Foundations: Axioms, Notation, and Historical Precursors

The mathematical architecture of music begins with a simple yet profound observation: musical pitch, when abstracted from timbre and dynamics, inhabits a discrete, cyclic space. This space, which we shall formalize as the pitch class set $\mathbb{Z}_{12}$, is not an arbitrary construct but an empirical condensation of centuries of acoustic discovery and theoretical refinement. To build a rigorous framework for the entire text, we must first establish the foundational axioms, notation, and historical precursors that have shaped the mathematical discourse on music. This chapter lays that groundwork by defining the primitive musical objects—pitch classes, durations, intervals—as elements of well-defined mathematical sets. We then trace the critical historical milestones that led to this formalization, from the Pythagorean discovery of simple integer ratios to the twentieth-century advent of set theory. Finally, we introduce the essential mathematical tools—modular arithmetic, logarithmic measurement, and combinatorial principles—that will serve as our analytical lingua franca. The objective is not merely to recount history or define terms, but to demonstrate how a consistent, axiomatic approach reveals the deep symmetries and generative principles underlying musical structure.

1.1 Axiomatic Foundations: Defining the Primitive Objects

Any mathematical treatment of music must begin with a clear specification of its elementary constituents. We adopt an axiomatic approach, positing a small set of undefined terms and then defining more complex structures in terms of them. Our primitive concepts are pitch, duration, and loudness. For the purposes of this text—which focuses on the architecture of pitch and time in common-practice and post-tonal repertoires—we will treat loudness as a secondary parameter, though its mathematical modeling (e.g., via decibel scales or psychoacoustic loudness models) is a rich field in its own right. Our primary focus is thus on the first two primitives.

1.1.1 Pitch and the Pitch Class Set

We define a pitch as a perceptual correlate of a fundamental frequency $f$ measured in Hertz (Hz). However, music theory operates at a higher level of abstraction. Humans perceive pitches that are related by simple frequency ratios as "equivalent" in a functional sense (e.g., all C's across octaves). This equivalence relation, called octave equivalence, partitions the infinite set of possible frequencies into equivalence classes. Each class is a pitch class.

Definition 1.1 (Pitch Class). A pitch class is the set of all pitches related by integer multiples of 2 (i.e., $f$ and $2^k f$ for any integer $k$). The perceptual basis for this equivalence is the cochlear mechanics of the basilar membrane and the phenomenon of periodicity pitch.

To represent pitch classes mathematically, we use the chromatic model derived from the division of the octave into 12 equal semitones, a convention that dominates Western art music. This model is not universal (e.g., Turkish makam or Indonesian pelog systems use different divisions), but it provides a powerful, standardized framework. We assign integers modulo 12 to these pitch classes, typically with $0 \equiv \text{C}$, $1 \equiv \text{C}\sharp/\text{Db}$, $2 \equiv \text{D}$, and so forth. This yields the set:

$$\mathbb{Z}_{12} = \{0, 1, 2, \dots, 11\}$$

where addition is performed modulo 12. This algebraic structure—a cyclic group of order 12—is the fundamental pitch class space. It is crucial to note that $\mathbb{Z}_{12}$ is a *model* of pitch class relations under the assumption of 12-tone equal temperament (12-TET). In other tuning systems, the interval relationships between adjacent pitch classes may not be equal, but the cyclic ordering remains. For a general, tuning-independent representation, we may use a continuous circle $S^1$ parameterized by a real number modulo 1 (representing the octave fraction), but the discrete $\mathbb{Z}_{12}$ model is indispensable for combinatorial and set-theoretic analysis.

The choice of integer 0 for C is conventional. Any pitch can serve as the origin; the mathematics is invariant under translation. However, the conventional mapping aligns with standard MIDI note numbers, where note 60 is Middle C ($C_4$), and pitch class is obtained by $pc(n) = n \bmod 12$. This bridge to digital music technology underscores the practical utility of our formalism.

1.1.2 Duration and the Rhythm Lattice

Duration, the temporal span of a musical event, is less standardized than pitch class. We can model duration as a positive real number (seconds), but musical practice often employs a discrete set of rhythmic values related by simple ratios (e.g., whole note = 2 half notes = 4 quarter notes). This suggests modeling duration multiplicatively.

A common approach, particularly in set theory and transformational theory, is to normalize all durations relative to a smallest unit (the "beat" or "tactus"). Let $u$ be this unit (e.g., a sixteenth note). Then any rhythmic value can be expressed as $k \cdot u$ for some positive integer $k$. The set of possible durations is then $\mathbb{N}^+ \cdot u$. However, this model fails to capture triplets or other irrational ratios.

A more flexible model uses the positive rational numbers $\mathbb{Q}^+$ as the duration space, with a reference unit $u$. A duration $d$ is then $d = q \cdot u$ with $q \in \mathbb{Q}^+$. This allows for any rational ratio (e.g., a triplet quarter note is $\frac{2}{3}$ of a regular quarter note). The operation is multiplicative: two consecutive durations add in time, but their ratio is multiplicative. For analytical purposes, we often take logarithms to convert multiplication to addition:

$$\log(d_2) - \log(d_1) = \log\left(\frac{d_2}{d_1}\right)$$

This logarithmic duration space is isomorphic to the additive real numbers, allowing us to apply linear techniques to rhythmic analysis.

In many analytical contexts, especially when dealing with pitch-class sets, duration is abstracted away entirely, and we focus on the temporal order of events, i.e., a sequence. This leads to the concept of a rhythmic pattern as a word over an alphabet of duration types.

1.1.3 Intervals: The Measure of Difference

Given our pitch class space $\mathbb{Z}_{12}$, the most fundamental relation is interval. Intervals measure the distance between pitch classes. There are two complementary perspectives:

1. Ordered interval (directed interval): Given two pitch classes $a, b \in \mathbb{Z}_{12}$, the ordered interval from $a$ to $b$ is $b - a \pmod{12}$. This is an element of $\mathbb{Z}_{12}$ itself. For example, from C (0) to E (4), the interval is 4 semitones (a major third). From E (4) to C (0), the interval is $0 - 4 = -4 \equiv 8 \pmod{12}$, a minor sixth. Ordered intervals capture melodic direction.

2. Unordered interval (interval class): The absolute distance, disregarding direction. We define the interval class $ic(x)$ for $x \in \mathbb{Z}_{12}$ as: $$ic(x) = \min(x, 12 - x)$$ This yields values in $\{0, 1, 2, 3, 4, 5, 6\}$. Interval class 0 is the unison, 1 the minor second/major seventh, 2 the major second/minor seventh, etc. Interval class 6 is the tritone, which is its own complement.

This duality is fundamental. In atonal set theory, pitch-class sets are analyzed up to transposition (addition of a constant $t \in \mathbb{Z}_{12}$) and inversion (multiplication by $-1$ modulo 12, i.e., $i(x) = -x \pmod{12}$). The interval class vector (or IC vector) of a set counts the occurrences of each interval class between all distinct pairs of elements. This vector is invariant under transposition and inversion, making it a primary normal form for set comparison.

For durations, an analogous concept exists: the rhythmic interval or ratio between two durations $d_1, d_2$ is $\frac{d_2}{d_1}$ (or its logarithm). In additive rhythmic