Cool Math for Hot Music

Joining mathematics and music for most of us creates a love-hate relationship, although historically, with the Pythagorean origin, these two fields of human knowledge and activity were united. In this book we don’t want to enforce their unification for two reasons: different evolution of these fields, and major creative interaction.

This chapter is a short overview of some important persons and movements in the history of the interaction of mathematics and music. It is far from complete, but should give the reader a first impression of this traditional and deeply “mathemusical” culture.

We present the theory of pure sets—the basis of all classical mathematics—in the form of the Zermelo-Fraenkel Axiomatics. This theory is illustrated with Iannis Xenakis’ composition Herma.

The elements of a set can be written in any order, and we don’t yet have a way to define ordered structures. So we now develop the theory of ordered structures. It also allows us to define the central concept of a function.

In this chapter, we present the fundamental concept of a cartesian product and its dual, the coproduct, which is obtained from the first one by reversing arrows. We also introduce exponentials and subobject classifiers. These concepts are broadly used in modern mathematics, because they describe constructions that appear everywhere in mathematical creativity.

Natural numbers are the first topic studied by all students in first years of elementary school. Here, the classic definitions of ordinal and natural numbers are entirely derived from set theory. The well-known five Peano axioms that define natural numbers are now presented as a theorem.

Recursion is a technique to define concepts that depend on natural numbers. First, such a concept is defined for n = 0. Then, the concept for n is supposed to be defined, and we use the concept for n to define the concept for n+1. Therefore, it is defined for all natural numbers n. This is the idea of recursion, namely the definition by induction. We will prove that this mathematical process is possible. We then apply recursion to create musical compositions.

In the previous chapter, we defined natural numbers. Now, we are interested in how to combine these numbers to obtain other natural numbers. In this chapter, we will define three operations: addition, multiplication, and exponentiation.

Romans used letters to denote natural numbers. Language of modern computers utilizes binary representation. Here we describe different ways to represent natural numbers, ending the chapter with an important theorem about prime numbers, already known to Euclid (300 BC).

Summer temperatures in Minnesota are around 30 Celsius, while in winter they go down to -30 Celsius. The first one is a natural number, and the second one is the same natural number with a minus sign. Both belong to the set of integer numbers (integers). We introduce these numbers to solve equations of type a+x = b not only for a ≤ b, but also in the case of a > b. The arithmetics developed so far for natural numbers will be extended to integers.

In Latin, ratio means rapport, division of two things. Here we introduce rational numbers as fractions of two integer numbers a and b. This procedure allows us to solve equations of type a・x = b with a≠ 0. Our strategy again follows the philosophy that the problem is the solution.

We have used the philosophy of the problem being the solution to construct integer and rational numbers when dealing with equations of type a + x = b or a • x = b. But there are many other equations, especially dealing with approximations in music theory, that cannot be solved with Z or Q. In this chapter we apply the above philosophy to find solutions of such problems, namely the real numbers.

Corollary 2 in Chapter 12 is crucial for the construction of some important structures for real numbers, such as general roots and logarithms. These are introduced in this chapter. We also discuss musical applications to pitch theory.

Square roots of negative real numbers are not defined yet. We introduce complex numbers to solve this problem. Essentially, we introduce an imaginary number i, the square root of −1, and thereby add a new dimension to the real numbers.

Up to now, we have been able to construct all basic number domains ℕ, ℤ, ℚ, ℝ, ℂ. But we have not considered geometric objects. This chapter begins to fill that gap. It introduces the most elementary geometric objects: graphs—systems of points and arrows connected by directed or undirected lines. We shall conclude part IV with the introduction of higher-dimension graphical objects that relate to coverings of sets by a system of subsets.

In biology, nerves connect different parts of a body. In music, we also can construct “nerves,” which are structures that connect different parts of a composition, making communication between such parts possible—similar to biology.

Monoids are the simplest type of algebraic structure, and for this reason they are omnipresent in mathematics. This situation is parallel to the hierarchy of numbers. The monoids will be extended structurally (not as sets!) to groups, rings, and modules later.

