Periodic scale: Difference between revisions

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A class is a category of all intervals spanning a specified number of scale degrees, such as seconds, thirds, fourths etc in diatonic, or the generalization to any kind of scale.

In mathematical terms, we can define a function class(''k'') on the integers which gives sets representing the ''generic intervals'' of a periodic scale. For some integer ''k'', the set class(''k'') consists of all intervals <math>s[k+i] - s[i]</math>. Since ''s'' is quasiperiodic, class(''P'') only contains the period ''O'', but the rest may contain multiple intervals.

In mathematical terms, we can define a function class(''k'') on the integers which gives sets representing the ''generic intervals'' of a periodic scale. For some integer ''k'', the set class(''k'') consists of all intervals <math>s[k+i] - s[i]</math>. Equivalently, it is all the intervals found on the same degree of the different modes of the scale, or all the intervals between notes a given number of scale steps apart. Since ''s'' is quasiperiodic, class(''P'') only contains the period ''O'', but the rest may contain multiple intervals.

=== Step form and cumulative form ===

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

Given a periodic scale, we may call the function defined above the "cumulative form", and we may define its ''step form'' as

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

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 A class is a category of all intervals spanning a specified number of scale degrees, such as seconds, thirds, fourths etc in diatonic, or the generalization to any kind of scale.
-In mathematical terms, we can define a function class(''k'') on the integers which gives sets representing the ''generic intervals'' of a periodic scale. For some integer ''k'', the set class(''k'') consists of all intervals <math>s[k+i] - s[i]</math>. Since ''s'' is quasiperiodic, class(''P'') only contains the period ''O'', but the rest may contain multiple intervals.
+In mathematical terms, we can define a function class(''k'') on the integers which gives sets representing the ''generic intervals'' of a periodic scale. For some integer ''k'', the set class(''k'') consists of all intervals <math>s[k+i] - s[i]</math>. Equivalently, it is all the intervals found on the same degree of the different modes of the scale, or all the intervals between notes a given number of scale steps apart. Since ''s'' is quasiperiodic, class(''P'') only contains the period ''O'', but the rest may contain multiple intervals.
 === Step form and cumulative form ===
-Given a periodic scale as defined above, we may define its ''step form'' as
+Given a periodic scale, we may call the function defined above the "cumulative form", and we may define its ''step form'' as
 <math>\Delta s[i] = s[i+1] - s[i],</math>