Xmas blog: Tuning systems by simple actions

• Introduction
• Pitch space as a group and simple actions
• Tuning systems of simple acctions
• Simple tuning systems are simple actions
• Disclaimer
• References

Introduction

In the previous blog of this series¹, I showed that building tuning systems as orbit of a group action given by $G$ over a single pitch space $\mathcal{S}$ from standard pitch $a$ has an equivalent counterpart as an action in homogenous spaces $G/G_a$. Both sets of actions are related by group-action isomorphism. The advangate of actions over homogenous spaces is that they can be defined almost exclusively using elements of the group $G$. However, this approach still depends in principle on the value of standard pitch $a$.

For most application in music like defining scales, it will be important for tuning systems to be invariant to the choice of standard pitch. In this blog I will show how to create tuning systems with this property. In the process of defining this tuning systems I will rely on another famous abstract algebra theorem: the First Isomorphism Theorem for Groups.

Pitch space as a group and simple actions

I start by enhancing the defition of simple pitch space² by assuming it is also a group $(\mathcal{S}, \cdot, e’)$.³ Using the added structure I write $\lambda_g(x)$ as the action given by

where $\phi: G \rightarrow \mathcal{S}$ is a function relating elements of the groups $G$ and $\mathcal{S}$. I refer to the actions as defined above as simple actions.

Group-action composition says that $\lambda_{gh}(x) = \lambda_g(\lambda_h(x))$ for all $g, h \in G$ and $x \in X$. It follows then

The last equality is possible if and only if

Group actions also require that $\lambda_e(x) = x$ or $\phi(e)x = x$ that implies by right multiplication by $x^{-1}$ that

Lastly, if $g = h^{-1}$ that is the inverse of $h \in G$, then $e = \phi(e) = \phi(h^{-1}h) = \phi(h^{-1}) \phi(h)$ implying that

The function $\phi: G \rightarrow \mathcal{S}$ that complies with all these properties is know as a group homomorphism⁴. Furthermore, you get a group isomorphism if a group homomorphic function is also bijective.

Tuning systems of simple acctions

I define a tuning system $\cal{T}$ by group action orbits $\lambda_G(a)$ for a given standard pitch $a$. I pointed out that a fundamental property of group-action orbit is that they are homogeneous spaces $\lambda_G(a) \simeq G/G_a$ where $G_a$ is the group action stabilizer.⁵ Just as reminder, the group action estibilizer $G_a$ are all the group action that transform $a$ element in to itself:

In case of a simple group actions, all elements on $g \in G$ such as $\lambda_g(a) = \phi(g) \cdot a = a$ are those elements $g$ in $G$ so $\phi(g) = e$ where $e$ is the identity of $G$. In other words, $G_a = \text{Ker}_{\phi}$ where $\text{Ker}_{\phi}$ is known as the kernel of the homorphism $\phi$ and it is given by

resulting in the following lemma.

There is even more. The orbit $\lambda_G(a)$ is group-action isomorphic to $G/\text{Ker}_{\phi}$ by bijective function $f:G/\text{Ker}_{\phi} \rightarrow \lambda_G(a); g\text{Ker}_{\phi} \mapsto \phi(g) \cdot a$.⁶ This function can be writen as the composition $f = f’’ \circ f’$ where $f’: G/\text{Ker}_{\phi} \rightarrow \phi(G); g\text{Ker}_{\phi} \mapsto \phi(g)$ and $f’’: \phi(G) \rightarrow \lambda_G(a); g \mapsto g \cdot a$. Therefore the composition implies that the homormorphism $f’$ (because $\phi$ is homomorphic) is also bijective resulting in the fact that $G/\text{Ker}_{\phi}$ is group isomorphic with $\phi(G)$ also notated as $G/\text{Ker}_{\phi} \simeq \phi(G)$ (see diagram below).⁷

In summary, the fact that tuning systems are homogenous spaces when apply to simple group actions reduces to what is know as the First Isomorphism Theorem for Groups.⁸

I more detailed explanation of the theorem and other important concept such normal subgroup, quotient group, etc is outside of the scope of this blog. For an intuitively explenation of the theorem and associated concepts can be found ⁹. All the details can be read at Smith’s wonderful book¹⁰.

The First Isomorphism Theorem for Groups provides as corollary that allows compute the size of the image $\phi(X)$ in case $X$ is finite.

In the context of tuning systems the theorem says that $\phi$ divides the group $G$ in set of equivalent elements. Those equivalent elements are mapped by $\phi$ to individual elements of $\mathcal{S}$. For finite group $G$, the previous corollary implies that the size of a tuning system $\vert \mathcal{T} \vert = \vert G \vert / \vert \text{Ker}_\phi \vert$.

Simple tuning systems are simple actions

For all all the simple tuning systems it is not hard to show that are simple actions with $\text{Ker}_{\phi} = \lbrace e \rbrace$ for all of them.¹¹

You can do the same argument to conclude that for

• real tuning system $\phi: \mathbb{R} \rightarrow \mathbb{R}; g \mapsto g$, resulting in pitch class is $\mathbb{R}$,
• equal temperament tuning system $\phi: \mathbb{Z} \rightarrow \mathbb{Z}; g \mapsto g$, and then pitch class is $\mathbb{Z}$,
• $p_n$-limit tuning system $\phi: \mathbb{Z}^n \rightarrow \mathbb{Q}; g \mapsto \prod^n_{i=1} p^{g_i}_i$ so the pitch class is $\mathbb{Z}^n$.

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