前言: 究竟音樂理論與數學是否有直接關係? 有很多同學做和聲習作時，根本不聆聽音響，而 是在計數。可是，老師就會在這時大大禁止他們這樣做。同學們就算計數做對了和聲答案，也不算甚麼。可是在音樂理論的研究裡，數學是否沒有任何地位? 如果是有，也是甚麼呢? 以下的閱讀文章評論，相信可以簡易的提出了答案。
Music Theory and Mathematics
The story of Pythagorean’s discovering the mathematical ratios illustrates not only the establishment of the underlying the science of harmonics but also a frame of reference in music-theoretical thought in the association between music and number. Indeed the relationship between music theory and mathematical models is not through number alone but through the more fundamental notions of universality and truth embedded in Pythagorean and Platonic mathematics and philosophy that one best begin to apprehend the broad range of interrelationships between music theory and mathematics. Catherine Nolan discusses such a relationship in several perspectives of topics, including numerical models, geometric imagery, combinatorics, set theory and group theory and transformational theory in this article.
The article is set out in the discussion of Pythagoreanism in two aspects: numbers are constituent elements of reality and numbers and their rations provide the key to explaining the order of nature and the universe. Nevertheless, central to Pythagorean mathematics is a theory of ratio, the relation of two quantities, and a theory of proportion, the relation of two or more ratios. These theories explain musical intervals in terms of ratios and combinations of ratios, correlating music theory with acoustic science. Through the ratios the dissonant intervals are computed in relation to the consonances.
The rich implications of Pythagorean and Platonic philosophy and mathematics, r4atios and magnitudes and their geometric representation, governed the science of music from the Middle Ages to the Renaissance. Ratio and proportion indeed are understood today in the form of algebraic terms. But they were conceived in Greek mathematics as a close association of arithmetic and geometry epitomized by proportional relations of lengths of vibrating strings. Pythagorean’s notion of ratio and numbers thus developed the diatonic tuning which was a music theory remained virtually unchallenged until the fifteen-century. In the monograph Le istitutioni harmoniche (1558), for example, Zarlino, a well-known Renaissance music theorist, appealed to this number theory for theoretical justification of the imperfect and perfect consonances.
While the representation of musical intervals through numbers has undoubtedly been most important to music theory, other kinds of mathematical models have also been adopted. This is the notion of geometric images which is used as heuristic devices to conceptualize music theory from number and proportion to logical and spatial representations of relations. Boethius, for example, utilized geometric figures, ideal universal shapes constructed mainly of lines, circles and arcs, to illustrate harmonic rations and divisions of the monochord. This seminal music theory indeed covered a long tradition extending back from Boethius and early medieval of monochord tunings of engaging geometric space to represent harmonic space to Descartes and the seventeenth-century Enlightenment of formulating a coordinate system and analytical geometry in La geometrie. However, another important mathematical branch, Combinatorics, which concerned with numeration, grouping and arrangements of elements in finite collections or sets, almost came parallel with the development of Descartes’ notion of analytical geometry in the same period. The mathematical combinatorics appeared in the form of ars combinatorial inspired numerous discourses on rational methods of musical composition by a variety of authors of theoretical treatises and practical manuals. Kirnberger, for example, described compositional decision-making by selection, using chance procedures from the total compilation of permutations of a given unit such as melodic or rhythmic figure or a two-part melodic-harmonic module. Thus, from the above discussion, music theory developed in relation to mathematical numbers, ratios and geometric imagery throughout the history was inextricably associated with the development of western science and humanism and rationalism in western culture.
The mainstream of music theory in relation to mathematics in the Modern period was undoubtedly the concept of Set theory, which was initiated by Milton Babbitt and Allen Forte. The algebraic structures of set theory and group theory are designed to explain harmonic innovations in the refractory repertoire of post-tonal music and extend to theoretical studies of other musical parameters and harmonic languages of systems. Even in the present academic field of music theory, there is a growing number of mathematicians and theorists continuing to explore and generalize the algebraic structure of the diatonic system and scales or tonal systems of disparate origins, ranging from diatonic or microtonal scale systems to medieval or non-Western modal systems. Furthermore, the music theorist David Lewin expounded a series of profound treatises, Generalized Musical Intervals and Transformations, in 1987 was also an important concept of mathematical music theory. His GIS covered a delineation of a formal space consisting of three elements: a set of musical objects, a mathematical group of generalized intervals and a function that maps all possible pairs of objects in the system into the group of intervals. Lewin’s model creates a transformation network recasting the role of generalized intervals, modeling actions upon or motions between objects, and is termed as “Transformation Theory”.
Therefore, mathematics brings to music theory not only the technical means to perform measurements and computations and the statistical means to correlate data, but also the conceptual means, symbols and vocabulary need in order to model musical relations of various kinds and to delineate levels of abstraction. The diversity of music theories developed throughout the western history, undoubtedly, tells the fact that from the ancient Greek to the postmodern present, the studies of numbers, ratios, sets, geometries are all inseparable from music and its related concepts. As such, to regard music theory as a fundamental mathematical paradigm is to enhance our understanding of one of the most abstract form of arts (music) in the world with a rather rational process – mathematical reasoning. In this sense, hearing music is no longer an irrational aesthetic process, which has also been the core of our understanding of music for many centuries in the western history of music.
David Leung (theorydavid)