Mathematical Foundations for Music Theory 数学音乐理论的基础

在音乐理论与数学的交叉领域，我们面临着如何用精确语言描述听觉现象的根本问题。音乐，作为一种时间艺术与声音艺术，其核心要素——音高、时值、和声——本质上都可以用数学结构来刻画。然而，长期以来，音乐理论依赖于模糊的描述性语言和直觉性的分析框架，这使得理论家们难以进行严谨的推理和跨时期的比较分析。本章将建立一套严格的数学工具，使我们能够精确地定义音乐理论中的基本概念，并在此基础上发展出能够应用于从巴洛克到二十世纪乃至当代音乐的统一分析框架。

The fundamental challenge at the intersection of music theory and mathematics lies in developing a precise language to describe auditory phenomena. Music, as both a temporal and sonic art form, has its core elements—pitch, duration, harmony—amenable to mathematical description. Yet music theory has long relied on vague descriptive language and intuitive analytical frameworks, making rigorous reasoning and cross-historical comparison difficult. This chapter establishes a rigorous mathematical apparatus enabling precise definitions of fundamental music-theoretic concepts, upon which we can develop a unified analytical framework applicable from the Baroque period through the twentieth century and contemporary music.

我们首先需要回答一个看似简单却至关重要的问题：什么是一个音高级（pitch class）？在音乐实践中，当我们听到钢琴上所有的C音时，无论它们来自低音区的C1、中音区的C4，还是高音区的C7，我们都能感知到某种共同的“音乐身份”。这种感知上的共性——即无论音高在物理上位于何处，只要它们产生相同的音名——正是音高级这一概念的心理物理基础。然而，数学化这一直觉需要我们仔细审视八度等价（octave equivalence）和等音等价（enharmonic equivalence）这两种不同的等价关系。

We must first answer what appears to be a simple yet crucial question: what exactly is a pitch class? In musical practice, when we hear all C's on the piano—whether from the low register's C1, middle register's C4, or high register's C7—we perceive some common "musical identity." This perceptual commonality—that notes sharing the same pitch name are recognized as essentially the same, regardless of their physical location—is the psychoacoustic foundation of the pitch class concept. However, mathematicalizing this intuition requires careful examination of two distinct equivalence relations: octave equivalence (C4 and C5 are "the same") and enharmonic equivalence (C♯ and D♭ are "the same").

集合论基础与音高级集合

Set-Theoretic Foundations and Pitch Class Collections

现代音乐理论中的集合论方法可以追溯到二十世纪中叶，随着十二音作曲技术的发展和序列主义理论的兴起，音乐理论家们开始寻求一种超越传统功能和声分析框架的通用描述工具。艾伦·福特（Allen Forte）于1963年发表的《音级集合的理论与实践》（"Structures of Atonal Music"）标志着这一领域的系统性开端，他引入的音级集合（set-class）概念至今仍是无调性音乐分析的核心工具。然而，这一方法的数学基础——作为现代数学基础概念的集合论——在音乐理论中的应用有着更为深刻的理论根源。

The set-theoretic approach in modern music theory traces back to the mid-twentieth century, when the development of twelve-tone compositional techniques and serialist theory prompted theorists to seek a universal descriptive tool transcending traditional harmonic functional analysis. Allen Forte's 1963 publication "Structures of Atonal Music" marked the systematic beginning of this field, introducing the set-class concept that remains central to atonal music analysis. Yet the mathematical foundation of this approach—set theory as a foundational concept in modern mathematics—has deeper theoretical roots in music theory.

让我们从最基本的形式化定义开始。在数学中，集合是对象的无序聚集，这些对象称为集合的元素。我们用大写字母表示集合，用小写字母表示元素，用符号“∈”表示成员关系。给定一个集合 $S$，当对象 $x$ 是 $S$ 的元素时，我们写作 $x \in S$；当 $x$ 不是 $S$ 的元素时，写作 $x \notin S$。集合可以通过列举其元素来定义，如 $S = \{0, 1, 2, 3\}$，也可以通过描述性质来定义，如 $S = \{x \in \mathbb{Z} : x \text{ 是偶数}\}$。

Let us begin with the most fundamental formal definitions. In mathematics, a set is an unordered collection of objects called elements. We denote sets with capital letters, elements with lowercase letters, and membership with the symbol "∈". Given a set $S$, when object $x$ is an element of $S$, we write $x \in S$; when $x$ is not an element of $S$, we write $x \notin S$. Sets can be defined by listing their elements, such as $S = \{0, 1, 2, 3\}$, or by describing properties, such as $S = \{x \in \mathbb{Z} : x \text{ is even}\}$.

在音乐理论的应用中，我们首先需要建立音高级的数学模型。传统上，西方音乐使用十二个音名：C、C♯/D♭、D、D♯/E♭、E、F、F♯/G♭、G、G♯/A♭、A、A♯/B♭、B。这些音名对应于钢琴键盘上的七个白键和五个黑键，共同覆盖了一个八度内的所有半音。数学上，我们可以建立音高级与整数模12之间的双射：

In applying set theory to music theory, we first need to establish a mathematical model for pitch classes. Traditionally, Western music uses twelve pitch names: C, C♯/D♭, D, D♯/E♭, E, F, F♯/G♭, G, G♯/A♭, A, A♯/B♭, B. These pitch names correspond to the seven white keys and five black keys on the piano keyboard, collectively covering all semitones within an octave. Mathematically, we can establish a bijection between pitch classes and integers modulo 12:

$$\text{PC} : \{C, C\sharp/D\flat, D, D\sharp/E\flat, E, F, F\sharp/G\flat, G, G\sharp/A\flat, A, A\sharp/B\flat, B\} \rightarrow \mathbb{Z}/12\mathbb{Z}$$

