Simple tuning systems

• Introduction
• Simple tuning systems
  • Positive-real tuning system
  • Real tuning system
  • Equal temperament tuning system
  • Equal ratio tuning system
  • $p_n$-limit tuning system
• Tuning system’s pitch class
• Disclaimer
• Reference

Introduction

In a previous blog, I introduced the concept of tuning systems. My goal in this blog is to present a few simple examples of these systems. Please be aware some of the names below are made up by me.

Simple tuning systems

Positive-real tuning system

Let $\mathcal{S}$ be the set set of real numbers $\mathbb{R}$ and with function $p: \mathbb{R}_{>0} \rightarrow \mathbb{R}; f \mapsto f/f_0$ such as the pitch $x = 1$ is the standard pitch of frequency $f_0$. Let me define a group action with group $(\mathbb{R}_{>0}, \cdot, 1)$ and map $\lambda_g: \mathbb{R}_{>0} \rightarrow \mathbb{R}; x \mapsto gx$ for any $g \in \mathbb{R}_{>0}$. The tuning systems is then given by the the orbit $\mathcal{T} = \lambda_{\mathbb{R}_{>0}}(1)$ and I call it positive-real tuning system.

The positive-real tuning system naturally appears when working with sound generators or synthesizers. This system takes a standard pitch and scales its frequency by positive real value $g$.

Real tuning system

Let $\mathcal{S}$ be the set set of real numbers $\mathbb{R}$ and with function $p: \mathbb{R}_{>0} \rightarrow \mathbb{R}; f \mapsto b\log_a(f/f_0)$ for some $a,b \in \mathbb{R}_{>0}$ such as the pitch $x = 0$ is the standard pitch of frequency $f_0$. Let me define the group action with group $(\mathbb{R}, +, 0)$ and a map $\lambda_g: \mathbb{R} \rightarrow \mathbb{R}; x \mapsto g + x$ for any $g \in \mathbb{R}$. The tuning systems is then given by the the orbit $\mathcal{T} = \lambda_{\mathbb{R}}(0)$ and I call it real tuning system.

The real tuning system is some way the same as the positive-real tuning system, where the sum gives the group action operation. This is done by exponentiation where $g$ of $\mathbb{R}$ can be transform to $g’ \in \mathbb{R}_{>0}$ of a positive-real tuning system by $g’ = \exp_a(g/b)$.

Equal temperament tuning system

Let $\mathcal{S}$ be the set of real numbers $\mathbb{R}$ and with function $p: \mathbb{R}_{>0} \rightarrow \mathbb{R}; f \mapsto b\log_a(f/f_0)$ for some $a,b \in \mathbb{R}_{>0}$ such as the pitch $x = 0$ is the standard pitch of frequency $f_0$. Let me define the group action with group $(\mathbb{Z}, +, 0)$ and a map $\lambda_n: \mathbb{R} \rightarrow \mathbb{R}; x \mapsto n + x$ for any $n \in \mathbb{Z}$. The tuning systems is then given by the the orbit $\mathcal{T} = \lambda_{\mathbb{Z}}(0)$ and I call it equal temperament tuning system.

The name of equal temperament is mostly historical. You can think of this system as a discreet version of the real tuning system divided into equal steps (equal temperaments) where each pitch is associated with an integer. As result, there are only a subset of frequencies are associated to this tuning system with values given by $f_n = a^{n/b} f_0$ such as $p(f_n) = n$ for $n \in \mathbb{Z}$.

Equal ratio tuning system

This system is analog to the discreet version for the positive-real tuning system. It is not commonly used as the equal temperament tuning system is perceived as a more natural option. This preference does not have practical consequences as equal ratio and temperament are equivalent throughout exponentiation.

