This blog is part of a series wth all my music notes. In the Series page you can find all the posts of the series.

Introduction

In a previous blog I introduced the notion of simple pitch space. This blog aims to build subsets of this space that can be useful to express musical concepts.

Standard pitch and actions

Music is built from a standard pitch1. Musicians use a standard pitch as a reference to define other pitches, often grouped in scales2. Luthiers tune instruments around a standard pitch so that any different pitch is somehow related to that standard pitch. It is easy to define a standard pitch from a simple pitch space as follows.

Definition (standard pitch): A standard pitch $a$ is an element of the simple pitch space $\mathcal{S}$ with an arbitrarely chosen frequency $f_0 \in \mathbb{R}_{>0}$, meaning $f(a) = f_0$.

Starting from a starndard pitch $a$ and a map $\lambda : \mathcal{S} \rightarrow \mathcal{S}$, I can construct another pitch $b \in \mathcal{S}$ as $b = \lambda(a)$. You can say that a pitch $b$ is the result of the action of the map $\lambda$ over the reference pitch $a$.

I can now extend this idea to create more than one pitch. Lets define a map $\lambda_x : \mathcal{S} \rightarrow \mathcal{S}$ for each element $x$ of a set $X$. Under these maps, each element of $X$ defines possibly different actions on the standard pitch. For example, give two elements $x, y \in X$, I can define two pitches $b, c \in \mathcal{S}$ by the following actions on the standard pitch $b = \lambda_x(a)$ and $c = \lambda_y(a)$. Therefore, it seems possible to use actions to build a set of pitches from a standard pitch useful for music representation.

One thing it would be nice to express is the trivial identity action that leaves the standard pitch untouched. The element $e \in X$ is the identity action if $\lambda_e(a) = a$ where $\lambda$ is the action maps described above.

Assume that you construct a pitch $b = \lambda_x(a)$ for an action on standard pitch with $x \in X$. I could now also create another pitch $c$ by acting on $b$ or $c = \lambda_y(b)$ for some $y \in X$. It would be desirable the pitch $c$ can be also the result of an action on the standard pitch or $c = \lambda_z(a)$ for a $z \in X$. This property implies that $\lambda_z(a) = \lambda_y(\lambda_x(a))$, meaning $\lambda_z$ is the function composition $\lambda_y \circ \lambda_x$. If this property holds for any combination of $x, y \in X$, it is possible to decribe every pitch as action on standard pitch, or alternative, as succession of actions starting from the standard pitch.

Groups

Actions with the above properties are known as group actions. Defining these actions requires first to explain what in algebra is known as a group.

Definition (Abstract group):

A group $(G, ., e)$ is a set $G$ with a binary operator (sometimes referred as group product) $\cdot$ satisfying the following properties:3

  1. Closure: $x \cdot y$ lies in $G$ for all $x, y \in G$;
  2. Associativity: $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ for all $x, y, z \in G$;
  3. Indentity: There is an element $e$ in $G$ with $e \cdot x = x \cdot e$ for all $x \in G$;
  4. Inverses: For each $x \in G$, there is a $x^{-1} \in G$ with $x \cdot x^{-1} = e = x^{-1} \cdot x$.

There are also sets that partially comply with some of the property to be a group. I am not planning to use much of these intermediated objects. However, their names often appear in the literature, so I think it is good to define them. Any set with a multiplication that satisfies properties 1-2 is known as semigroups4. A monoid is a semigroup with an identity element, meaning it satisfy properties 1-35.

Let me provide some examples of a group that will be important later on.

Example (Real numbers under addition): The real numbers form a group $(\mathbb{R}, +, 0)$ with addition as commutative operator. The inverse or additive inverse of a real number r is its negation -r.6
Example (Multiplication on nonzero reals): Under multiplication, the nonzero real numbers form a commutative group $(\mathbb{R}^{*}, \cdot, 1)$.7
Remark (Group multiplication): The group binary operator $\cdot$ is called group multiplication or simply multiplication. However, as shown in the previous examples, this operator is not always the traditional multiplication operator.

