Pitch Class Sets

The concept of pitch class sets typically arises in studies of post-tonal music theory. A noteworthy resource is Joseph N. Straus's Introduction to Post-Tonal Theory, which significantly shaped my understanding of this subject. The phrase "post-tonal" might be misleading, as this mathematical perspective on music theory is not solely tied to atonal compositions or 12-tone serialism.

In Western music, the 12-tone scale is constructed by dividing an octave into 12 distinct segments. To simplify our mathematical discussions, we can assign numerical values to these pitches: C becomes 0, C# is 1, D is 2, and so forth, culminating with B as 11. When we reach the next octave, we wrap around, treating another C as 0. This process aligns with modulo 12 arithmetic.

To visualize this concept, imagine a regular dodecagon, a 12-sided polygon where all sides are of equal length.

Consider a C major chord represented as 0, 4, 7. If we transpose it upward by three semitones, we simply add 3 to each pitch, yielding 3, 7, and 10. For any collection of notes, we can apply the transposition operation defined as follows:

Tₙ: x → (x + n) mod 12

This means we can think of 12 as 0, 13 as 1, 14 as 2, etc., since they represent the same notes.

Additionally, we can perform an operation known as inversion. In musical terms, this involves inverting the intervals. Mathematically, this translates to negating each number and then applying modulo 12. For instance, the inversion of the C major chord [0, 4, 7] results in [0, -4, -7] = [0, 8, 5]. Here, the order of the numbers doesn't matter, allowing us to recognize that this is equivalent to an F minor chord.

We denote this operation as I for "inversion." Visualizing this on the dodecagon illustrates how applying Tₙ across all values of n to [0, 4, 7] generates all 12 major chords, while applying both I and Tₙ produces all 12 minor chords.

The combined operations of transposition and inversion form a mathematical group. Specifically, the group generated by these operations is known as D₁₂, the Dihedral group of symmetries of the dodecagon, which contains 24 elements.

We refer to an unordered collection of numbers ranging from 0 to 11 as a pitch set, and we discover that D₁₂ acts on these pitch sets. Let’s explore this idea further.

Group Actions on Pitch Sets

We have shown that the orbit of the chord [0, 4, 7] under this group action includes all major and minor triads, which are foundational in Western music. Notably, none of the triads map onto themselves. In other words, a non-trivial symmetry involving transpositions and inversions always yields a unique new triad. Mathematicians might express this notion more formally: the set of major and minor triads behaves as a torsor under the TI-action.

Interestingly, this phenomenon is somewhat generic. When selecting a pitch set at random, there’s a greater than 50% chance it will have the property of yielding 24 distinct new chords through transposition and inversion. We can say that it lacks TI-symmetry.

For example, if we take the set [0, 6] and invert it, we end up with [0, -6], which is equivalent to [0, 6] modulo 12. Thus, this inversion does not produce a new pitch set. The reason [0, 4, 7] does not exhibit this behavior is that it lacks such a symmetry.

We define a k-chord (an unordered collection of k notes) as TI-symmetric if there exists some non-trivial transposition and inversion that maps it to itself. Although these examples are less common, there is always at least one k-chord exhibiting this property for any value of k.

A trivial example is [0, 1, 2, …, k], which will always serve as such a chord. For a more complex example, consider the whole-tone scale [0, 2, 4, 6, 8, 10]. Translating by 2 will return the same set.

This leads us to question the nature of k-chords that exhibit TI-symmetry in tonal systems with n notes. For instance, can we derive a simple formula for counting TI-symmetric k-chords for a given n? More intriguingly, for each n, which k yields the maximum number of TI-symmetric k-chords?

Notably, excluding trivial instances of TI-symmetry, this inquiry holds practical significance. TI-symmetry has historically influenced composition, as seen in chords such as the augmented triad, the French augmented sixth, and the diminished seventh, along with notable instances in Stravinsky's Petrushka.

The Geometry of Pitch Class Sets

Now, let's outline how to construct a space that categorizes pitch class sets. We will not delve into specifics from my previous discussion on the Yoneda Lemma; simply trust that we can define a "generalized space" that retains essential information about the relationships among these objects.

Recall that a pitch set (or chord) converts notes into numerical representations: 0 for C, 1 for C#, 2 for D, etc. This collection of pitches can be expressed in a more useful notation when we are not confined to a specific key. For instance, a C major chord can be denoted as (0, 4, 7).

You might wonder why I’ve switched from [0, 4, 7] to (0, 4, 7). This change emphasizes the class set (sometimes referred to as an equivalence class) of chords derived from applying translations and inversions to [0, 4, 7].

A pitch class set thus represents collections of chords we consider equivalent. This notion is supported by several music-theoretic principles. For one, our designation of 0 is arbitrary; we could have assigned it to A, and the resulting music theory would remain unchanged. This identification also extends to sets that remain equivalent post-inversion.

Previously, I demonstrated that labeling the vertices of a dodecagon corresponds to a reflection symmetry. The combinations of reflections and translations generate the dihedral group, allowing it to act on all number tuples ranging from 0 to 11, where each number appears only once and in increasing order. Therefore, a pitch class set is merely an equivalence class of a chord under this group action.

Although I won’t focus here, every class has a unique representative known as the "prime form," which is the most compact representation starting from 0. For further insights into this concept, refer to Straus’s work, as it serves as a standard reference for post-tonal theory.

The Geometric Space

The collection of all "chords" should possess a meaningful topology. For example, the chord [0, 1, 2, 3] should relate closely to [0, 1, 2, 4] since they differ by just one note. It’s unclear whether a straightforward distance measurement based on coordinates would suffice. However, constructing a lattice of open sets based on intuition may clarify the definition.

Let’s denote this space of chords as X. With a group action established, one might consider forming the quotient space X/G. The quotient mapping X → X/G will generally be 24 to 1 at most points, although it will also lose track of which chords were invariant under the group’s elements. This retention of information is a crucial aspect of music theory.

Thus, I propose creating a quotient stack [X/G]. This concept, which I previously mentioned, serves as the moduli space of pitch class sets. While it may appear complex, it offers significant advantages.

This construction results in a "space" where points correspond to pitch class sets. If a class contains 24 distinct chords, that point is a "true" point devoid of additional information. The fiber of the quotient map will consist of those 24 chords, accessible through actions performed by the group elements.

Now, consider the pitch class set [0, 2, 4, 6, 8, 10]. The fiber of the quotient map comprises elements such as (0, 2, 4, 6, 8, 10) and (1, 3, 5, 7, 9, 11). The stack will label these points with D₆, the subgroup of symmetries that leave this chord unchanged.

At this point, I anticipate some skepticism regarding its simplicity. To clarify, the foundational structure is the space I’ve described, where each point is associated with its prime form representative and the subgroup of symmetries that uphold the class. This framework may appear straightforward yet encapsulates the intricate music theory discussed earlier. If the topology is properly defined, exploring this space could reveal profound insights into the relationships between symmetries of the classes.

Further Reading

To my knowledge, I am the first to propose this specific construction, though it seems to align with Guerino Mazzola's framework. He employs topos theory in his remarkable book The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Dmitri Tymoczko has a related approach, creating orbifolds by examining one key and chord at a time in his book A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice. This could serve as an excellent introduction for those interested in these spaces without delving deeply into topos theory.

The first video, Music Theory: Set Theory, Part 1, provides an engaging overview of set theory in music, highlighting its foundational concepts and applications.

The second video, Set Theory: An Introduction, serves as an excellent introduction to the principles of set theory, linking it to various music theory topics.