'''[''a''], '''s'''[''b'']), which we will abbreviate as d(''a'', ''b''), to be {{nowrap\|'''P''' − #S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''}})}}.

	=Properties=		== Properties ==
	The structure metric has the following properties:		The structure metric has the following properties:

	1. d(a, a) = 0		1. {{nowrap\|d(''a'', ''a'') {{=}} 0}}

	#S(\|'''s'''[a] - '''s'''[a]\|, \|a - a\|) = #S(0, 0) = '''P'''.		{{nowrap\|#S({{abs\|'''s'''[''a''] − '''s'''[''a'']}}, {{abs\|''a'' − ''a''}}) {{=}} #S(0, 0)}} = '''P'''.

	2. d(a, b) ≥ 0		2. {{nowrap\|d(''a'', ''b'') ≥ 0}}

	The cardinality of #S(c, j) cannot exceed '''P''', since ~~0≤i~~<'''P'''.		The cardinality of #S(''c'', ''j'') cannot exceed '''P''', since {{nowrap\|0 ≤ i < '''P'''}}.

	3. d(a, b) = 0 implies a ~~equals~~ b.		3. {{nowrap\|d(''a'', ''b'') {{=}} 0}} implies {{nowrap\|''a'' {{=}} ''b''}}.

	If a ≠ b and d(a, b) = 0 then #S(\|'''s'''[a] - '''s'''[b]\|, \|a - b\|)) = '''P''', so \|a - b\| is a period, and \|'''s'''[a] - '''s'''[b]\| is an interval of repetition. However, '''P''' is the smallest period, contradiction.		If {{nowrap\|a ≠ b}} and {{nowrap\|d(''a'', ''b'') {{=}} 0}} then {{nowrap\|#S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''\|}}) {{=}} '''P'''}}, so {{abs\|''a'' − ''b''}} is a period, and {{nowrap\|{{abs\|'''s'''[''a''] − '''s'''[''b'']}}}} is an interval of repetition. However, '''P''' is the smallest period, leading to a contradiction.

	4. d(a, b) = d(b, a)		4. {{nowrap\|d(''a'', ''b'') {{=}} d(''b'', ''a'')}}

	d(a, b) ~~equals~~ '''P''' - #S(\|'''s'''[a] - '''s'''[b]\|, \|a - b\|~~) equals~~ '''P''' - #S(\|'''s'''[b] - '''s'''[a]\|, \|b - a\|~~) equals~~ d(b, a).		{{nowrap\|d(''a'', ''b'') {{=}} '''P''' − #S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''}})}} {{nowrap\|{{=}} '''P''' − #S({{abs\|'''s'''[''b''] − '''s'''[''a'']}}, {{abs\|''b'' − ''a''}})}} {{nowrap\|{{=}} d(''b'', ''a'')}}.

	5. d(a, c) ≤ d(a, b) + d(b, c)		5. {{nowrap\|d(''a'', ''c'') ≤ d(''a'', ''b'') + d(''b'', ''c'')}}

	Suppose X is the ~~[https://en.wikipedia.org/wiki/Indicator_function~~ indicator function] (characteristic function) for the set S(\|'''s'''[a] - '''s'''[b]\|, \|a - b\|), Y for the set S(\|'''s'''[b] - '''s'''[c]\|, \|b - c\|), and Z for the set S(\|'''s'''[a] - '''s'''[c]\|, \|a - c\|), which we may regard as vectors in ℝ^'''P'''. Let J be the '''P'''-dimensional vector [1, 1, ..., 1] of all 1s. Then what we wish to prove may be rewritten '''P''' - Z.J ≤ ('''P''' - X.J) + ('''P''' - Y.J). This may be rewritten again as Z.J ≥ (X + Y - J).J. Every index contributing to X.Y counts as one of Z, and hence Z.J ≥ X.Y. The vector X + Y - J is 1 at an index where both X and Y are 1, is -1 when neither is 1, and 0 otherwise. Hence (X + Y - J).J is X.Y - (J - X).(J - Y), and so is less than or equal to X.Y, and hence less than or equal to Z.J.		Suppose ''X'' is the {{w\|indicator function}} (characteristic function) for the set {{nowrap\|S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''}})}}, ''Y'' for the set {{nowrap\|S({{abs\|'''s'''[''b''] − '''s'''[''c'']}}, {{abs\|''b'' − ''c''}})}}, and ''Z'' for the set {{nowrap\|S({{abs\|'''s'''[''a''] − '''s'''[''c'']}}, {{abs\|''a'' - ''c''}})}}, which we may regard as vectors in {{nowrap\|ℝ<sup>'''P'''</sup>}}. Let ''J'' be the '''P'''-dimensional vector {{nowrap\|[1, 1, ..., 1]}} of all 1s. Then what we wish to prove may be rewritten {{nowrap\|'''P''' − ''Z'' · ''J'' ≤ ('''P''' − ''X'' · ''J'') + ('''P''' − ''Y'' · ''J'')}}. This may be rewritten again as {{nowrap\|''Z'' · ''J'' ≥ (''X'' + ''Y'' − ''J'') · ''J''}}. Every index contributing to {{nowrap\|''X'' · ''Y''}} counts as one of ''Z'', and hence {{nowrap\|''Z'' ⋅ ''J'' ≥ ''X'' ⋅ ''Y''}}. The vector {{nowrap\|''X'' + ''Y'' − ''J''}} is 1 at an index where both ''X'' and ''Y'' are 1, is −1 when neither is 1, and 0 otherwise. Hence {{nowrap\|(''X'' + ''Y'' − ''J'') · ''J'' {{=}} ''X'' · ''Y'' − (''J'' − ''X'') · (''J'' − ''Y'')}}, and so is less than or equal to {{nowrap\|''X'' · ''Y''}}, and hence less than or equal to {{nowrap\|''Z'' · ''J''}}.

