A pair of pitches with a twist

• Parallel and contrarian motions
• The twist
• Disclaimer
• References

Parallel and contrarian motions

In the post ordered pair of pitches I described ordered pair of pitches using group actions. However, in harmony it is more common to refer to pair of pitches based on their intervals regardless their order. In order to understand this point, we start by redefining real pitch pair using a different parametrization as follows.

Let the pitch space $\mathcal{S} \subset \mathbb{R}^2$ be the set of points in two-dimensional plane, and define the 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 parameters $\mathbf{a},\mathbf{b} \in \mathbb{R}^2_{>0}$ such that the pitch $\mathbf{x} = (0,0)$ correspondes to the standard pitch with associated frequencies $(\hat{f}_1, \hat{f}_2)$. Define the group $G = (\mathbb{R}^2, \mathbf{+}, \mathbf{0})$ and the action $\lambda_{(\mu, \delta)}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined as

This map parametrizes actions as parallel motions (controlled by $\mu$)git and contrary motions (controlled by $\delta$). In terms of the previous real pitch pair definition, $\mu = g_1 + g_2$ and $\delta = g_2 - g_1$. The figure below shows the group actions and thier effect on pitch space. The figure also illustrates how group operations results in the composition of transformation in the pitch space. In particular, the example illustrates the composition rule resulting of group actions, namely that any trajectory that starts and ends at the same group element (here the identity) must also start and end at the same pitch pair (in this case the standard pitch pair).

The twist

In harmony, the interval between two pitches is given by the distances between them regardless their order. This means the interval between two pitches is given by absolute value of $\vert\delta\vert$. For example, the action $A = (1.0, 0.5)$ transforms the standard pitch pair into the pair with coordinates $(0.75,0.25) \in \mathcal{S}$. In harmony, usually this is treated as equivalent to it twisted twin given by $A^{\prime} = (1.0, -0.5)$ which transforms the standard pitch pair to $(0.25,0.75)$. In music this kind of objects are called dyads¹.

One way to identify as both action given by $A$ and $A^{\prime}$ is to require their transformations send the pitch same point in $\mathcal{S}$. I say that $A$ and $A^{\prime}$ are equivalent relative to standard pitch $0 \in \mathcal{S}$ if $\lambda_A(0) = \lambda_{A^{\prime}}(0)$ where I refer as $A^{\prime}$ as image of $A$ or viseversa. It is not hard to implement an action $\lambda_{(\mu, \delta)}: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that the necessary requirements for a real pitch dyad as follow

Below the figure illustrates the result of this transformation over the same action trajectory as above.

From the plot we must notice that

• Only points in the half-plane above the diagonal line are reachable from standard pitch.
• This is by convention; it would be easy to define a transformation to reaches the lower half-plane as well.
• The action from $A \in G$ to its equivalent $A^{\prime} \in G$ results in a bounce on the diagonal.
• The previous composition rule does not hold $\lambda_{1+2+3}(0) {=}\mathllap{/\,} \lambda_{0}(0)$ !

However, we can make $\lambda_{1+2+3}(0) = \lambda_0(0)$ by an appropiate choice of action as shown in the next graph.

However, composition rule derived from group actions is not valid in general. This means real pitch dyads cannot be represented by group actions. This means that identifying dyads by interval behaves very different from identifying, for example, octaves that was possible by defining group stebilizer action over homogeneous spaces. We will need a new type of object to decribe the algebra of dyads and its derived structures!

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