A Group-Theoretic Framework for Musical Tuning Systems
This article explores the underlying mathematical structures of a specific type of musical tuning systems through the lens of group theory, offering a unified framework for understanding their generation and properties. While a multitude of tuning systems exist, many share a common foundation, historically referred to as "chaining/stacking and reducing/folding" or its linguistic equivalents (e.g., "encadenamiento y cancelación" in Spanish).
This method, exemplified in the Pythagorean tuning system, involves repeatedly adding intervals, specifically fifths, and reducing the results by octaves (a 1:2 ratio). This principle finds its parallel in the ancient Mesopotamian and Chinese musical systems demonstrating a universal approach to generating scales and temperaments.
For example, the 3-limit 12-tone Pythagorean scale, (also the shí’èr lǜ (十二律) or “twelve-pitch” system), takes twelve consecutive powers of 3 and reduces them by octave 1:2. The scales are then presented as the size-ordered values in the \((1, s_n]\) range and can be seen as its minimal generating set, \(S\). These values are used to compute the rest of the ratios for a given instrument based on a fixed reference frequency \(f\), the complete set of pitch ratios \(P\) generated by the system is given by:
\[ P = \bigcup_{k \in \mathbb{Z}} \{ s_n^k \times s \times f \mid s \in S, f \in \mathbb{R/Z} \} \]
This makes the system a group-like structure, it is not only a minimal generating set but a subgroup—a quotient by an equivalence class \(Q/{\sim}\), usually the octave, the free generator.
To illustrate this further (to keep examples shorter), consider the 3-limit pentatonic scale. This scale is generated using the first five powers of 3, reduced by octaves, resulting in the following frequency ratios:
9/8, 81/64, 3/2, 27/16, 2/1
(These ratios are usually presented relative to a fixed base, typically 9/8, giving: 9/8, 4/3, 3/2, 16/9, 2/1)
Each pitch within the pentatonic scale can be expressed as a product of powers of the generators. Representing this with a group presentation:
\( \text{Pentatonic} = \langle 2, 3 \,|\, 3^5 = 1 \rangle \)
Where each pitch \(p\) can be defined as:
\( p = (2^n \times 3^{m \bmod 5}) \) with \( n, m \in \mathbb{Z} \).
This notation, analogous to finitely generated abelian groups like \( G = \langle a, b\, |\, b^k = 1 \rangle \), captures the fundamental elements of the system. Here, the pentatonic group is isomorphic to the direct sum of a cyclic group of order 5 (representing the modulo-constrained generator) and an infinite cyclic group:
\( \langle \text{Pentatonic}\rangle = \mathbb{3_{5}\oplus 2}\cong \mathbb{Z/5Z \oplus Z} \).
The minimal generating set, often referred to as the "basic region" or "fundamental domain", in this case, corresponds to:
{ (1, 9/8, 81/64, 3/2, 27/16, 2/1] }
(The 1 is usually omitted in presentations like synthesizer tuning files.)
This set embodies the distinct elements of the pentatonic group, excluding octave duplicates.
1 = (2^0 * 3^(0%5)),
9/8 = (2^-3 * 3^(2%5)),
81/64 = (2^-6 * 3^(4%5)),
3/2 = (2^-1 * 3^(1%5)),
27/16 = (2^-4 * 3^(3%5)),
2/1 = (2^1 * 3^(5%5)),
Note that this is only one subgroup, as many subsets could generate different scales.
The underlying group structure can be exploited to understand the mathematical properties of these systems. For example, an equivalence relation, where octaves are treated as equal, leads to the definition of cosets. Let \(a\) represent the octave interval (2/1) and \(x, y\) denote any two pitches in the pentatonic scale. The equivalence relation is defined as:
\( x \sim y \Leftrightarrow x = y \times a^k\) with \(k \in \mathbb{Z}\).
The set of equivalence classes, \( P/{\sim} \), forms a subgroup isomorphic to \(\mathbb{Z/5Z}\):
\( P/{\sim} = \{\,\{\, x_{i\bmod 5} \times 2^k\,\vert\,k,i \in \mathbb{Z}\,\}\,|\, x_i \in P\,\}\cong \mathbb{Z/5Z} \subset P \)
This implies that the pentatonic scale, with the octave equivalence, reduces to a five-element cyclic group. Crucially, in this group structure, each pitch is associated with a unique element.
