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2019 | OriginalPaper | Chapter

Contrapuntal Aspects of the Mystic Chord and Scriabin’s Piano Sonata No. 5

Authors : Octavio A. Agustín-Aquino, Guerino Mazzola

Published in: Mathematics and Computation in Music

Publisher: Springer International Publishing

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Abstract

We present statistical evidence for the importance of the “mystic chord” in Scriabin’s Piano Sonata No. 5, Op. 53, from a computational and mathematical counterpoint perspective. More specifically, we compute the effect sizes and perform \(\chi ^{2}\) tests with respect to the distributions of counterpoint symmetries in the Fuxian, mystic, Ionian and representatives of the other three possible counterpoint worlds in two passages of the work, which provide evidence of a qualitative change between the Fuxian and the mystic worlds in the sonata.

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Appendix
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Footnotes
1
All the musical events of the work, harmonic as well as melodic and contrapuntal, are essentially within this six-tone complex: “[...] all the melodic voices are on the sounds of the accompanying harmony, all counterpoints are subordinated to the same principle”.
 
2
The statement is correct: The basic harmony of Prometheus is no longer understood and treated by Scriabin as dissonance: “This is a basic harmony, a consonance”.
 
3
It is interesting to note that the mystic chord can be seen as the first of a sequence of dichotomies that are always strong in microtonal equitempered tunings, as described in [2, Proposition 2.4 and Remark 3].
 
4
This name stems from the fact that this representative consists in all proper (non-vanishing) intervals in the Ionian mode, when counted from the tonic.
 
5
Historical musical objects may not be relevant in terms of this counterpoint model. For example, the Lydian dichotomy induced by the pc-set \(\{2,4,6,7,9,11\}\) could be seen as a good candidate for a system of consonances, but is not strong because it has as a non-trivial inner affine symmetry.
 
6
We denote with \(T^{x}\) the transposition by x.
 
7
The pairs (x, y) can also be written as \(x+\epsilon .y\) with \(\epsilon ^{2}=0\) in commutative algebra. The reason to introduce this algebraic structure is that it describes tangent vectors (see [12, Section 7.5] for further details).
 
8
This, by the way, leads to a natural concept of dissonant counterpoint [5].
 
9
In Renaissance counterpoint the notion of resolution is understood only by stepwise movement of voices [10, p. 131], but the model can trivially be restricted to fulfill this requirement.
 
10
A counterpoint world is a directed graph, where each vertex is a counterpoint interval and there is an arrow connecting for each valid step. See [1, Chapter 4] for further details.
 
11
The effect size we take is the so-called Cohen’s d, which is the mean difference on the means between the two variables divided by the pooled standard deviation. See [6] for further details.
 
12
In fact, the kurtosis of the distribution of the number of symmetries per step in the Fuxian world are 4.70239 and 7.55462 for part 1 and 2, respectively. This means that the second distribution deviates less from its mean, and thus in this case Cohen’s d does not explain the change sufficiently because the mean of both distributions is very close to the general one. This was also observed in the first-species fragments of Misae Papae Marcelli by G. P. Palestrina against the general distribution; see [13] for details.
 
Literature
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3.
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Eberle, G.: Zwischen Tonalität und Atonalität: Studien zur Harmonik Alexander Skrjabins, Berliner Musikwissenschaftliche Arbeiten, vol. 14. Musikverlag Emil Katzbichler, München-Salzburg (1978)
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Fux, J.J.: Gradus Ad Parnasum. W. W. Norton, New York (1965)
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Kostka, S., Payne, D.: Tonal Harmony, 3rd edn. McGraw-Hill Inc., Boston (1995)
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Nieto, A.: Una aplicación del teorema de contrapunto. B. Sc. thesis, ITAM (2010)
14.
Scriabine, A.: Cinquème sonate pour piano, Op. 53. Self-published, Lausanne (1907)
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Trombley, R.: Impressionism in music. In: DiMartino, D., et al. (ed.) Music in the 20th Century, vol. 2, p. 305. Routledge (1999)
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Warner, R.M.: Applied Statistics. SAGE, California (2012)
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Wise, H.H.: The relationship of pitch sets to formal structure in the last six piano sonatas of Scriabin. Ph.D. thesis, University of Rochester (1987)
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Zydek, S.: The harmonic and melodic language in Alexander Scriabin’s Sonata No. 5, Op. 53 (2010). essay for Dr. Janners’ course Music Theory II
Metadata
Title
Contrapuntal Aspects of the Mystic Chord and Scriabin’s Piano Sonata No. 5
Authors
Octavio A. Agustín-Aquino
Guerino Mazzola
Copyright Year
2019
Book
Mathematics and Computation in Music

Print ISBN: 978-3-030-21391-6

Electronic ISBN: 978-3-030-21392-3

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-21392-3_1

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