Ordered pairs of pitches

• Introduction
• Direct product of two groups
• Real-pitch pair tunning system
• Octave-equivalent real-pitch pair
  • Tunning system
  • Octave-equivalent real-pitch pair is a Torus
• Summary
• Disclaimer
• References

Introduction

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

Direct product of two groups

I build the concept of pitch class by using group actions on a single-pitch space that I called tuning systems. Moreover, I show how the most helpful tuning system can be described using simple actions defined as homomorphism between groups.

The following proposition allows me to combine two groups to create a new group that will help represent the pitch pairs.

Using this proposition, we can combine any tuning system in this summary to represent two pitches. 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 the simples example 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 $p: \mathbb{R}^2_{>0} \rightarrow \mathbb{R}^2; p(f_1,f_2) = b_1\log_{a_1}(f_1/\hat{f}_1) \mathbf{e}_1 + b_2\log_{a_2}(f_2/\hat{f}_2)\mathbf{e}_2$ for some points $\mathbf{a},\mathbf{b} \in \mathbb{R}^2_{>0}$ such as the pitch $\mathbf{x} = (0,0)$ is the standard pitch with associated frequencies $(\hat{f}_1, \hat{f}_2)$.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; see the plot below. 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.

This pitch class can be parametrized by the orbit $o:\mathbb{R}^2 \rightarrow \mathbb{R}^2$; $o(g_1, g_2) = g_1 \mathbf{e}_1 + g_2 \mathbf{e}_2$. For example, a real pitch pair with action given by coordinates $A = (0.25, 0.75)$ transforms the standard pitch from $(0,0)$ to $f(A) = (0.25,0.75)$, see figure below. 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 $p(f_1,f_2) = (0.25, 0.75)$.

Octave-equivalent real-pitch pair

Tunning system

Next, we would like to find an octave-equivalent version of the real pitch pair. Using the same approach as above, we derive the following definition.

Let pitch space $\mathcal{S}$ be the set of two dimensional complex numbers $\mathbb{C}^2$ and with function $p: \mathbb{R}^2_{>0} \rightarrow \mathbb{C}^2; p(f_1,f_2) = e^{i2\pi b_1\log_{a_1}(f_1/\hat{f}_1)} \mathbf{e}_1 + e^{i2\pi b_2\log_{a_2}(f_2/\hat{f}_2)}\mathbf{e}_2$ for some points $\mathbf{a},\mathbf{b} \in \mathbb{R}^2_{>0}$ such as the pitch $\mathbf{x} = \mathbf{1} = (1,1)$ is the standard pitch of frequency $(\hat{f}_1, \hat{f}_2)$.⁴ Let me define the group action with group $(\mathbb{R}^2, \mathbf{+}, \mathbf{0})$ and a map $\lambda_{\mathbf{g}}: \mathbb{C}^2 \rightarrow \mathbb{C}^2; \lambda_{\mathbf{g}}(\mathbf{x}) = e^{i2\pi g_1} x_1 \mathbf{e}_1 + e^{i2\pi g_2} x_2 \mathbf{e}_2$ 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 $\mathcal{T} = \lambda_{\mathbb{R}^2}\left(\mathbf{1}\right)$ that forms a octave-equivalent real pitch pair.

The tuning system is just the direct product of two octave-equivalent real tuning systems⁶ in which the pitch class is $\mathbb{R}/\mathbb{Z}$ isomorphic with group circle $\mathbb{T}$. As a result, the pitch class of an octave-equivalent real pitch pair is simply $\mathbb{T}^2$. This pitch class can be parametrized by the orbit $o: \mathbb{R}^2 \rightarrow \mathbb{C}^2$; $o(g_1, g_2) = e^{i2\pi g_1} \mathbf{e}_1 + e^{i2\pi g_2} \mathbf{e}_2$. One way to visualize this tuning system is by representing the pitch space as a pair of to unit circles. Technically, those circles are a pair of unit circles in complex plan $\mathbb{C}$. An arrow from their center represents any point of those circles circumferences. An action given by coordinates $A = (0.25, 0.75)$ transforms the standard pitch $(1,1)$ into to $o(A) = (e^{i\pi/2},e^{i3\pi/2})$, see figure below. Assuming a pitch-space function given by $p(f_1,f_2) = e^{i2\pi \log_2(f_1/f_0)} \mathbf{e}_1 + e^{i2\pi \log_2(f_2/f_0)}\mathbf{e}_2$, then the collection of frequency $f_1 = 2^{0.25+n}f_0$ and $f_2 = 2^{0.75+n}f_0$ for $n \in \mathbb{Z}$ are related to pitch pair because $p(f_1,f_2) = (e^{i\pi/2},e^{i3\pi/2})$.

You should also notice that for the given point A, a set of points (images) $A + \mathbb{Z}^2$ results in the same action over the standard pitch. The figure below shows how I can represent in a two-dimensional space. The light blue square represents points that induce unique actions to the standard pitch. This region is called the fundamental domain, and it is one of choice of infinite countable domains.⁷ The dark blue arrows are known as identification lines because those element induces equivalent actions. In this case, in particular, the action leaves any pitch on the tuning system $x \in \mathcal{T}$ unchanged, meaning $\lambda_{\mathbb{Z}^2}(x) = x$.

Octave-equivalent real-pitch pair is a Torus

Alternatively, we can thinking about the action over the pitch space as elements on pitch-class $\mathbb{T}^2$⁵. The direct product of the group circle $\mathbb{T}^2$ is isomorphic to the shape of a Torus embedded in a 3D space under the transformation

where the major radius $R$ is the distance from the center of the tube to the center of the torus and the minor radius $r$ is the radius of the tube under the condition $0 < r < R$⁸. Therefore, the action given by coordinates $(0.25, 0.75)$ can be represented as points on a torus’s surface, as shown in the figure below.

Summary

I define the space of two-dimensional real pitch spaces without and with the assumption of the equivalence of octaves. These definitions are simple extensions of one-dimensional cases because they are the direct product of two groups. Then, I show how I can understand pitch space by studying the geometry features of group use to define actions over the standard pitch.

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