To visualize the relationship between pitches, we can construct an "interval matrix". Each row in the matrix corresponds to a different starting note, and each column represents an interval relative to that note. This matrix highlights the specific relationship between pitches within the system:
\(\langle x_0 \rangle =\) { 9/8, 81/64, 3/2, 27/16, 2/1 }
\(\langle x_1 \rangle =\) { 9/8, 4/3, 3/2, 27/16, 2/1 }
\(\langle x_2 \rangle =\) { 9/8, 4/3, 3/2, 16/9, 2/1 } Natural Mode
\(\langle x_3 \rangle =\) { 32/27, 4/3, 3/2, 16/9, 2/1 }
\(\langle x_4 \rangle =\) { 32/27, 4/3, 128/81, 16/9, 2/1 }
Each row corresponds to the starting note \(x_i\), and the set represents all its related notes generated from the \(P/{\sim}\) system.
The analysis above illustrates a framework for examining any tuning system, regardless of its specific construction. A a common variation of the diatonic scale, for instance, which uses the generator \(5\) in addition to \(2\) and \(3\), calculates 3 fifths and a major third for each except the last one, can also be represented as:
\( \text{Diatonic} = \langle 2, 3, 5\,|\, 3^4 = 1, 5^2 = 1\rangle \)
remove, traditionally, the major third of the Re; its relative \([\text{45}]\) harmonic class :
\( D/{\langle 2^n\times 3^3\times 5^1 \,\vert\, n \in \mathbb{Z} \rangle^D}\)
Fa ← Do → Sol → Re [1/3] ← [1] → [3] → [9]
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
La Mi Si Fa# [1/5] [5] [15] [45]
These demonstrate that, with the use of additional generators and relations, various traditional and modern scales can be represented as groups.
While it may seem practical to describe these tunings as groups using Equave/Octave Reduction, \( a \cdot b = \text{Octave Reduction}(a \times b)\) this is not strictly necessary. Standard multiplication can also be employed as the group operation, provided the correct relations are specified.
This group-theoretic perspective offers a powerful framework for analyzing and understanding the intricate structure of musical tuning systems. It also clarifies why common musical manipulations such as transposition, retrograde, and inversion can be performed in an abstract manner without needing to work directly with frequencies. However, systems with a single generator (isotropic* or equally divided), like the 12-tone equal temperament \( 12{\text{ed}}2 = \langle 2^{n/12} \rangle\) with \( n \in \mathbb{Z},\, \langle12{\text{ed}}2\rangle = \{\ldots, 2^{−1/12}, 1, 2^{1/12}, \ldots\} \), are simpler to describe from a construction standpoint (see chromas). In these systems, equivalence relations lead to shifts (cosets), which are invariant and congruent. Unlike other systems, there is no fundamental region, only the generator itself, which results in a trivial analysis as shown by its simple interval matrix. However, subscales derived from any number of equal divisions can still be considered potential group-like structures.
For example, the pentatonics scales derived from 12EDO are also isomorphic to \(\mathbb{Z/5Z}\).
Each element or step \( g_k = (\, k +(\,(\,p \times (n \bmod 5)) \bmod 12)\) with \(n,k \in \mathbb{Z},\,p \in \{5,7\} \)
\(5\) and \(7\) are the only non-trivial generators of the additive group \(\mathbb{Z/12Z}\), constrained to a 5-cycle:
(the first 5 classes in the "circle of fifths/fourths")
...
k =-1 { 11, 1, 3, 6, 8 }, \(\flat\)
k = 0 { 0, 2, 4, 7, 9 },
k = 1 { 1, 3, 5, 8, 10 }, \(\sharp\)
...
Applying group theory offers a unified mathematical language to express the essential properties and relations of these seemingly disparate approaches, making this method interesting for research and exploration in the field of music theory.
\( \text{Golden Harmonics} = \langle \sqrt{\phi}, \sqrt{5}\,|\, (\sqrt\phi)^{10} = 1\rangle \)
*While most systems are typically presented as either Just Intonation or Equal Divisions, this dichotomy becomes problematic when encountering variations. Just Intonation traditionally involves rational intervals, and Equal Divisions involve irrational ones, but this distinction isn't always clear-cut; there are equal divisions of rational intervals and vice versa. A more descriptive categorization is (a spectrum of) non-isotropic and isotropic, abstracting away the nature of generators. Determining the exact categories and properties might be subjective, but these terms broaden the definition, reduce ambiguity, and preserve the original meanings of established terms.
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