# Color and Timbre Gestures: An Approach with Bicategories and Bigroupoids

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## Abstract

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## 1. Introduction

**ff**dynamic, rather than a flute playing with a

**pp**dynamic or a diapason). These analogies are not related with cultural associations between colors/sound and semantic meaning [10].

#### 1.1. Mix and Comparisons

#### 1.2. Some Empirical Evidence

## 2. Categorical Depictions of Color and Timbre Gestures

In dimension 0 points $p,\phantom{\rule{0.166667em}{0ex}}q$ of X will be identified if they can be joined by a path, i.e., a continuous map $f:I\to X$ from the unit interval $I=[0,1]$ of real numbers such that $f\left(0\right)=p$ and $f\left(1\right)=q$. This gives rise to the set of path-components, ${\Pi}_{0}\left(X\right)$, of X. In dimension 1 the points of X will be retained, but paths $f,{f}^{\prime}$ between fixed points $p,\phantom{\rule{0.166667em}{0ex}}q$ will be identified if there is a homotopy of rel end points between them. This gives rise to the fundamental groupoid, ${\Pi}_{1}\left(X\right)$, of X. The class of a path will be called a 1-track. Hence, the most natural approach to 2-dimensional homotopical algebra of a space X is to retain points and paths between them and identify homotopies’ rel end points under a suitable homotopy relation. This gives rise to the notion of a 2-track. In this way we obtain a two-dimensional structure with points in dimension 0 (0-cells), paths in dimension 1 (1-cells), and 2-tracks in dimension 2 (2-cells). […] Horizontal pasting is neither strictly associative, nor do we have strict identities. However, horizontal pasting is still reasonably well-behaved in the sense that associativity does hold and strict inverses do exist up to coherent isomorphisms. Thus, we obtain a bicategory, ${\Pi}_{2}\left(X\right)$, in the sense of Bénabou. The bicategory ${\Pi}_{2}\left(X\right)$ has the additional feature that the 2-cells are strictly invertible with respect to vertical pasting and the 1-cells are invertible up to coherent isomorphism, that is, ${\Pi}_{2}\left(X\right)$ is a bigroupoid which will be called the homotopy bigroupoid of the topological space X.

- There is a set of objects of COLOR, the points, that is, the 0-paths, or 0-cells (and similarly for TIMBRE);
- For each pair of objects in COLOR (TIMBRE), there is a 1-path between them, that is, an arrow or 1-cell;
- A morphism between two 1-paths exists and it is a 2-cell, here called a color band (timbre band);
- The composition of two 2-paths $\beta :{path}_{1}\Rightarrow {path}_{2}$ and ${\beta}^{\prime}:{path}_{2}\Rightarrow {path}_{3}$ is additive: $\beta +{\beta}^{\prime}:{path}_{1}\Rightarrow {path}_{3}$. In fact, we can add a color band to another adjacent one, creating a larger color band (similarly for timbre bands);
- The identity element exists and it is a 1-cell: it corresponds to the lazy path for colors (timbres);
- For each triple of color points (color1, color2, color3), there is a composition functor $path({color}_{1},{color}_{2})\times path({color}_{2},{color}_{3})\to path({color}_{1},{color}_{3})$ (same for timbres);
- The identity 2-cell (the identity 2-arrow) can be considered as a mapping from a 1-cell to itself (as the identity 1-cell maps a 0-cell to the same 0-cell);
- For each quadruple $({color}_{1},{color}_{2},{color}_{3},{color}_{4})$, there are natural isomorphisms, the associativity isomorphisms: $\alpha :{path}_{1}\xb7({path}_{2}\xb7{path}_{3})\Rightarrow ({path}_{1}\xb7{path}_{2})\xb7{path}_{3}$, where ${path}_{1}:{color}_{1}\to {color}_{2}$, ${path}_{2}:{color}_{2}\to {color}_{3}$, ${path}_{3}:{color}_{3}\to {color}_{4}$ (same for timbre paths);
- For each pair $({color}_{1},{color}_{2})$ of objects of COLOR, there are two natural isomorphisms, the left and right identities: $\lambda :\left[identity\phantom{\rule{0.166667em}{0ex}}arrow\phantom{\rule{0.166667em}{0ex}}on\phantom{\rule{0.166667em}{0ex}}{color}_{2}\right]:{path}_{1}\Rightarrow {path}_{1}$ and $\rho :f\xb7\left[identity\phantom{\rule{0.166667em}{0ex}}arrow\phantom{\rule{0.166667em}{0ex}}on\phantom{\rule{0.166667em}{0ex}}{color}_{1}\right]\to {path}_{1}$ (same for timbres);
- Isomorphisms $\alpha ,\phantom{\rule{0.166667em}{0ex}}\lambda ,\phantom{\rule{0.166667em}{0ex}}\rho $ satisfy pentagonal and triangular identities, similarly to the conditions required for monoidal categories.