具体对应关系如下：C对应0，C♯/D♭对应1，D对应2，D♯/E♭对应3，E对应4，F对应5，F♯/G♭对应6，G对应7，G♯/A♭对应8，A对应9，A♯/B♭对应10，B对应11。这一对应关系看似简单，却蕴含了音乐理论中两个关键等价关系的数学化：C对应0意味着所有C音——无论频率为261.63 Hz的C4还是频率为523.25 Hz的C5——都映射到同一个等价类；而C♯和D♭都映射到1则实现了等音等价。

The specific correspondence is as follows: C maps to 0, C♯/D♭ maps to 1, D maps to 2, D♯/E♭ maps to 3, E maps to 4, F maps to 5, F♯/G♭ maps to 6, G maps to 7, G♯/A♭ maps to 8, A maps to 9, A♯/B♭ maps to 10, B maps to 11. This correspondence, though seemingly simple, encapsulates the mathematical formalization of two crucial equivalence relations in music theory: C mapping to 0 means that all C notes—whether the C4 with frequency 261.63 Hz or C5 with frequency 523.25 Hz—map to the same equivalence class; while both C♯ and D♭ mapping to 1 realizes enharmonic equivalence.

值得强调的是，模12对应关系的建立包含了一个重要的音乐理论假设：我们假设等音调律（equal temperament），即所有半音之间的频率比相等（比为 $2^{1/12}$）。在纯律（just intonation）或其他非等音律制中，等音等价不再成立——C♯和D♭代表不同的音高，频率比分别为 $\frac{16}{15}$ 和 $\frac{9}{8} \cdot \frac{10}{9} = \frac{25}{24}$。然而，即使在这些律制中，音高级的概念仍然有意义，只是其数学模型需要从 $\mathbb{Z}/12\mathbb{Z}$ 变为更复杂的商群结构。本书的分析框架主要基于等音调律，因为这已成为现代西方音乐的理论基础和实践标准。

It is crucial to emphasize that establishing the modulo-12 correspondence involves an important music-theoretic assumption: we assume equal temperament, where all semitone intervals have equal frequency ratios (the ratio $2^{1/12}$). In just intonation or other non-equal temperaments, enharmonic equivalence does not hold—C♯ and D♭ represent different pitches with frequency ratios $\frac{16}{15}$ and $\frac{9}{8} \cdot \frac{10}{9} = \frac{25}{24}$ respectively. However, even in these temperaments, the pitch class concept remains meaningful; its mathematical model simply changes from $\mathbb{Z}/12\mathbb{Z}$ to a more complex quotient group structure. This book's analytical framework is primarily based on equal temperament, as it has become the theoretical foundation and practical standard of modern Western music.

一旦我们建立了音高级作为 $\mathbb{Z}/12\mathbb{Z}$ 的元素的数学模型，就可以定义音高级集合（pitch class set）这一核心概念。音高级集合是 $\mathbb{Z}/12\mathbb{Z}$ 的一个子集，记作有序或无序的元素列举。例如，钢琴大调音阶 C-D-E-F-G-A-B 可以表示为集合 $\{0, 2, 4, 5, 7, 9, 11\}$，而小三和弦 C-E♭-G 则表示为 $\{0, 3, 7\}$。这里需要特别注意的是，音高级集合中的元素是模12的剩余类，因此“0”和“12”被视为同一元素，我们在列举时只取 $0, 1, 2, \ldots, 11$ 范围内的代表元。

Once we establish pitch classes as elements of $\mathbb{Z}/12\mathbb{Z}$, we can define the core concept of pitch class set. A pitch class set is a subset of $\mathbb{Z}/12\mathbb{Z}$, denoted by an ordered or unordered listing of elements. For example, the piano C major scale C-D-E-F-G-A-B can be represented as the set $\{0, 2, 4, 5, 7, 9, 11\}$, while the C minor triad C-E♭-G is $\{0, 3, 7\}$. A crucial point: elements in a pitch class set are modulo-12 residue classes, so "0" and "12" are considered the same element; when listing, we only take representatives in the range $0, 1, 2, \ldots, 11$.

集合论提供了丰富的运算和关系来描述音高级集合之间的联系。两个集合的并集（union）$A \cup B$ 包含所有属于 $A$ 或 $B$ 的元素，交集（intersection）$A \cap B$ 包含所有同时属于 $A$ 和 $B$ 的元素，而差集（difference）$A \setminus B$ 包含所有属于 $A$ 但不属于 $B$ 的元素。对于任意集合 $A$，其补集（complement）$\overline{A}$ 包含所有不属于 $A$ 的音高级，即 $\overline{A} = \mathbb{Z}/12\mathbb{Z} \setminus A$。特别地，任意六音的音高级集合 $A$ 都满足 $|A| = 6$ 且 $|\overline{A}| = 6$，这使得六音集合与其六音补集之间存在一种特殊的关系。

Set theory provides rich operations and relations to describe connections between pitch class sets. The union of two sets $A \cup B$ contains all elements belonging to $A$ or $B$; the intersection $A \cap B$ contains all elements belonging to both $A$ and $B$; the difference $A \setminus B$ contains all elements belonging to $A$ but not $B$. For any