Monoids that have only invertible elements are called groups. Groups are the most important single structure in algebra and have enormous applications in physics as well.

In musical creativity, actions are very important. They always deal with two components: the agent who acts in a determined way, and the object on which this action is being performed. Of course, not just any action can be performed on any object, so one has to specify agent/action as well as domain of objects that are suitable for a determined action. In this chapter we develop the formalism of group actions and provide introductory examples.

This chapter deals with the study of the permutation groups S_n. We also give examples of compositional methods using permutation groups.

This chapter deals with the mathematics of the third torus group ℤ₃ × ℤ₄ and its symmetries and then applies these structures to music theory, in particular to counterpoint.

This chapter deals with an analysis of John Coltrane’s famous composition Giant Steps from 1959, released in 1960 on the synonymous LP.

Modulation is a central theme in tonal musical composition. It means the transition from one tonality to another. Of course such a situation is not omnipresent since either the very concept of a tonality is not given (in the composer’s or in the theorist’s mind), or when present, there is no real theory of how to modulate. This chapter dös not intend to present the one and only modulation theory, but is written to prove that precise mathematical conceptualization and the application of mathematical methods can lead to explicit and efficient models of modulation.

Rings are the basic structures for algebra. We already have many examples of rings: the integers, real and complex numbers, and the structure of addition and multiplication that was defined on ℤ_n in the chapter about the third torus and its geometry.

Prime numbers play a crucial role in music theory, and in particular in the theory of tuning. In this chapter, we prove uniqueness of prime decomposition for the integers and polynomial rings.

Matrices are a very classical tabular form to represent data, for example in accounting. They are built from columns that are juxtaposed and can be split horizontally into a stack of rows. The novelty in mathematics is that matrices that are built from numbers can be used to perform calculations that are of general benefit to mathematics.

Many core structures in algebra are richer than groups but poorer than rings. For example, an ideal I ⊂ R in a commutative ring is an additive subgroup, but not a ring because it has no 1 in general. However, one may multiply elements of I with any ring elements. Also, the set M_m,n(R) is an additive group, but not a ring for n ≠ m. Its structure as a cartesian product ring Rmn is rarely considered. But again, one may multiply a matrix by a “scalar” from R. These structures remind us of vector calculus in high school. This is what we now want to investigate for the sake of music theory. The structure of this type is called a “module”, and we want to give a short and very incomplete account of the theory of modules, which plays a major role in mathematical music theory.

Western tuning systems all are selections of subsets of pitches within EulerSpace. We only discuss two typical tunings here: just tuning and 12-tempered tuning.

We have often referred to certain types of Fig. 29.1. Categories are the smiley of contemporary mathematics. structures—sets, monoids, groups, rings, digraphs, or modules—where there was a shared structural characteristic: All of these structures have objects (such as sets, monoids, groups, etc.) and a type of “function” (set functions, monoid morphisms, digraph morphisms, etc.). And all of these functions can be composed if domains and codomains can be ‘concatenated.’ The common denominator of these structures is the concept of a category.

Despite the rich algebraic formalism of monoids, groups, rings, and modules, we lack a type of analysis that does not compare objects by their transformational relations, such as symmetries or module homomorphisms, but by their similarity—referring to the paradigm of deformation. This type of relationship is what topology, the mathematics of continuity, is about.

Differentiability is stronger than continuity in that for a differentiable curve, we need to have a slope line at every point, i.e., the curve must not have corners. This chapter deals with this concept and its application to music.

Performance is understood to be the transformation of a symbolic musical object of notes—as represented in a score of Western tradition—to a physical object composed of sound events. Going beyond the common description of performance, we shall present a mathematical theory of this type of transformation.

Gestures are complex in their common understanding. The concept of a gesture has never been thoroughly defined to this date, although gestures are very important in humancomputer interface design, human expressivity in and beyond common language, and above all in the arts. Painting, dance, music, theater, and film would not be understood without gestural concepts and processes. We therefore will give a short introduction to the first mathematical theory of gestures.

This chapter contains the solutions of the mathematical and musical exercises. Each solution number corresponds to the exercise number.