ABSTRACT
The study of music by means of mathematics goes back several thousand years. Well documented are, for instance, mathematical and philosophical studies by the Pythagorean school in ancient Greece (see e.g. van der Waerden 1979). Advances in mathematics, computer science, psychology, semiotics, and related fields, together with technological progress (in particular computer technology) lead to a revival of quantitative thinking in music in the last two to three decades (see e.g. Archibald 1972, Solomon 1973, Schnitzler 1976, Balzano 1980, Go¨tze and Wille 1985, Lewin 1987, Mazzola 1990a, 2002, Vuza 1991, 1992a,b, 1993, Keil 1991, Lendvai 1993, Lindley and Turner-Smith 1993, Genevois and Orlarey 1997, Johnson 1997; also see Hofstadter 1999, Andreatta et al. 2001, Leyton 2001, and Babbitt 1960, 1961, 1987, Forte 1964, 1973, 1989, Rahn 1980, Morris 1987, 1995, Andreatta 1997; for early accounts of mathematical analysis of music also see Graeser 1924, Perle 1955, Norden 1964). Many recent references can be found in specialized journals such as Computing in Musicology, Music Theory Online, Perspectives of New Music, Journal of New Music Research, Inte´gral, Music Perception, and Music Theory Spectrum, to name a few. Music is, to a large extent, the result of a subconscious intuitive “pro-
cess”. The basic question of quantitative musical analysis is in how far music may nevertheless be described or explained partially in a quantitative manner. The German philosopher and mathematician Leibniz (1646-1716) (Figure 1.5) called music the “arithmetic of the soul”. This is a profound philosophical statement; however, the difficulty is to formulate what exactly it may mean. Some composers, notably in the 20th century, consciously used mathematical elements in their compositions. Typical examples are permutations, the golden section, transformations in two or higher-dimensional spaces, random numbers, and fractals (see e.g. Scho¨nberg, Webern, Barto´k, Xenakis, Cage, Lutoslawsky, Eimert, Kagel, Stockhausen, Boulez, Ligeti, Barlow; Figures 1.1, 1.4, 1.15). More generally, conscious “logical” construction is an inherent part of composition. For instance, the forms of sonata and symphony were developed based on reflections about well balanced proportions. The tormenting search for “logical perfection” is well
documented in Beethoven’s famous sketchbooks. Similarily, the art of counterpoint that culminated in J.S. Bach’s (Figure 1.2) work relies to a high degree on intrinsically mathematical principles. A rather peculiar early account of explicit applications of mathematics is the use of permutations in change ringing in English churches since the 10th century (Fletcher 1956, Price 1969, Stewart 1992, White 1983, 1985, 1987, Wilson 1965). More standard are simple symmetries, such as retrograde (e.g. Crab fugue, or Canon cancricans), inversion, arpeggio, or augmentation. A curious example of this sort is Mozart’s “Spiegel Duett” (or mirror duett, Figures 1.6, 1.7 ; the attibution to Mozart is actually uncertain). In the 20th century, composers such as Messiaen or Xenakis (Xenakis 1971; figure 1.15) attempted to develop mathematical theories that would lead to new techniques of composition. From a strictly mathematical point of view, their derivations are not always exact. Nevertheless, their artistic contributions were very innovative and inspiring. More recent, mathematically stringent approaches to music theory, or certain aspects of it, are based on modern tools of abstract mathematics, such as algebra, algebraic geometry, and mathematical statistics (see e.g. Reiner 1985, Mazzola 1985, 1990a, 2002, Lewin 1987, Fripertinger 1991, 1999, 2001, Beran and Mazzola 1992, 1999a,b, 2000, Read 1997, Fleischer et al. 2000, Fleischer 2003). The most obvious connection between music and mathematics is due to
the fact that music is communicated in form of sound waves. Musical sounds can therefore be studied by means of physical equations. Already in ancient Greece (around the 5th century BC), Pythagoreans found the relationship between certain musical intervals and numeric proportions, and calculated intervals of selected scales. These results were probably obtained by studying the vibration of strings. Similar studies were done in other cultures, but are mostly not well documented. In practical terms, these studies lead to singling out specific frequencies (or frequency proportions) as “musically useful” and to the development of various scales and harmonic systems. A more systematic approach to physics of musical sounds, music perception, and acoustics was initiated in the second half of the 19th century by path-breaking contributions by Helmholz (1863) and other physicists (see e.g. Rayleigh 1896). Since then, a vast amount of knowledge has been accumulated in this field (see e.g. Backus 1969, 1977, Morse and Ingard 1968, 1986, Benade 1976, 1990, Rigden 1977, Yost 1977, Hall 1980, Berg and Stork 1995, Pierce 1983, Cremer 1984, Rossing 1984, 1990, 2000, Johnston 1989, Fletcher and Rossing 1991, Graff 1975, 1991, Roederer 1995, Rossing et al. 1995, Howard and Angus 1996, Beament 1997, Crocker 1998, Nederveen 1998, Orbach 1999, Kinsler et al. 2000, Raichel 2000). For a historic account on musical acoustics see e.g. Bailhache (2001). It may appear at first that once we mastered modeling musical sounds
by physical equations, music is understood. This is, however, not so. Music is not just an arbitrary collection of sounds – music is “organized sound”.
Physical equations for sound waves only describe the propagation of air pressure. They do not provide, by themselves, an understanding of how and why certain sounds are connected, nor do they tell us anything (at least not directly) about the effect on the audience. As far as structure is concerned, one may even argue – for the sake of argument – that music does not necessarily need “physical realization” in form of a sound. Musicians are able to hear music just by looking at a score. Beethoven (Figures 1.3, 1.16) composed his ultimate masterpieces after he lost his hearing. Thus, on an abstract level, music can be considered as an organized structure that follows certain laws. This structure may or may not express feelings of the composer. Usually, the structure is communicated to the audience by means of physical sounds – which in turn trigger an emotional experience of the audience (not necessarily identical with the one intended by the composer). The structure itself can be analyzed, at least partially, using suitable mathematical structures. Note, however, that understanding the mathematical structure does not necessarily tell us anything about the effect on the audience. Moreover, any mathematical structure used for analyzing music describes certain selected aspects only. For instance, studying symmetries of motifs in a composition by purely algebraic means ignores psychological, historical, perceptual, and other important issues. Ideally, all relevant scientific disciplines would need to interact to gain a broad understanding. A further complication is that the existence of a unique “truth” is by no means certain (and is in fact rather unlikely). For instance, a composition may contain certain structures that are important for some listeners but are ignored by others. This problem became apparent in the early 20th century with the introduction of 12-tone music. The general public was not ready to perceive the complex structures of dodecaphonic music and was rather appalled by the seemingly chaotic noise, whereas a minority of “specialized” listeners was enthusiastic. Another example is the
