Abstract
This article contributes to the study of diatonicity in tunings with N equal divisions of the octave. The new definition we are proposing of a diatonic scale is based on two concepts. The first is well known since it concerns generated scales studied by Norman Carey and David Clampitt. The second is much less known. It is based on the sets of progressive transposition introduced by the French composer Alain Louvier. From these two concepts, we formulate a new definition of microdiatonic scales which is entirely characterized by two fundamental parameters. Then we define the majorness of a scale, by introducing an interval equivalent of the tritone by using limited transposition sets. We conclude this article by observing that under these conditions diatonicity and majorness are two different characters which do not necessarily exist in all tunings.
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
Similar content being viewed by others
References
Agmon, E.: A mathematical model of the diatonic system. Journal of Music Theory 33, 1–25 (1989)
Agmon, E.: The Languages of Western Tonality. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39587-1
Balzano, G.J.: The group-theoretic description of 12-fold and microtonal pitch systems. Comput. Music. J. 4, 66–84 (1980)
Broué, M.: Les tonalités musicales vues par un mathématicien. In: Le temps des savoirs, Le code, Dominique Rousseau, Michel Morvan (dir.), vol. 4, pp. 41–82. Odile Jacob, Revue de l’Institut universitaire de France, Paris (2001)
Carey, N., Clampitt, D.: Aspects of well-formed scales. Music Theory Spectr. 11, 187–206 (1989)
Clough, J., Douthett, J.: Maximally even sets. J. Music Theory 35, 93–173 (1991)
Clough, J.: Diatonic interval cycles and hierarchical structure. Perspect. New Music 32(1), 228–53 (1994)
Clough, J., Engebretsen, N., Kochavi, J.: Scales, sets, and interval cycles: a taxonomy. Music Theory Spectr. 21(1), 74–104 (1999)
Douthett, J., Kranz, R.: Construction and interpretation of equal-tempered scales using frequency ratios, maximally even sets, and P-cycles. J. Acoust. Soc. Am. 107, 2725–34 (2000)
Douthett, J., Hyde, M.M., Smith, C.J.: Music Theory and Mathematics, Chords, Collections and Transformations. University of Rochester Press, Rochester (2008)
Jedrzejewski, F.: Generalized diatonic scales. J. Math. Music 2(1), 21–36 (2008)
Jedrzejewski, F.: Hétérotopies musicales. Modèles mathématiques de la musique. Hermann, Paris (2019)
Johnson, T.A.: Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Scarecrow Press, Lanham (2008)
Lewin, D.: A formal theory of generalized tonal functions. J. Music Theory 26, 23–60 (1982)
Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)
Louvier, A.: Recherche et classification des modes dans les tempéraments égaux. Musurgia 4(3), 119–131 (1997)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Jedrzejewski, F. (2022). New Insights on Diatonicity and Majorness. In: Montiel, M., Agustín-Aquino, O.A., Gómez, F., Kastine, J., Lluis-Puebla, E., Milam, B. (eds) Mathematics and Computation in Music. MCM 2022. Lecture Notes in Computer Science(), vol 13267. Springer, Cham. https://doi.org/10.1007/978-3-031-07015-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-031-07015-0_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-07014-3
Online ISBN: 978-3-031-07015-0
eBook Packages: Computer ScienceComputer Science (R0)