On Quotient Spaces in Relation to Music with Dmitri Tymoczko

Dmitri Tymoczko

Dmitri Tymoczko

On Math

On Topology

On Symmetry

On Math

On Geometry of Music

On Quotient Space and Symmetry in Music

Music

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AI

Dmitri Tymoczko uses quotient spaces to model how musicians and listeners abstract away from specific musical details—such as which octave a note is in, the order of notes in a chord, or the specific transposition of a chord. By "gluing together" points in a high-dimensional space that represent equivalent musical situations, he creates complex, low-dimensional geometric shapes called orbifolds. These spaces, often called OPTIC spaces (Octave, Permutation, Transposition, Inversion, Cardinality), help visualize and analyze musical voice leading and harmonic relationships. [1, 2, 3, 4, 5, 6]

Here is how Tymoczko uses quotient spaces to model music:

1. Abstracting Musical Information (The OPTIC Relations)

Tymoczko forms quotient spaces by applying five standard equivalence relations, often referred to in collaboration with Clifton Callender and Ian Quinn as OPTIC: [1, 2, 3]

• O (Octave): Identifying pitches that are an octave apart, creating a circular "pitch-class" space (\(R/12Z\)).
• P (Permutation/Order): Identifying chords that contain the same notes in a different order (e.g., C-E-G is the same as G-E-C). This creates a quotient space \(T^n/S_n\).
• T (Transposition): Treating chords that are transpositions of one another as similar, identifying them in a single space.
• I (Inversion): Identifying chords that are inversions of each other as the same point.
• C (Cardinality): Treating chords with different numbers of notes as similar (less frequently used, but part of the general model). [1, 2, 3, 4]

2. Constructing Musical Orbifolds

These mathematical quotients result in shapes known as orbifolds (spaces with singularities or "kinks" where the local geometry is not flat): [1, 2, 3]

• Two-note chords (Intervals): Represented as a Möbius strip, a 2D surface with a twist that identifies inversional pairs.
• Three-note chords: Modeled as cones or other higher-dimensional shapes, where the "vertex" represents chords with multiple identical notes (singular points).
• Four-note chords: Often modeled using a cone over the real projective plane, specifically a 4D space that helps visualize harmonic relationships. [1, 2, 3, 4, 5]

3. Visualizing Voice Leading

In these quotient spaces, chords are points, and voice leadings (how individual notes move from one chord to another) are paths connecting those points. [1]

• Efficient voice leading: Short paths in these spaces represent efficient voice leading (small movements of voices).
• Singularities as Mirrors: Some boundaries of the orbifolds act like mirrors, meaning the shortest path between two chords may "bounce" off the boundary, representing a specific type of voice-leading. [1, 2, 3, 4, 5]

4. Examples of Use

• Chord Space (2006): Tymoczko demonstrated that musical chords can be represented in these spaces, allowing for a geometric understanding of why certain chords (like triads) are considered similar and how they can be connected.
• Constrained Spaces (2015): He analyzed "power chords" and other specific, restricted voicing scenarios using these spaces, often finding that they resemble the larger, unconstrained space.
• Contrapuntal Analysis (2020): He used these spaces to model how voice-leading possibilities change based on how we define "similar" chords. [1, 2, 3, 4, 5]

Tymoczko’s work allows musicians to see that the "rules" of music theory are deeply connected to the geometric structure of these quotient spaces, helping to explain why traditional music (using "nearly even" chords) has developed the way it has. [1, 2]

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On Quotient Space

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