There are many different music schools and an infinite number of methods for creating musical tuning systems, but most use a similar process. Known since antiquity, with slight variations and mutations through the years, it is now referred to (in modern Western music theory) as "chaining and reducing/folding" (and similarly in other languages: encadenamiento y cancelación).
From the Pythagorean school, the method is provided by Boethius, the algorithm which involved calculating seven fifths in one direction and five in the other, (Guido d'Arezzo refers to the "chaining" as "connuctio") is identical to the sanfen sunyi (三分損益) method, known in english as up-down generation. There is an older appearance of this method from Mesopotamia that generates notes similarly and arrives to the same system.
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 \(f\).
\[ \bigcup_{k \in \mathbb{Z}} \{ s_n^k \times s \times f \mid s \in S, f \in \mathbb{R/Z} \} \] This makes the system a group-like structure, no longer abstract. It is not a minimal generating set but a subgroup—a quotient by an equivalence class \(Q/{\sim}\), usually the octave, the free generator.
To keep examples shorter, consider the 3-limit pentatonic scale with 5 notes, the first 5 powers of 3 reduced by octaves:
9/8, 81/64, 3/2, 27/16, 2/1
(usually found in base(9/8): 9/8, 4/3, 3/2, 16/9, 2/1)
Every pitch in the Pythagorean or Pentatonic scale can be expressed as the product of powers of its generators.
Let \( \text{Pentatonic} = \langle 2, 3 \,|\, 3^5 = 1 \rangle \) where each pitch \(p = (2^n \times 3^{m \bmod 5}) \) with \( n, m \in \mathbb{Z} \).
With an identity, inverses, associativity, and closure, this is similar to a finitely generated abelian group \( G = \langle a, b\, |\, b^k = 1 \rangle \), a direct sum of a cyclic group of order k (a modulo-constrained generator) and an infinite cyclic group or free abelian group.
So, \( \langle \text{Pentatonic}\rangle = \mathbb{3_{5}\oplus 2}\) is isomorphic to \( \mathbb{Z/5Z \oplus Z} \).
Its minimal generating set (also called the "basic region" or "fundamental domain") expressed as ratios is:
{ (1, 9/8, 81/64, 3/2, 27/16, 2/1] }
(The 1 is usually omitted in presentations like synthesizer tuning files.)
These correspond to the elements:
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)),
which correspond to elements of a subgroup of <Pentatonic>, a quotient by an equivalence class, keeping one representative for each.
\( x \sim y \Leftrightarrow x = y \times a^k\) where \(k \in \mathbb{Z}\) and \(a = 2\) (the unconstrained generator).
So, \( P/{\sim} = \{\,\{\, x_{i\bmod 5} \times 2^k\,\vert\,k,i \in \mathbb{Z}\,\}\,|\, x_i \in P\,\}\cong \mathbb{Z/5Z} \subset P \)
while in a group with abstract generators there aren't "consecutive elements", here elements can be ordered by size, and since the system is infinite in both directions indexing uses a reference, for example, to the identity element, in this case: (2^0 * 3^(0%5)) = \( x_0 = 1 = e = \text{Unison} \)
The interval matrix is constructed from all \( P/{\sim_{x_i}} \) with each \(x_{i_j}\) taken as \(x_{i_j}/{x_{i_0}}\)
\(\langle x_0 \rangle =\) { 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 }
Other tuning systems are generated similarly, with more complex structures incorporating additional generators and relations. For example, a common variation of the diatonic scale uses two generators, calculates 3 fifths and a major third for each except the last one.
So, \( \text{Diatonic} = \langle 2, 3, 5\,|\, 3^4 = 1, 5^2 = 1\rangle \) then
\( D/{\langle 2^n\times 3^3\times 5^1 \,\vert\, n \in \mathbb{Z} \rangle^D}\) remove, traditionally, the major third of the Re; the \([\text{135}]\) harmonic class :
Fa ← Do → Sol → Re
↓ ↓ ↓
La Mi Si
The 3-limit subgroup (as in Western music theory) is a closed-infinite example:
\( \text{3-limit}=\langle 2,3\,\vert\rangle \cong\mathbb{Z\oplus Z}\)
General form:
\(G = \langle a_1, a_2, \ldots, a_n\, \vert\, a_{i_1}^{k_1}=e,a_{i_2}^{k_2}=e,\ldots, a_{i_j}^{k_j}=e \rangle \cong \mathop{\LARGE\oplus}_{i=1}^n \langle a_i \rangle \)
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 perspective is valuable for understanding the structure and nature of pitches, but musical functions such as transposition, retrograde, and inversion can be performed abstractly within specific classes, rather than directly on pitches. 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 }, \(\sharp\)
...
Some other systems don't directly fit within this generative process or group paradigm, but all are subsets of \( \mathbb{R}^{+} \). Nevertheless, many functions can still be applied to generate scales and chord progressions.
Incorporating group theory concepts into tuning theory enhances manageability and categorization, given the vast array of possibilities (Serialism appreciates this).
*While the direct sum Z/nZ × Z might appear trivial from a purely group-theoretic standpoint, it is used as a generalization of the "chain and reduce". Unlike abstract group theory where elements are represented by generators and their powers (e.g., a^-2 * b^1), in tuning we deal with concrete frequency ratios (e.g., 2^-2 * 3^1 = 3/4).
*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 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. This approach aligns with the accepted notion that consonance is linked to timbre, pitch, and function.
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