Periodic scale: Difference between revisions

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=== Classes ===

We may define an important function class(''i'') on the integers which gives the ''generic intervals'' of a periodic scale. This is defined by s[''j''] - s[''i''] is in class(''k'') if ''j'' - ''i'' = ''k''. Since s is quasiperiodic, class(''nP'') consists only of {''nO''}, but the rest define sets of numbers in terms of which we can define some important scale properties.

=== Step form and cumulative form ===

Given a periodic scale as defined above, we may define its ''step form'' as

<math>\Delta s[i] = s[i+1] - s[i],</math>

and <math>\Delta s</math> has the property <math>\Delta s [i + P] = \Delta s[i] \ \forall i \in \mathbb{Z}.</math>

The step form <math>\Delta s</math> and the ''cumulative form'' <math>s</math> of a periodic scale are related by the fundamental theorem of finite-difference calculus:

<math>\sum_{i=n_0}^{n_1} \Delta s[i] = s[n_1+1]-s[n_0].</math>

Thus, we may equivalently define a periodic scale as a periodic function (in the usual mathematical sense) of step sizes.

== Scale properties ==

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 === Classes ===
 We may define an important function class(''i'') on the integers which gives the ''generic intervals'' of a periodic scale. This is defined by s[''j''] - s[''i''] is in class(''k'') if ''j'' - ''i'' = ''k''. Since s is quasiperiodic, class(''nP'') consists only of {''nO''}, but the rest define sets of numbers in terms of which we can define some important scale properties.
+=== Step form and cumulative form ===
+Given a periodic scale as defined above, we may define its ''step form'' as
+<math>\Delta s[i] = s[i+1] - s[i],</math>
+and <math>\Delta s</math> has the property <math>\Delta s [i + P] = \Delta s[i] \ \forall i \in \mathbb{Z}.</math>
+The step form <math>\Delta s</math> and the ''cumulative form'' <math>s</math> of a periodic scale are related by the fundamental theorem of finite-difference calculus:
+<math>\sum_{i=n_0}^{n_1} \Delta s[i] = s[n_1+1]-s[n_0].</math>
+Thus, we may equivalently define a periodic scale as a periodic function (in the usual mathematical sense) of step sizes.
 == Scale properties ==