Structure metric: Difference between revisions

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__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=

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 __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&lt;'''P''', then we may define the specific interval set S(c, j) to be {i|'''s'''[i+j] - '''s'''[i] = c} with 0≤i&lt;'''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 &le; ''i'' &lt; '''P'''}}, then we may define the specific interval set S(''c'',&nbsp;''j'') to be {{nowrap|{i {{!}} '''s'''[''i'' + ''j''] − '''s'''[''i''] {{=}} ''c''}<nowiki/>}} with {{nowrap|0 &le; ''i'' &lt; '''P'''}}, that is, indicies for the set of intervals with specific, chromatic size ''c'' and generic, scalar interval ''j''. If #S(''c'',&nbsp;''j'') is the cardinality of S(''c'',&nbsp;''j''), then we set d('''s'''[''a''],&nbsp;'''s'''[''b'']), which we will abbreviate as d(''a'',&nbsp;''b''), to be {{nowrap|'''P''' − #S({{abs|'''s'''[''a''] − '''s'''[''b'']}}, {{abs|''a'' − ''b''}})}}.
 =Properties=