# Generalizing the Orbifold Model for Voice Leading

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**together with the projection**$X\to X/G$ is a developable orbifold (sometimes called a good orbifold). The unqualified term orbifold includes spaces in which every point has a neighborhood that is a developable orbifold, but the spaces X and the groups G may vary among such neighborhoods, and the orbifold structures are required to be compatible where the neighborhoods intersect. A developable orbispace is an orbit space $X/G$ together with its corresponding projection $X\to X/G$ (note that X need not be a manifold and G need not act properly discontinuously), and an orbispace is a space in which every point has a neighborhood that is a developable orbispace, and the orbispace structures are compatible where the neighborhoods intersect. All of the orbifolds and orbispaces in this work (and other voice-leading literature) are developable, so for brevity we will usually dispense with the adjective. In most cases, X will be the simply connected universal orbispace cover of the quotient. In all cases, it is extremely important to note that orbifolds and orbispaces carry more information than their corresponding quotient spaces; the projection maps are part of the defining data.

#### 2.1. The Spaces

**R**. We note that $C\mathbf{R}$ is not a manifold, since no neighborhood of O is homeomorphic to an open subset of ${\mathbf{R}}^{2}$. Nor is $C\mathbf{R}$ a manifold with boundary, for similar reasons. Note that deleting the cone point O would yield a manifold, but doing so would also obviate our main motivation. We also note that $C\mathbf{R}$, like all cones, is contractible and thus lacking topological structure; a recurring theme in this work is that structure detailed enough to be useful comes not from topology per se but from the additional data with which an orbispace, by definition, comes equipped.

#### 2.2. The Paths

- The projection $p:X\to X/G$ has the path lifting property: if $c:I\to X/G$ is a path, then there is a path $\tilde{c}:I\to X$ such that $p\circ \tilde{c}=c$.
- If $x\in X$, then x has an open neighborhood ${U}_{x}$ such that
- (a)
- If $g\in G$ does not belong to the isotropy group ${G}_{x}$ of x, then ${U}_{x}\cap (g\xb7{U}_{x})=\varphi $;
- (b)
- If a and b are paths in ${U}_{x}$ beginning at x and such that $p\circ a$ and $p\circ b$ are homotopic rel endpoints in $X/G$, then there is an element $g\in {G}_{x}$ such that $g\xb7a$ and b are homotopic in X rel endpoints.

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

**Proof.**

#### 2.3. The Groupoids

**Definition**

**3.**

**Theorem**

**2.**

**Proof.**

**Definition**

**4.**

**Theorem**

**3.**

**Proof.**

## 3. Results

#### 3.1. Voice Leadings as Groupoid Elements

- Choose an element ${\gamma}_{0}$ of G that moves the given initial voicing $\tilde{x}$ to the basepoint voicing ${\tilde{x}}_{0}$ of the initial chord x (i.e., ${\gamma}_{0}$ that moves $\tilde{x}$ into the chosen fundamental domain).
- Apply ${\gamma}_{0}$ to the given terminal voicing $\tilde{y}$ of the terminal chord y, effectively completing the uniform operation of ${\gamma}_{0}$ on the path from $\tilde{x}$ to $\tilde{y}$. Note that since the path may end in a fundamental domain other than the one where it originated, ${\gamma}_{0}\xb7\tilde{y}$ will not necessarily be equal to the basepoint voicing ${\tilde{y}}_{0}$ of y (see Figure 3).
- Since $\tilde{y}$, ${\gamma}_{0}\xb7\tilde{y}$, and ${\tilde{y}}_{0}$ all lie in the fiber over x and hence in the same orbit under the action of G (i.e., they are all voicings of the same chord), we can choose $\gamma \in G$ such that $\gamma \xb7({\gamma}_{0}\xb7\tilde{y})={\tilde{y}}_{0}$ (i.e., $\gamma $ that moves ${\gamma}_{0}\xb7\tilde{y}$ into the chosen fundamental domain).
- The associated element of the groupoid ${\pi}^{\mathrm{o}rb}(X/G)\cong T(X/G)\u22caG$ is then $(x,y,\gamma )$.

