Balanced word: Difference between revisions

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{{distinguish|perfect balance}}{{Todo|rework|inline=1|text=this can probably be explained without invoking "words"}}

An abstract scale pattern is '''balanced''' if it satisfies a certain (quite strong) restriction on how much the intervals within any of the scale's interval classes can differ; by one characterization of the property, it stipulates that for any step size, no two ''k''-steps can differ too much in how many times the step size occurs in them. The simplest non-trivial examples of balanced scales are [[MOS scales]], and balanced words are one of many possible generalizations of [[MOS scale]]s to scales with three or more step sizes.

== Mathematical definition ==

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== Generalizations ==

# If <math> \mathsf{block\_balance}(s) \leq m,</math> then we say that ''s'' is ''m''-'''block-balanced'''{{idiosyncratic}}.

# The following stronger property implies ''m''-block-balancedness for ''m'' ≥ 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(w') = len(''w'') + ''m'' + 1 satisfies |w'|''a'' ~~≥~~ |''w''|''a'' ~~+ 1~~.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word proves that for binary scales, balanced implies 1-chain-balanced.)

# The following stronger property implies ''m''-block-balancedness for ''m'' ≥ 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(w') = len(''w'') + ''m'' + 1 satisfies |w'|''a'' > |''w''|''a''.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word proves that for binary scales, balanced implies 1-chain-balanced.)

Block and chain balancedness are equivalent for balanced scales (which are 1-balanced in both senses) and ternary billiard ones, but ''m''-chain-balancedness is stronger than ''m''-block-balancedness in the general case.

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-{{distinguish|perfect balance}}
+{{distinguish|perfect balance}}{{Todo|rework|inline=1|text=this can probably be explained without invoking "words"}}
 An abstract scale pattern is '''balanced''' if it satisfies a certain (quite strong) restriction on how much the intervals within any of the scale's interval classes can differ; by one characterization of the property, it stipulates that for any step size, no two ''k''-steps can differ too much in how many times the step size occurs in them. The simplest non-trivial examples of balanced scales are [[MOS scales]], and balanced words are one of many possible generalizations of [[MOS scale]]s to scales with three or more step sizes.
 == Mathematical definition ==
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 == Generalizations ==
 # If <math> \mathsf{block\_balance}(s) \leq m,</math> then we say that ''s'' is ''m''-'''block-balanced'''{{idiosyncratic}}.
-# The following stronger property implies ''m''-block-balancedness for ''m'' &ge; 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(<i>w'</i>) = len(''w'') + ''m'' + 1 satisfies |<i>w'</i>|<sub>''a''</sub> &ge; |''w''|<sub>''a''</sub> + 1.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word proves that for binary scales, balanced implies 1-chain-balanced.)
+# The following stronger property implies ''m''-block-balancedness for ''m'' &ge; 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(<i>w'</i>) = len(''w'') + ''m'' + 1 satisfies |<i>w'</i>|<sub>''a''</sub> > |''w''|<sub>''a''</sub>.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word proves that for binary scales, balanced implies 1-chain-balanced.)
 Block and chain balancedness are equivalent for balanced scales (which are 1-balanced in both senses) and ternary billiard ones, but ''m''-chain-balancedness is stronger than ''m''-block-balancedness in the general case.