.svg){kind=logo}
ON REPRESENTING CONSONANCE STRUCTURES
Keywords:
consonance, tuning, Bach.DOI:
https://doi.org/10.17654/0972087124002Abstract
We consider the representation of consonance structures in music, and consider two families of examples. We thus derive pieces of music which are similar to existing ones, via generalized tunings.
In the first family of examples, certain representations define interpretations of two part compositions, written on a stave. The relevant consonance structures are quivers, built from consonances between notes that are close in the score, as determined by a certain algorithm. We present some interpretations of Bach’s Invention No. 9 in F minor. Special cases of our interpretations are certain tunings to equal temperament and just intonation.
In the second family of examples, we represent consonance structures found in one part compositions written on the stave with contrapuntal pieces, with harmonics of notes in the one part composition corresponding to motifs in the contrapuntal pieces. We present an example where the one part composition is the first twelve crotchets of the folk song ‘The False Bride’.
Received: August 29, 2023;
Revised: October 18, 2023;
Accepted: October 19, 2023
References
D. J. Benson, Music: A Mathematical Offering, Cambridge University Press, 2006.
A. S. Crans, T. M. Fiore and R. Satyendra, Musical actions of dihedral groups, Amer. Math. Monthly 116(6) (2009), 479-495.
P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71-103.
G. Hadjeres, F. Pachet and F. Nielsen, DeepBach: a steerable model for bach chorales generation, Proceedings of the 34th International Conference on Machine Learning, PMLR 70, 2017, pp. 1362-1371.
P. J. Higgins, Notes on Categories and Groupoids, Van Nostrand Reinhold Math. Studies 32, 1971.
D. Lewin, Generalized Musical Intervals and Transformations, Yale University Press, New Haven, 1987.
G. Mazzola and M. Andreatta, Diagrams, gestures and formulae in music, Journal of Mathematics and Music 1(1) (2007), 23-46.
K. Stange, C. Wick and H. Hinrichsen, Playing music in just intonation: a dynamically adaptive tuning scheme, Computer Music Journal 42(3) (2018), 47-62.
The False Bride, Roud 154. We use a version collected by C. Sharp from L. White in 1904, transposed down by a perfect fourth.
https://www.vwml.org/record/CJS2/10/75.
W. Turner, Algebraic pure tone compositions constructed via similarity. http://homepages.abdn.ac.uk/w.turner/pages/.
W. Turner, Examples 1, 2, 3a, 3b, 3c, 4.
http://homepages.abdn.ac.uk/w.turner/pages/.
W. Turner, Some scales that are similar to the chromatic scale.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________
Attribution: Credit Pushpa Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
Contact Puspha Publishing House for more info or permissions.