- for each pair $({color}_{1},\phantom{\rule{0.166667em}{0ex}}{color}_{2})$ of objects of COLOR, the bicategory COLOR$({color}_{1},\phantom{\rule{0.166667em}{0ex}}{color}_{2})$ is a groupoid, that is, any 2-cell is invertible (same for TIMBRE);
- for each pair $({color}_{1},\phantom{\rule{0.166667em}{0ex}}{color}_{2})$ of COLOR, there is a (covariant) functor$${F}^{-1}:COLOR({color}_{1},{color}_{2})\to COLOR({color}_{2},{color}_{1})$$(and similarly for TIMBRE);
- for each pair $({color}_{1},\phantom{\rule{0.166667em}{0ex}}{color}_{2})$ of COLOR (same for TIMBRE) there are two natural isomorphisms, the cancellation isomorphisms: $\iota :{path}_{1}^{-1}\xb7{path}_{1}\Rightarrow {identity}_{{color}_{1}}$ and ${\iota}^{\prime}:{path}_{1}\xb7{path}_{1}^{-1}\Rightarrow {identity}_{{color}_{2}}$, where ${path}_{1}:{color}_{1}\to {color}_{2}$ is a 1-cell, that is, a 1-path, with the composition of $path,\phantom{\rule{0.166667em}{0ex}}{path}^{-1}$ verifying the pentagonal relationship of Diagram (3), where p stands for ${path}_{1}$, i is the identity arrow from and to ${color}_{1}$, ${i}^{\prime}$ is the identity arrow from and to ${color}_{2}$, ${0}_{p}$ is the identity 2-cell ${0}_{p}:i\to i$.

- Our structure is a bigroupoid, and thus it is a special case of bicategory;
- We can define a tensor functor as $\otimes :COLOR\times COLOR\to COLOR$, which in our case can be the following: adding two colors in COLOR gives as output another color in COLOR obtained as the weighted algebraic sum of the first two colors;
- The tensor product of a color (with itself) is the color itself: mixing white with white gives white;
- The tensor product of two objects ${color}_{1},\phantom{\rule{0.166667em}{0ex}}{color}_{2}$ is ${color}_{1}\otimes {color}_{2}$;
- We have the associator $a:({color}_{1}\otimes {color}_{2})\otimes {color}_{3}\to {color}_{1}\otimes ({color}_{2}\otimes {color}_{3})$;
- The pentagonator of [47] (p. 272) is verified;
- Similarly, the associahedron [47] (p. 273) is verified as well, because there is no dependency on the organization of brackets;
- There is a monoidal unit I, for example, in this case mixing white with a transparent color, as a transparent acrylic;
- There are two unitor elements $l:I\otimes {color}_{1}\to {color}_{1}$, $r:{color}_{1}\otimes I\to {color}_{1}$;
- There are two unitor invertible modifications $\lambda ,\phantom{\rule{0.166667em}{0ex}}\mu ,\phantom{\rule{0.166667em}{0ex}}\rho $ verifying triangular correspondences as shown in the Reference [47] (p. 275);
- There are four equations of modifications as shown in the Reference [47] (p. 276).

