Ordered pairs of pitches

• Introduction
• Direct product of two groups
• Combining two real tuning systems
  • Approach
  • Real-pitch pair tunning system
• Ploting real-pitch pair tuning system
• Disclaimer
• References

Introduction

In this blog, I will explore ordered pairs of pitches (or simply pitch pairs) using the same techniques we apply to learn about single-pitch spaces. This post is my first step toward the main result summarized by the great book Dmitri Tymoczko’s “A Geometry of Music.”¹

Direct product of two groups

We build the concept of pitch class by using group actions on a single-pitch space name as a tuning system. Moreover, I show how the most helpful tuning system can be described using straightforward actions. Simple actions are defined using groups and homomorphism between those groups. The following proposition allows me to combine two groups to create a new group helpful to represent the pitch pairs.

Using this proposition, we can combine any tuning system in this summary to represent two pitches.

Combining two real tuning systems

Approach

For example, combining two real tuning systems using their direct product and the appropriated componentwise operations and definitions is straightforward. All the concepts and theorems we define for single-pitch space follow trivially. As a result, the pitch class is $\cal{T} \simeq \mathbb{R}^2$. For completeness, I will define this group as a template for combining two common tuning systems.³

Real-pitch pair tunning system

Let pitch space $\mathcal{S}$ be the set of point in a two dimensional plane $\mathbb{R}^2$ and with function $p: \mathbb{R}^2_{>0} \rightarrow \mathbb{R}^2; p(f_1,f_2) = b_1\log_{a_1}(f_1/f_0) \mathbf{e}_1 + b_2\log_{a_2}(f_2/f_0)\mathbf{e}_2$ for some points $\mathbf{a},\mathbf{b} \in \mathbb{R}^2_{>0}$ such as the pitch $\mathbf{x} = (0,0)$ is the standard pitch of frequency $f_0$.⁴ Let me define the group action with group $(\mathbb{R}^2, \mathbf{+}, \mathbf{0})$ and a map $\lambda_{\mathbf{g}}: \mathbb{R}^2 \rightarrow \mathbb{R}^2; \lambda_{\mathbf{g}}(\mathbf{x}) = \mathbf{g + x}$ for any $\mathbf{g} \in \mathbb{R}^2$ with $\mathbf{+}$ componentwise operator and $\mathbf{0}$ representing the componentwise identity element $(0,0)$. The tuning systems pitch-class is the orbit $\mathcal{T} = \lambda_{\mathbb{R}^2}(\mathbf{0})$ that forms a real pitch pair.

Please observe that the stabilizer around identity is the same as the identity itself $G_\mathbf{0} = \lbrace \mathbf{0} \rbrace$. It follows then $\lambda_{\mathbb{R}^2}(\mathbf{0})$ is group-action isomorphic to homogenous space $\mathbb{R}^2/ \lbrace \mathbf{0} \rbrace$ that is a simple representation of $\mathbb{R}^2$. This means that the pitch-class $\mathcal{T} = \lambda_{\mathbb{R}^2}(\mathbf{0}) \simeq \mathbb{R}^2$.⁵

Therefore, we can represent this tuning system as a two-dimensional plane where each point parametrized the action from the standard to an arbitrary pitch pair. Alternatively, we could use the pitch space $\cal{S}$ itself because for this tuning system has no practical differences. However, understanding pitch class algebra or geometry will become more relevant as we apply less trivial actions over the standard pitch pair.

Ploting real-pitch pair tuning system

In the following plot, we show as an example a representation of the real pitch pair with action given by coordinates $(0.25, 0.75)$. As a result of the action, we transform the standard pitch from $(0,0)$ to $(0.25,0.75)$ within $\cal{S}$. Moreover, if I choose the pitch-space function to be $p(f_1,f_2) = \log_{2}(f_1/f_0) \mathbf{e}_1 + \log_{2}(f_2/f_0)\mathbf{e}_2$, then following frequency values $f_1 = 2^{0.25}f_0$ and $f_2 = 2^{0.75}f_0$ are associated to that pitch pair or $p(f_1,f_2) = (0.25, 0.75)$.

Disclaimer

Expect conceptual errors and constant updates to the whole blog series. Be skeptical and critical of this series content. Ultimately, you are responsible for the information you consume.

References