nLab chord orbifold -- section

Chord orbifolds in music theory

Idea

In music theory, the configuration space of unordered musical chords, quotentied by acoustic equivalences like octave repetition and voice permutation, may be thought of $[$Tymoczko 2006, Calender, Quinn & Tymoczko 2008] as an orbifold (specifically a toroidal orbifold) whose singular strata correspond to distinct voices, as follows:

Pitch-Class Space as a Circle. In Western music, pitch may be modeled continuously as a real number $p\in\mathbb{R}$, where integers represent semitones (e.g., piano keys). However, human hearing exhibits octave equivalence: a “C” played in one octave sounds functionally equivalent to a “C” played in the next octave, 12 semitones away. Therefore considering pitches modulo octave equivalent yields the pitch-class quotient space, topologically a circle:

(1)$$\mathbb{R}/12\mathbb{Z} \cong S^{1} \mathrlap{\,.}$$

Ordered Chords Space as a Torus. An ordered $n$-note chord is an $n$-tuple of such pitch classes. The space of all such ordered $n$-note chords is therefore the Cartesian product of $n$ circles, which is the $n$-torus:

(2)$$T^{n} = \big(S^1\big)^{\times_n} \mathrlap{\,.}$$

Unordered Chord Space as an Orbifold. But music theory typically treats chords as unordered sets. For example, the chord $\{C, E, G\}$ is functionally the same chord as $\{E, G, C\}$. To formalize this permutation invariance, one is to consider the further quotient of the torus $T^{n}$ (2) by the permutation action of the symmetric group $Sym_{n}$. But since this action is not free (it has fixed points), the proper incarnation of this is as the global quotient orbifold

(3)$$T^n \sslash Sym_n \,.$$

The singular strata of this orbifold correspond to the chords where voices merge (cross).

Further Symmetries. There are further symmetries that may be quotiented out:

• Transposition: Shifting all notes up or down by a constant interval $c$. This acts as a continuous translation along the main diagonal of $T^{n}$. Factoring out transposition yields a space locally isomorphic to $\mathbb{R}^{n-1}$.

• Inversion: Reflecting pitch space ($p\mapsto-p$), which corresponds to turning chords upside down.

The global quotient orbifold of the $n$-torus by permutation, transposition, and inversion simultaneously results in a farily complex orbifold. For example, the space of 3-note chords modulo transposition and inversion is a Möbius strip with a boundary of singular points.

Of course, there is mathematical idealization involved in this, which may not necessarily always be reflected in musical practice (cf. Tymoczko 2007)

Applications

These orbifolds are used to study voice leading — the transition from one chord to another over time. A piece of music can be modeled as a continuous path winding through this orbifold. Efficient voice leading (where voices move by the smallest possible intervals) translates to finding geodesics on this Riemannian orbifold, where singular boundaries act like acoustic “mirrors”; paths that bounce off the singularities represent musical voices crossing or temporarily merging into a unison.

References

The original articles:

• Dmitri Tymoczko: The Geometry of Musical Chords, Science 313 5783 (2006) 72–74 [doi:10.1126/science.1126287, pdf]

• Clifton Callender, Ian Quinn, Dmitri Tymoczko: Generalized Voice-Leading Spaces, Science 320 5874 (2008) 346–348 [doi:10.1126/science.1153021, pdf]

Further discussion:

• Dmitri Tymoczko: Response to Comment on “The Geometry of Musical Chords”, Science 315 5810 (2007) 330 [doi:10.1126/science.1134163]

• Dmitri Tymoczko: Generalizing Musical Intervals, Journal of Music Theory 53 2 (2009) 227–254 [doi:10.1215/00222909-2010-003, jstor:40925744]

• Dmitri Tymoczko: A Geometry of Music – Harmony and Counterpoint in the Extended Common Practice, Oxford University Press (2011) [isbn:9780195336672, webpage]

Exposition:

• Dmitri Tymoczko: Mathematical Moments: Putting Music on the Map, AMS News & Outreach (July 2006) [mp3]

• Mattia G. Bergomi: Musical modeling through graphs and orbifolds (2014) [pdf]

• Jon Latané: Topologica: Jazz, Orbifolds, and Your Event-Sourced, Flux-Driven Dream Code, blog post (2017)

• Wikipedia: Orbifold – Applications – Music Theory