#### 2.1. Timbre Spaces

## 3. Mapping of Color Classes onto Timbre Classes

- A map $F:Ob\left(\mathbf{S}\right)\to Ob\left(\overline{\mathbf{S}}\right)$ for the objects;
- For each pair $(p,\phantom{\rule{0.166667em}{0ex}}q)$ of objects of $\mathbf{S}$, a functor $F:\mathbf{S}(p,q)\to \overline{\mathbf{S}}(Fp,Fq)$ for the morphisms.

## 4. Morphing versus Mixing

#### 4.1. On Orchestration, Morphing and Hybridization

#### 4.2. Assisted Orchestration and Sound Colors

## 5. The Souvenir Theorem

#### On Computer-Assisted Orchestration

## 6. An Audio Example

## 7. Cultural Influences

English | Blue | English | Yellow |

Yoruba | Olomi Aro (blue) | Yoruba | Elezuru (yam special) |

Igbo | Oji (something dark) | Igbo | Onashara (light white) |

Tiv | Kwar Kwaodo (like sky) | Tiv | Oyha (like banana fruit) |

Owan | Iblue (sky) | Owan | (paint of banana) |

Urhobo | Oda dibo (paint of banana) | Ijaw | Pinapina (just like white) |

English | Green | English | Red |

Yoruba | Alawo ewe (color of leaves) | Yoruba | Pupa |

Igbo | Akwukwo Ndu | Igbo | Uhie (color of blood) |

Tiv | Ngu-er-ka Ikya uwer nahan | Tiv | Nyian |

Owan | Ebesugbo (leaf) | Heusa | Ja |

Ijaw | Deibide (like a particular cloth) | Urhobo | Oda Obara (blood-like) |

Hausa | Igreen | Ijaw | Kwekwe |

## 8. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Grey’s timbre space. Each point represents a musical instrument sound, such that similar timbres are close together and dissimilar timbres are farther apart. Reproduced from [36], with the permission of the Acoustical Society of America.

**Figure 4.**

**Top**,

**left**: a sequence and partial superposition of colors;

**bottom**,

**left**: a similar sequence and partial superposition of timbres;

**bottom**,

**right**: the corresponding timbre gesture;

**Top**,

**right**: the corresponding color gesture.

**Figure 5.**The functor which maps points in the space of colors to points in the space of timbres, and color gestures to timbre gestures, implies the existence of a similar perception which justifies the association. The experiment described in Section 3 aimed to show the association between color points (with loop arrows) and equivalence classes of timbre points (with arrows between them). The color cluster is one of clusters actually found in the experiment.

**Figure 6.**

**Left**side: a sketch of strings glissandi (mm. 309–314) from Xenakis’ Metastaseis.

**Right**side: a photo of the Philips Pavilion by Le Corbusier.

**Figure 9.**A change of parameters of sound morphing gives different “timbre gestures”, that is, transitions from a timbre to another one.

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**MDPI and ACS Style**

Mannone, M.; Santini, G.; Adedoyin, E.; Cella, C.E.
Color and Timbre Gestures: An Approach with Bicategories and Bigroupoids. *Mathematics* **2022**, *10*, 663.
https://doi.org/10.3390/math10040663

**AMA Style**

Mannone M, Santini G, Adedoyin E, Cella CE.
Color and Timbre Gestures: An Approach with Bicategories and Bigroupoids. *Mathematics*. 2022; 10(4):663.
https://doi.org/10.3390/math10040663

**Chicago/Turabian Style**

Mannone, Maria, Giovanni Santini, Esther Adedoyin, and Carmine E. Cella.
2022. "Color and Timbre Gestures: An Approach with Bicategories and Bigroupoids" *Mathematics* 10, no. 4: 663.
https://doi.org/10.3390/math10040663