The Mathematical in Music Thinking

“The Mathematical in music thinking” is based on Heidegger’s understanding of “the Mathematical” as the basic assumption of the knowledge of the things. Heidegger’s ideas are combined with Peirce’s classification of sciences, in particular, to distinguish between the Mathematical from the less abstract logical thinking and the more abstract mathematical thinking. The aim of this paper is to make understandable the role of the Mathematical in music. The paper concentrates on three domains: the rhythmic of music, the doctrine of music forms, and the theory of tonal systems. The theoretical argumentations are assisted by musical examples: the Adagio of Mozart’s string quartet C major (KV 465), the second movement of Webern’s Symphony op.21, and a cadence illustrating the problem of the harmony of second degree. 1 Music Thinking and The Mathematical ”Musica est exercitium arithmeticae occultum animi” (”Music is a hidden arithmetical exercise of the soul”) this statement was written by the philosopher, mathematician, and scientist Gottfried Wilhelm Leibniz on April 17, 1712, in a letter to the mathematician and diplomat Christian von Goldbach. Leibniz referred with his statement to the astonishing phenomenon of the correspondence between musical tones and numbers which has been already demonstrated by the pythagoreans on their monochord. This phenomenon has been extensively described by the German musicologist Martin Vogel in his book “Die Lehre von den Tonbeziehungen”; there he writes: “Each interval used in music corresponds to a certain numerical proportion and, since each melody and each harmonic connection can be composed by numerically described intervals, each composition can finally be understood and analytically recognized as an arrangement of uniquely determined relations of numbers” ([18], p.9). If one wants to comprehensively understand the role of mathematics in music thinking, then the numerical relations in music compositions pointed out by Vogel do not suffice. In particular, the numerical relations cannot suitably grasp the more extended set semantics basic for modern mathematics. For our theme we use the understanding of “the Mathematical” which Martin Heidegger worked out in his 1935/36 lecture on “Basic Questions of Metaphysics” (published in [11]). For Heidegger “the Mathematical” is not derivable out of mathematics, but mathematics itself is at the time a historically, socially, and culturally determined formation abstracted from the Mathematical. Heidegger deduced his c © Radim Belohlavek, Sergei O. Kuznetsov (Eds.): CLA 2008, pp. 167–180, ISBN 978–80–244–2111–7, Palacký University, Olomouc, 2008. understanding of “the Mathematical” from the ancient Greeks: τὰ μαθήματα means “the learnable”. Learning the learnable is a kind of “taking”, by which the taker takes only such things which, strictly speaking, he already has. According to Heidegger it follows: “τὰ μαθήματα, the Mathematical, is what of the things we actually already know, which we therefore do not first take out of the things, but which we already bring with us in a certain way” ([11], p.57); or phrased in another way: “The Mathematical is that basic position to the things by which we take on the things according to that which the things have already been given to us. The Mathematical is therefore the basic assumption of the knowledge of the things” ([11], p.58). For Heidegger this makes clear the central significance of the Mathematical for modern thinking, because “a will of reformation and self-foundation of the knowledge form as such” lies in the character of the Mathematical as distinctive conception ([11], p.75). But how can we recognize the Mathematical? A promising approach is to abstract logical forms of thinking to mathematical forms of thinking which gives rise to rich mathematical theory developments retroacting, in particular, the logical forms and in this way enriching also the logical thinking (cf. [23]). To capture the Mathematical in music thinking, it suggests itself to identify first of all the logical in music, for instance in a manner as articulated by the musicologist Hans-Peter Reineke in referring to musical hearing; he writes: “Certain regulatives in musical hearing constitute and preserve music as a logical being that must sound plausibly out of itself if it shall be accepted” [17]. During the ending 18th century the term “musical logic” was linked to the idea “that music is an art which is autonomous, resting in itself, and submitted only to its own law of form; in particular, its right to exist needs not to be justified extramusically” ([3], p.66). But, inspite of numerous efforts (here, first of all, the musicologist Hugo Riemann has to be named), a musical logic has never been really established in musicology. Nevertheless, to identify the Mathematical in music thinking, the connection between logical and mathematical thinking shall be discussed more extensively. The philosopher and scientist Charles Sanders Peirce has convincingly described the connection between logical and mathematical thinking in the frame of his philosophy of science. In his classification of sciences from 1903 ([16], 258ff.), in which he ordered the sciences by the degree of their abstractness, mathematics as the most abstract science of all sciences is positioned at the most abstract level. As the only hypothetical science, mathematics has the task to develop a cosmos of forms of potential realities. All other sciences, under which philosophy is the most abstract, relate to actual realities. According to Peirce’s classification, philosophy partitions into phenomenology, normative science, and metaphysics while normative science divides further into esthetics, ethics, and logic. Musicology has to be classified such as history under the descriptive science. In Peirce’s classification the sciences are ordered in a manner that each science – refers, according to its general principles, exclusively to the sciences which are more abstract than itself, and 168 Rudolf Wille, Renate Wille-Henning