- The basepoint voicing of the first chord is (D4, F4, A4), so the element ${\gamma}_{0}\in G$ taking the first given voicing to the basepoint voicing is the cyclic permutation $\left(\right)$ followed by raising the third voice by an octave: ${\gamma}_{0}=(0,0,1)\circ \left(\right)open="("\; close=")">\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}$.
- We apply ${\gamma}_{0}$ to the second voicing to obtain (C4, E4, A4), which we note in this case happens to be the basepoint voicing.
- The corresponding element $\gamma \in G$ for this voice leading is thus the identity element $1=(0,0,0)\circ \left(\right)open="("\; close=")">\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\end{array}$.
- We therefore associate the groupoid element (Dm, Am, 1) to the first voice leading.

- To move (A3, C4, E4) to the basepoint voicing, we again need to apply ${\gamma}_{0}=(0,0,1)\circ \left(\right)open="("\; close=")">\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}$.
- We apply ${\gamma}_{0}$ to (C4, E4, G4) to obtain (E4, G4, C5), which we note is not the basepoint voicing.
- To move (E4, G4, C5) to the basepoint voicing (C4, E4, G4), we apply the group element $(-1,0,0)\circ \left(\right)open="("\; close=")">\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}$.
- Thus, we associate the groupoid element (Am, C, $(-1,0,0)\circ \left(\right)open="("\; close=")">\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}$) to the second voice leading. Note that the “forward” version would take the basepoint voicing of Am to the basepoint voicing of C and then apply the inverse of the associated group element, which is $\left(\right)open="("\; close=")">\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}$. This illustrates the remark made above on factoring a given voice leading between two different chords as a fixed, chosen (i.e., basepoint) voice leading from the first chord to the second, followed by a voice leading of the second chord to itself.

#### 3.2. Chords with Rests or Doublings, and Their Associated Subspaces

**Definition**

**5.**

**Theorem**

**4.**

**Proof.**

#### 3.3. Visualization and Braids

#### 3.3.1. Basic Definitions and Terms; the Classical Case

#### 3.3.2. Singular Braids, Symmetric Products, and Orbispace Path Classes

#### 3.3.3. Musical Interpretations and Examples

## 4. Discussion

#### 4.1. The Paths: Basic Definitions

#### 4.2. The Subspaces: Relevant Restrictions

#### 4.3. Moves of Musical Interest

#### 4.4. Transpositional and Inversional Set-Class Spaces

- NF1.
- Its first element is 0;
- NF2.
- It is in ascending order, with the final element less than or equal to the octave $\mathbf{o}$;
- NF3.
- Its smallest interval lies between its first two notes (including the “wraparound” interval $\mathbf{o}-{x}_{n-1}$ among the chord’s intervals).

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Author’s transcription of an excerpt from “Any Way the Wind Blows” from Hadestown showing the use of solos with resting voices, harmonized voices, and unison voices in quick succession. In some Broadway performances, the solo in the third measure is taken by Fate 2. The orbifold path model can distinguish the two different assignments.

**Figure 3.**A schematic illustration of the process by which a groupoid element is associated with a voice leading. Diamond-shaped regions depict fundamental domains in the universal orbispace cover X; green shading indicates the preferred fundamental domain. Solid blue curves are paths; dashed orange lines represent group actions.

**Figure 4.**The first three chords of the fifth measure of the Hadestown excerpt; all voices treble clef.

**Figure 11.**The Hadestown example as a singular braid. The blue strand is Fate 1, the red strand is Fate 2, and the green strand is Fate 3. Resting voices are on the central axis; sounding voices are on the curved outer surface.

**Figure 12.**Figure 27 from [17].

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Hughes, J.R.
Generalizing the Orbifold Model for Voice Leading. *Mathematics* **2022**, *10*, 939.
https://doi.org/10.3390/math10060939

**AMA Style**

Hughes JR.
Generalizing the Orbifold Model for Voice Leading. *Mathematics*. 2022; 10(6):939.
https://doi.org/10.3390/math10060939

**Chicago/Turabian Style**

Hughes, James R.
2022. "Generalizing the Orbifold Model for Voice Leading" *Mathematics* 10, no. 6: 939.
https://doi.org/10.3390/math10060939