Balanced word: Difference between revisions
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where |''u''|<sub>''x''<sub>''i''</sub></sub> is the number of occurrences of the letter ''x''<sub>''i''</sub> in the word ''u''. A scale is '''balanced''' if it is 1-balanced. | where |''u''|<sub>''x''<sub>''i''</sub></sub> is the number of occurrences of the letter ''x''<sub>''i''</sub> in the word ''u''. A scale is '''balanced''' if it is 1-balanced. | ||
A balanced word in ''N'' letters has a [[maximum variety]] bound of <math> N \choose {\lceil N/2 \rceil}</math>. | A balanced linear or circular word in ''N'' letters has a [[maximum variety]] bound of <math> N \choose {\lceil N/2 \rceil}</math>. | ||
[[Category:Scale]][[Category:Terms]] | [[Category:Scale]][[Category:Terms]] | ||
Revision as of 00:31, 24 December 2023
Let d ≥ 0. A linear or circular word s (representing a periodic scale if circular) is d-balanced if its balance satisfies the following:
[math]\displaystyle{ \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq d, }[/math]
where |u|xi is the number of occurrences of the letter xi in the word u. A scale is balanced if it is 1-balanced.
A balanced linear or circular word in N letters has a maximum variety bound of [math]\displaystyle{ N \choose {\lceil N/2 \rceil} }[/math].