Word: Difference between revisions

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A '''word''', in mathematical terminology, is a finite or infinite sequence of letters from some finite alphabet, such as "aabaaab" or "LLsLLLs." The mathematical field of combinatorics on words often studies words that are infinite in one or both directions. In xenharmonic music theory, words are often used to represent an abstract scale template with some specified pattern of step sizes, such as when working with [[MOS]] scales. Two words that are rotations of one another, such as "aabaaab" and "abaaaba", are said to be ''conjugate'', and ~~an equivalence class~~ of ~~words under conjugacy~~ is termed a '''circular word''' or a '''~~necklace~~'''. A word ''w'' is ''primitive'' if it is not the case that ''w'' = ''u''<sup>''k''</sup> for any shorter word ''u'' and any integer ''k'' > 1.

A '''word''', in mathematical terminology, is a finite or infinite sequence of letters from some finite alphabet, such as "aabaaab" or "LLsLLLs." The mathematical field of combinatorics on words often studies words that are infinite in one or both directions. In xenharmonic music theory, words are often used to represent an abstract scale template with some specified pattern of step sizes, such as when working with [[MOS]] scales. Two words that are rotations of one another, such as "aabaaab" and "abaaaba", are said to be ''conjugate'', and the set consisting of a word and all its rotations is termed a '''circular word''', a '''necklace''', or a '''scale word''' in a specifically music-related context. A word ''w'' is ''primitive'' if it is not the case that ''w'' = ''u''<sup>''k''</sup> for any shorter word ''u'' and any integer ''k'' > 1. A necklace is primitive if any (therefore every) representative of it is primitive.

Words may be used to represent the step pattern of a [[periodic scale]], in which case the distinction between words and necklaces may be elided when not referring to a specific mode of the scale. For example, the word "aabaaab" can be thought of as representing a '''scale template''' where "a" represents one kind of step, and "b" represents a different kind of step. It is often useful to specify the relative sizes of the steps as well, imposing a total order on the alphabet, leading to the two related words "LLsLLLs" and "ssLsssL," where L means "large" and s means "small." Of these, (assuming octave equivalence) LLsLLLs corresponds to the diatonic scale, and ssLsssL to the anti-diatonic scale; aabaaab can be thought of a more generic word incorporating both into a single template.

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	A '''word''', in mathematical terminology, is a finite or infinite sequence of letters from some finite alphabet, such as "aabaaab" or "LLsLLLs." The mathematical field of combinatorics on words often studies words that are infinite in one or both directions. In xenharmonic music theory, words are often used to represent an abstract scale template with some specified pattern of step sizes, such as when working with [[MOS]] scales. Two words that are rotations of one another, such as "aabaaab" and "abaaaba", are said to be ''conjugate'', and ~~an equivalence class~~ of ~~words under conjugacy~~ is termed a '''circular word''' or a '''~~necklace~~'''. A word ''w'' is ''primitive'' if it is not the case that ''w'' = ''u''<sup>''k''</sup> for any shorter word ''u'' and any integer ''k'' > 1.		A '''word''', in mathematical terminology, is a finite or infinite sequence of letters from some finite alphabet, such as "aabaaab" or "LLsLLLs." The mathematical field of combinatorics on words often studies words that are infinite in one or both directions. In xenharmonic music theory, words are often used to represent an abstract scale template with some specified pattern of step sizes, such as when working with [[MOS]] scales. Two words that are rotations of one another, such as "aabaaab" and "abaaaba", are said to be ''conjugate'', and the set consisting of a word and all its rotations is termed a '''circular word''', a '''necklace''', or a '''scale word''' in a specifically music-related context. A word ''w'' is ''primitive'' if it is not the case that ''w'' = ''u''<sup>''k''</sup> for any shorter word ''u'' and any integer ''k'' > 1. A necklace is primitive if any (therefore every) representative of it is primitive.

	Words may be used to represent the step pattern of a [[periodic scale]], in which case the distinction between words and necklaces may be elided when not referring to a specific mode of the scale. For example, the word "aabaaab" can be thought of as representing a '''scale template''' where "a" represents one kind of step, and "b" represents a different kind of step. It is often useful to specify the relative sizes of the steps as well, imposing a total order on the alphabet, leading to the two related words "LLsLLLs" and "ssLsssL," where L means "large" and s means "small." Of these, (assuming octave equivalence) LLsLLLs corresponds to the diatonic scale, and ssLsssL to the anti-diatonic scale; aabaaab can be thought of a more generic word incorporating both into a single template.		Words may be used to represent the step pattern of a [[periodic scale]], in which case the distinction between words and necklaces may be elided when not referring to a specific mode of the scale. For example, the word "aabaaab" can be thought of as representing a '''scale template''' where "a" represents one kind of step, and "b" represents a different kind of step. It is often useful to specify the relative sizes of the steps as well, imposing a total order on the alphabet, leading to the two related words "LLsLLLs" and "ssLsssL," where L means "large" and s means "small." Of these, (assuming octave equivalence) LLsLLLs corresponds to the diatonic scale, and ssLsssL to the anti-diatonic scale; aabaaab can be thought of a more generic word incorporating both into a single template.