Balanced word: Difference between revisions
mNo edit summary |
mNo edit summary |
||
| Line 1: | Line 1: | ||
{{distinguish|perfect balance}} | {{distinguish|perfect balance}} | ||
An abstract scale pattern is '''balanced''' if it satisfies a certain (quite strong) restriction on how much the intervals in within any of the scale's interval classes can differ; by one characterization of the balancedness, for any step size, the property stipulates that no two ''k''-steps can differ too much in how many times the step size occurs in them. | An abstract scale pattern is '''balanced''' if it satisfies a certain (quite strong) restriction on how much the intervals in within any of the scale's interval classes can differ; by one characterization of the balancedness, for any step size, the property stipulates that no two ''k''-steps can differ too much in how many times the step size occurs in them. Balanced words are one of many possible generalizations of [[MOS scale]]s to scales with three or more step sizes. | ||
== Mathematical definition == | |||
Let ''a'' be a letter in a [[word]] or [[necklace]] ''s''. Define | |||
<math> \mathsf{block\_balance}(s, a) := \max \big\{ \big| |w|_{a} - |w'|_{a} \big| : w, w'\text{ are length-}k \text{ subwords of } s\big\},</math> | <math> \mathsf{block\_balance}(s, a) := \max \big\{ \big| |w|_{a} - |w'|_{a} \big| : w, w'\text{ are length-}k \text{ subwords of } s\big\},</math> | ||