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 .
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Domínguez, M., Clampitt, D., Noll, T.: Well-formed scales, maximally even sets and Christoffel Words. In: Klouche, T., Noll, T. (eds.) MCM 2007. Communications in Computer and Information Science, vol. 37, pp. 477–488. Springer, Heidelberg (2009)
Berstel, J., Lauve, A., Reutenauer, C., Saliola, F.: Combinatorics on Words: Christoffel Words and Repetition in Words. American Mathematical Society CRM Monograph Series, vol. 27 (2008)
Berthé, V., de Luca, A., Reutenauer, C.: On an involution of Christoffel words and Sturmian morphisms. European Journal of Combinatorics 29(2), 535–553 (2008)
Lothaire, M.: Combinatorics on Words. Cambridge Math. Lib. Cambridge Univ. Press, Cambridge (1997)
Lothaire, M.: Algebraic Combinatorics on Words. Encylopedia Math. Appl., vol. 90. Cambridge Univ. Press, Cambridge (2002)
de Luca, A.: Sturmian words: Structure, combinatorics, and their arithmetics. Theoretical Computer Science 183(1), 45–82 (1997)
Carey, N.: On coherence and sameness, and the evaluation of scale candidacy claims. Journal of Music Theory 46, 1–56 (2002)
Carey, N.: Coherence and sameness in well-formed and pairwise well-formed scales. Journal of Mathematics and Music 1(2), 79–98 (2007)
Clampitt, D., Noll, T.: Modes, the height-width duality, and Handschin’s Tone Character. Music Theory Online 17(1), (forthcoming), http://user.cs.tu-berlin.de/%7Enoll/HeightWidthDuality.pdf
Allouche, J.-P., Baake, M., Cassaigne, J., Damanik, D.: Palindrome complexity. Journal of Theoretical Computer Science 292(1), 9–31 (2003)
Brlek, S., Hamel, S., Nivat, M., Reutenauer, C.: On the palindromic complexity of infinite words. International Journal of Foundations of Computer Science 15(2), 293–306 (2004)
Droubay, X., Pirillo, G.: Palindromes and Sturmain words. Theoretical Computer Science 223, 73–85 (1999)
Carey, N., Clampitt, D.: Structural properties of musical scales (unpublished manuscript)
Carey, N., Clampitt, D.: Regions: A theory of tonal spaces in early medieval treatises. Journal of Music Theory 40, 113–147 (1996)
Carey, N., Clampitt, D.: Self-similar pitch structures, their duals, and rhythmic analogues. Perspectives of New Music 34(2), 62–87 (1996)
Singler, F.: Zur Dualität zwischen doppelter Periodizität und binärer Intervall-Struktur in der Theorie der Tonregionen. Thesis. Hochschule für Musik und Theater ,, Felix Mendelssohn Bartholdy”, Leipzig (2008) http://www.qucosa.de/fileadmin/data/qucosa/documents/2536/Dualit%C3%A4t_Tonregionen_Sept09.pdf
Carey, N.: Distribution Modulo 1 and Musical Scales. Ph.D. Dissertation. University of Rochester (1998)
Slater, N.B.: Gaps and steps for the sequence \(n\theta \bmod{1}\). Proceedings of the Cambridge Philosophical Society 63, 1115–1122 (1967)
Sós, V.T.: On the distribution mod 1 of the sequence nα. Annales Universitatis Scientarium Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica 1, 127–134 (1958)
Carey, N., Clampitt, D.: Aspects of well-formed scales. Music Theory Spectrum 11(2), 187–206 (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carey, N. (2011). On a Class of Locally Symmetric Sequences: The Right Infinite Word Λ θ . In: Agon, C., Andreatta, M., Assayag, G., Amiot, E., Bresson, J., Mandereau, J. (eds) Mathematics and Computation in Music. MCM 2011. Lecture Notes in Computer Science(), vol 6726. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21590-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-21590-2_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21589-6
Online ISBN: 978-3-642-21590-2
eBook Packages: Computer ScienceComputer Science (R0)