	These properties mean that the structure metric defines a ''finite metric space''. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.		These properties mean that the structure metric defines a ''finite metric space''. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.

ArrowHead294: /* Definition */ Formatting

2026-01-26T02:46:11Z

Definition: Formatting

← Older revision		Revision as of 02:46, 26 January 2026
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	__FORCETOC__		__FORCETOC__
	=Definition=		== Definition ==
	The ''structure metric'' is a ~~[https://en.wikipedia.org/wiki/Metric_~~(mathematics) distance function] on the notes of a [[~~Periodic_scale\|~~periodic scale]] within a single period, which give to it the property of being a ~~[https://en.wikipedia.org/wiki/Metric_space~~ finite metric space]. If '''s''' is a periodic scale with quasiperiod '''P''', and if c is an interval '''s'''[i+j] - '''s'''[i] with ~~0≤i~~<'''P''', then we may define the specific interval set S(c, j) to be {i\|'''s'''[i+j] - '''s'''[i] = c} with ~~0≤i~~<'''P''', that is, indicies for the set of intervals with specific, chromatic size c and generic, scalar interval j. If #S(c, j) is the cardinality of S(c, j), then we set d('''s'''[a], '''s'''[b]), which we will abbreviate as d(a, b), to be '''P''' - #S(\|'''s'''[a] - '''s'''[b]\|, \|a - b\|).		The ''structure metric'' is a {{w\|metric (mathematics)\|distance function}} on the notes of a [[periodic scale]] within a single period, which give to it the property of being a {{w\|metric space\|finite metric space}}. If '''s''' is a periodic scale with quasiperiod '''P''', and if ''c'' is an interval {{nowrap\|'''s'''[''i'' + ''j''] − '''s'''[''i'']}} with {{nowrap\|0 ≤ ''i'' < '''P'''}}, then we may define the specific interval set S(''c'', ''j'') to be {{nowrap\|{i {{!}} '''s'''[''i'' + ''j''] − '''s'''[''i''] {{=}} ''c''}<nowiki/>}} with {{nowrap\|0 ≤ ''i'' < '''P'''}}, that is, indicies for the set of intervals with specific, chromatic size ''c'' and generic, scalar interval ''j''. If #S(''c'', ''j'') is the cardinality of S(''c'', ''j''), then we set d('''s'''[''a''], '''s'''[''b'']), which we will abbreviate as d(''a'', ''b''), to be {{nowrap\|'''P''' − #S({{abs\|'''s'''[''a''] − '''s'''[''b'']}}, {{abs\|''a'' − ''b''}})}}.

	=Properties=		=Properties=

Sintel: Inaccessible, spelling

2025-03-26T15:52:27Z

Inaccessible, spelling

← Older revision		Revision as of 15:52, 26 March 2025
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			{{inacc}}

	__FORCETOC__		__FORCETOC__
	=Definition=		=Definition=
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	p = ∞ otonal and utonal tetrad; this implies the space is ultrametric		p = ∞ otonal and utonal tetrad; this implies the space is ultrametric

	==~~Sparcity~~==		==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 ''~~sparcity~~'' of the space is S/(2(n-1)^2(n-2)). The ~~sparcity~~ 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 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.

	Tempering will often shrink distances and so increase density. For example, the duodene has a ~~sparcity~~ 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 ~~sparcity~~ becomes 0. To give another example, [[pentadekany2\|pentadekany2]], which is Cps([2,3,5,7,9,11], 3), has a ~~sparcity~~ 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 ~~sparcity~~ 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\|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.

	An invariant related to ~~sparcity~~ 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 spread(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with spread(0) = '''P''', the number of notes in the scale and therefore points in the space, and spread(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 ~~sparcity~~ 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 sp(n) = ∑ t^d(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function spread(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with spread(0) = '''P''', the number of notes in the scale and therefore points in the space, and spread(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.

	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, spread(Euler(35)) = 4/(t^3 + 2t^2 +1), spread(Euler(35*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(pentadekany) = 16/(6t^14 + 8t^11 + 1), spread(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, spread(Euler(35)) = 4/(t^3 + 2t^2 +1), spread(Euler(35*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), spread(hexany) = 6/(t^6 + 4t^4 + 1), spread(dekany) = 10/(3t^9 + 6t^7 + 1), spread(pentadekany) = 16/(6t^14 + 8t^11 + 1), spread(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).

BudjarnLambeth: link

2024-12-03T08:50:47Z

link

← Older revision		Revision as of 08:50, 3 December 2024
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	==Centrality==		==Centrality==
	The ''eccentricity'' of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in John O'Sullivan's scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center. A more refined measure than eccentricity is the ''distance degree'' of a point, which is the sum of the distances from that point to other points; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {1, 3/2}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.		The ''eccentricity'' of a point x of a metric space (and therefore of a note of our scale) is its maximum distance from any other point in the space. The minimum eccentricity is the radius of the space, and the maximum eccentricity is the diameter. The center of the space is the set of points whose eccentricity equals the radius. This can be the whole space, and hence the whole scale, but more often it singles out some notes as of particular importance in the scale. For instance in [[John O'Sullivan]]'s scale Blue, 1-15/14-9/8-6/5-5/4-4/3-7/5-3/2-8/5-5/3-9/5-15/8-2, {1, 6/5, 5/4, 3/2} is singled out as the center. A more refined measure than eccentricity is the ''distance degree'' of a point, which is the sum of the distances from that point to other points; we can use the minimum of this to define the distance degree center. In the case of Blue, that would be {1, 3/2}. Note that the importance of these notes is not derived from tuning considerations but purely from the structure of the scale.

	==Roundness==		==Roundness==