Let $\mathcal{S}$ be the set of real numbers $\mathbb{R}$ and with function $p: \mathbb{R}_{>0} \rightarrow \mathbb{R}; f \mapsto f/f_0$ such as the pitch $x = 1$ is the standard pitch of frequency $f_0$. Let me define a group action with group $(a^{\mathbb{Z}b^{-1}}, \cdot, 1)$ and map $\lambda_g: \mathbb{R}_{>0} \rightarrow \mathbb{R}; x \mapsto gx$ for any $g \in a^{\mathbb{Z}b^{-1}}$¹. The tuning systems is then given by the the orbit $\mathcal{T} = \lambda_{a^{\mathbb{Z}b^{-1}}}(1)$ and I call it Equal ratio tuning system.

As in the case of equal temperament, there are only a subset of frequencies are associated to this tuning system with values given by $f_n = a^{n/b} f_0$ such as $p(f_n) = a^{n/b}$ for $n \in \mathbb{Z}$. The name of equal ration is coming from the fact ration between two concecutive pitches $a^{(n+1)/b}$ and $a^{n/b}$ is the constant $a^{1/b}$.

$p_n$-limit tuning system

Let $\mathcal{S}$ be the set of real numbers $\mathbb{R}$ and with function $p: \mathbb{R}_{>0} \rightarrow \mathbb{R}; f \mapsto f/f_0$ such as the pitch $x = 1$ is the standard pitch of frequency $f_0$. Let me define a group action with group $(\mathbb{Z}^n, +, 0)$ and map $\lambda_g: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}; x \mapsto \prod^n_{i=1} p^{g_i}_i x$ for any $g \in \mathbb{Z}^n$ and $p_1,…, p_n$ the first n-th prime numbers. I call $p_n$-limit tuning system to the orbit $\mathcal{T} = \lambda_{\mathbb{Z}^n}(1)$.

The tuning system is commonly associated with the tradition of just intonation². As in the equal temperament case, only a discreet set of frequencies are associated to this tuning system with values given by $f_g = \prod^n_{i=1} p^{g_i}_i f_0$. It is a rather complex tuning system to generate a simple pitch space. Each pitch is associated with a n-dimentional vector of integers or $\mathbb{Z}^n$. Therefore, it is customary to reduce the complexity of this system by limiting the dimension, for example, in the case of the five-limit tuning³.

Tuning system’s pitch class

A natural question from the previous tuning systems $\mathcal{T}$ is what kind of set they form. How the tuning systems $\mathcal{T}$ are related to the group, $G$ used to generate them. How does the group structure of the tuning-system group manifest in the tuning system set? This blog will answer these questions informally for the previous tuning systems.

By definition, any element $b$ of the positive-real tuning system $\mathcal{T}$ there is an $x \in \mathbb{R}_{>0}$ such as $b = x$. The relationship between $b$ (element of the tuning system) and $x$ (element of the tuning-system group) is one-to-one because both are basically represented by the same real number. Therefore, I would expect that positive-real tuning system has as pitch class the set of positive real number using the notation $\mathcal{T} \simeq \mathbb{R}_{>0}$, or $\lambda_{\mathbb{R}_{>0}}(1) \simeq \mathbb{R}_{>0}$.

Following the same heuristic about the other examples of tuning systems, it is not hard to see that for real tuning system $\lambda_{\mathbb{R}}(0) \simeq \mathbb{R}$ (or its pitch class is $\mathbb{R}$), for equal temperament system $\lambda_{\mathbb{Z}}(0) \simeq \mathbb{Z}$ (or its pitch class is $\mathbb{Z}$), for equal ratio system $\lambda_{a^{\mathbb{Z}b^{-1}}}(1) \simeq a^{\mathbb{Z}b^{-1}}$ (or its pitch class is $a^{\mathbb{Z}b^{-1}}$), and for $p_n$-limit tuning system $\lambda_{\mathbb{Z}^n}(1) \simeq \mathbb{Z}^n$ (or its pitch class is $\mathbb{Z}^n$). Finally, a lot of the pitch classes presented here are equivalent by exponentation resulting in the summary table below.

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.

Reference