Abstract
Norman Carey and David Clampitt observed in [4] that each region has two well-formed scales as its prefixes. If one looks at this finding from the viewpoint of word theory, one observes that regions are central words and the two prefixes are their independent periods. More precisely, each region, understood as a word in a two-letter alphabet, contains two distinct prefixes, both of which represent well-formed scales. One period is a special standard word, and the other period is a non-special standard word. Thomas Noll proposed in [13] to generalize the authentic Ionian mode through special standard words. He showed that the property of divider incidence characterizes these words among their conjugates. Thus there are two parallel lines of generalization which can be further enriched by observations from [7], [8], as well as by further combinatorial connections between central and standard words.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berstel, J., de Luca, A.: Sturmian words, Lyndon words and trees. Theoretical Computer Science 178, 171–203 (1997)
Berstel, J., Lauve, A., Reutenauer, C., Saliola, F.V.: Combinatorics on words: Christoffel words and repetitions in words. CRM Monograph Series, vol. 27. American Mathematical Society, Providence (2009)
Berthé, V., de Luca, A.: Christophe Reutenauer. On an involution of Christoffel words and Sturmian morphisms. European Journal of Combinatorics 29(2), 535–553 (2008)
Carey, N., Clampitt, D., Regions: A theory of tonal spaces in early medieval treatises. Journal of Music Theory 40(1), 113–147 (1996)
Carey, N., Clampitt, D.: Aspects of well-formed scales. Music Theory Spectrum 11(2), 187–206 (1989)
Clampitt, D., Noll, T.: Modes, the height-width duality, and divider incidence. In: Society for Music Theory national conference, Nashville, TN (2008)
Clampitt, D.: Double neighbor polarity (unpublished paper) (2008)
Clampitt, D.: Sensitive intervals: major third analogues in standard well-formed words. In: Workshop on Mathematical Music Theory, South Bristol, ME (2008)
Clampitt, D., Noll, T.: Regions within enveloping standard modes (unpublished paper) (2009)
DomÃnguez, M., Clampitt, D., Noll, T.: Well-formed scales, maximally even sets and Christoffel words. In: Proceedings of the MCM 2007, Berlin, Staatliches Institut für Musikforschung (2007)
Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)
Kassel, C.: From Sturmian morphisms to the braid group B 4. In: The Workshop on Braid Groups and Applications, Banff International Research Station (2004)
Noll, T.: Ionian theorem. Journal of Mathematics and Music 3(3) (to appear, 2009)
Noll, T., Clampitt, D., DomÃnguez, M.: What Sturmian morphisms reveal about musical scales and tonality. In: Proceedings of the WORDS 2007 Conference, Marseille (2007), http://iml.univ-mrs.fr/words2007/
Singler, F.: Zur Dualität zwischen doppelter Periodizität und binärer Intervall-Struktur in der Theorie der Tonregionen. Thesis (final paper). Hochschule für Musik und Theater Felix Mendelssohn Bartholdy, Leipzig (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Clampitt, D., Noll, T. (2009). Regions and Standard Modes. In: Chew, E., Childs, A., Chuan, CH. (eds) Mathematics and Computation in Music. MCM 2009. Communications in Computer and Information Science, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02394-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-02394-1_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02393-4
Online ISBN: 978-3-642-02394-1
eBook Packages: Computer ScienceComputer Science (R0)