On a Class of Locally Symmetric Sequences: The Right Infinite Word Λ _θ

Abstract

The Nicomachus Triangle, a two-dimensional representation of powers of 2 and 3, provides a starting point for the development of an infinite class of right infinite Lambda words, Λ _θ . The word is formed by encoding differences in the sequence $\{\mathcal{M}_{\theta}\}_{i} = \{a+b\theta\}_{i}$, a, b ∈ ℕ. Although the word is on an infinite alphabet, it is traversable via environments containing no more than three letters. When θ = ϑ = log₂3, the word encodes all well-formed scales and regions generated by the intervals octave and perfect twelfth. The study sheds additional light on the role of palindromes in musical tone structures. Regions are palindromes on two letters, and form the largest palindromes in the Lambda word, as it develops. The regions have a significant dual representation, connecting them to the palindromic prefixes of a characteristic Sturmian word. The Lambda word is rich in palindromes beyond regions. In particular, a palindrome is formed between any two successive appearances of the same letter. Although Λ _ϑ is of particular importance musically, Lambda words are interesting in their own right as word theoretic objects. The paper ends with a brief look at the Fibonacci Lambda word, Λ _φ .