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.
This work was partially supported by a grant from the Niels Hendrik Abel Board.
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- 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.
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Acknowledgments
We thank Thomas Noll at Escola Superior de Música de Catalunya, Daniel Tompkins at Florida State University, and the anonymous referees for their valuable feedback.
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Appendix: Source Code and Data
Appendix: Source Code and Data
The following code implements a function in Octave (version 4.2.0) to calculate the number of counterpoint symmetries per step in a sequence of counterpoint intervals, encoded as columns of a matrix.
The listed arrays contains the analyzed intervals extracted from Scriabin’s sonata, one for each part.
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Agustín-Aquino, O.A., Mazzola, G. (2019). Contrapuntal Aspects of the Mystic Chord and Scriabin’s Piano Sonata No. 5. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_1
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