Abstract
This paper examines one aspect of Ligeti’s approach to writing music that is neither tonal nor atonal—the use of complementary collections to achieve what Richard Steinitz has termed combinatorial tonality. After a brief introduction, the paper explores properties of the intervallic content both within and between complementary collections, which I term the intra- and inter-harmonies. In particular, the inter-harmonies are useful in understanding harmonic control in works based on complementary collections, as demonstrated by revisiting Lawrence Quinnett’s analysis of Ligeti’s first Piano Étude, Désordre.
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Notes
- 1.
My thanks to Nancy Rogers and an anonymous reviewer for comments that greatly improved this paper.
- 2.
The reader is strongly directed to Amiot [1] for an excellent and detailed presentation of the interval function and its relation to recent applications of the discrete Fourier transform in music theory.
- 3.
Multiplicity of pc interval i is indicated by the \(i^{th}\) component of the interval function, which begins with pc interval 0.
- 4.
The distinctiveness of a collection, A, can also be measured in terms of the magnitude of its interval content, \(\Vert \text {IC}_A \Vert _2\). (See Callender [4].) For the present purposes, the standard deviation is preferable.
- 5.
- 6.
Measuring the distance between intra- and inter-harmonies using other metrics, such as angular (or cosine) distance, yields similar relative distances. (See Rogers [11].) The Euclidean metric is sufficient and advantageous for the present purposes.
- 7.
Statistical analysis of Désordre was greatly aided by Cuthbert’s music21 [5], which is a Python toolkit for computer-aided musicology.
- 8.
Comparison of interval counts with the inter-harmonies in Désordre in both Quinnett’s treatise and the current paper stem from our conversations while Quinnett was a student at Florida State University.
- 9.
Note that because there are more than a single degree of freedom in the data, \(\phi \) is not normalized to a maximum of 1.
References
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Callender, C. (2017). Complementary Collections in Ligeti’s Désordre . In: Agustín-Aquino, O., Lluis-Puebla, E., Montiel, M. (eds) Mathematics and Computation in Music. MCM 2017. Lecture Notes in Computer Science(), vol 10527. Springer, Cham. https://doi.org/10.1007/978-3-319-71827-9_26
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