Group actions

Having the definition of group, we can now define action group as follow.

Definition (group action):

Let $X$ be a set and let $(G, \cdot, e)$ be a group. Suppose that a map $\lambda_g: X \rightarrow X$ is defined for each element $g \in G$. Then there is a group action of $G$ on $X$ if and only if

  1. $\lambda_e(x) = x$ and
  2. $\lambda_{g \cdot h}(x) = \lambda_g(\lambda_h(x))$

for all $x \in X$ and $g, h \in G$.8

In the next section, I will show that knowing the subset of all elements of $X$ given by all actions of $G$ on one element $x \in X$ is essential. This concept is captured the notion of group action orbits.

Definition (orbit): Let $(X, G, \lambda)$ be an action of a group $G$ on a set X. For an element $x \in X$, the orbit of $x$ is the set \(\lambda_G(x) = \big\lbrace \lambda_g(x) \vert g \in G \big\rbrace\) of images of $x$ under the actions $\lambda_g$ of the elements $g \in G$.9

Tuning systems

The plan is to use action group orbits to define an interesting subset of pitches from a single pitch space. I plan to do that to explain what I call a tuning system. This system is the only concept that is my own. It is my way of applying group action to describe a collection of pitches.

Definition (tuning system): Let $(\mathcal{S}, G, \lambda)$ be an action of a group $G$ on the single pitch space $\mathcal{S}$. Given a standard pitch $a \in \mathcal{S}$, I say that the tuning system notated $\mathcal{T}$ is the orbit $\lambda_G(a)$.

The idea with tuning system is to use a given group action $(\mathcal{S}, G, \lambda)$ with a group $G$ with its binary operator to carve a subset of pitches $\mathcal{T}$ by all possible actions generated by elements of $G$ starting from a standard pitch $a \in \mathcal{S}$, meaning $\mathcal{T} = \lambda_G(a)$. Tuning systems are then an example of action groups restricted to an orbit where we can define $(\mathcal{T}, G, \lambda)$ as an action group of $G$ on $\mathcal{T}$.10

An important property of action groups $(\mathcal{T}, G, \lambda)$ is that they are transitive. This means that $\mathcal{T}$ is not empty and for each pair of elements $b, c \in \mathcal{T}$ there is an element $g \in G$ such $c = \lambda_g(b)$.11 This is exactly the property we identified a few sections back as important to describe meaningful subsets of the single pitch space!

Summary

Starting from single pitch space $\mathcal{S}$ I do the following.

  • I choose a standard pitch $a$ of $\mathcal{S}$.
  • I introduce the group action $(\mathcal{S}, G, \lambda)$ of $G$ on $\mathcal{S}$.
  • I define a tuning system as the set $\mathcal{T} \subseteq \mathcal{S}$ given by orbit $\lambda_G(a)$.
  • I show there is also action group $(\mathcal{T}, G, \lambda)$ of $G$ on $\mathcal{T}$ that is transitive!

In the next blog, I will show some examples of tuning systems that will be useful to define interesting collections of pitches.

Disclaimer

Disclaimer: THE "KNOWLEDGE" IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE "KNOWLEDGE" OR THE USE OR OTHER DEALINGS OF THE "KNOWLEDGE".

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

  1. I like to warn the reader that most of this blog centers around the western musical tradition that I experienced as I grew up. Western music tradition is my current focus on interest. I expect some of concepts I discussed can be extended to other traditions. However, I am not sure of it. ↩

  2. Wiki entry: Scale(music) ↩

  3. Definition 4.14 of 12. ↩

  4. Definition 4.1 of 12. ↩

  5. Definition 4.7 of 12. ↩

  6. Example 4.16 of 12. ↩

  7. Example 4.17 of 12. ↩

  8. Proposition 10.5 of 12. ↩

  9. Section 10.2 “Orbits” from 12. ↩

  10. Example 10.18 of 12. ↩

  11. Section 10.3 “Transitive actions” from 12. ↩

  12. Smith, Jonathan DH. Introduction to abstract algebra. Vol. 31. CRC Press, 2015. ↩ ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9