A glimpse into the world of Voice Leading

Visual examples

From a musical perspective, we’ll be dealing with chords (finite sets of at least two distinct notes) and their transpositions (translating them by some distance), but, from a mathematical perspective, we’ll be working with points in a circle (the border, not the inside) and rotating them by some angle.

For simplicity, we’ll work with the unitary circle, the circle with radius 1 and whose center is the origin of the Cartesian Plane. Each point in the unitary circle can be thought of as a possible musical note, if we conveniently match them.

In that sense, let’s work with the following pairing:

Let’s start with a simple case: Rotating the chord “C major” by ${90}^{°}.$

Such chord is comprised of the notes C, E and G; notes which we can represent in the Cartesian Plane (given the note $\equiv$ angle pairing we have already established) as the following points:

C : $(\cos (0^{°}),\sin (0^{°}))$

E : $(\cos ({120}^{°}),\sin ({120}^{°}))$

G : $(\cos ({210}^{°}),\sin ({210}^{°}))$

However, it’s much simpler to represent chords not by the coordinates of the points which comprise such chord, but by the unique angles, inside $[0^{°};{360}^{°}[$, associated to those points.
In that sense, let’s work with the following equivalence:
C major $\equiv$ {C; E; G} $\equiv (0,120,210),$
where the angles in the vector representation of the chord will be written in degrees and from lowest to highest angle.

Now, let’s rotate each of the points which comprise C major by ${90}^{°}$ with respect to the origin, as we can see in the following video, where the initial points shown (in green) correspond to the notes C, E and G, in that order.

From a musical perspective, we’ve transposed the C major chord by
${90}^{°}=\frac{{360}^{°}}{4}\equiv \frac{12\text{ semitones}}{4}=3\text{ semitones }$ and landed on Eb major, but, mathematically, we can think of having rotated the initial chord.

The set of trajectories that the notes/points from C major take along the unitary circle to get to the notes in Eb major is called the voice leading between C major and Eb major.

Let me clarify that the voice leadings which we’ll be dealing with in this post are precisely those which satisfy that no pair of distinct notes from the initial chord end up at the same note in the transposed chord. In that sense, when we tranpose/rotate any chord made up of $n$ notes, the number of possible voice leadings between such chord and its transposition is precisely $n!=n\cdot (n-1)\cdot (\dots )\cdot 2\cdot 1$.
That’s because every note $x$ in the initial chord is moving (in a voice leading) to a note $y$ in the transposed chord, unique to $x$. That means that no other note from the initial chord is also moving to $y$; therefore, combinatorics is all we need.
Curious fact: We actually only need to check $n$ special kinds of voice leadings, not $n!$, in order to find one which is, at least, a minimal voice leading. The justification comes from a mathematical theorem which we’ll explore further in the next post, where it will play a key role. If you are interested, you can read about it in this article .

Going back to our concrete case, let’s see some voice leading examples for the transposition from C major to Eb major in the following video:

Notice from the video that the trajectories which define the voice leading correspond to the shortest of the two paths that a point could have taken to get to the other. In the case where both paths have the same length, that is, for a ${180}^{°}$ rotation, we’ll choose, as a convention, the anticlockwise trajectory to be a part of the voice leading.

In this post, we’ll focus on two properties that we’d like a voice leading (VL) between a chord and some of its transpositions to have:

1. The sum of the length of the trajectories which define VL, also known as the voice leading size, is the smallest one out of all the possible voice leadings between the initial chord and its transposition.
   Any voice leading which satisfies this property is said to be a minimal voice leading.
2. The trajectories which define VL share the same direction, that is, they are all clockwise or all anticlockwise.
   Any voice leading which satisfies this property is said to have no contrary motion.

Partly, what I’ve shown you in the initial graphs is which chords, of some fixed chord size, satisfy that after transposing/rotating them by a certain amount, there exists a voice leading, between the initial and transposed chord, that is a minimal voice leading and has no contrary motion. If some chord satisfies such property, we’ll say that it has an unidirectional minimal voice leading.

I wouldn’t blame you if you were just thinking the following:
“Why would a minimal voice leading be unidirectional?.
The properties 1. and 2. do NOT seem related at all”.
You’d be partly right, in fact, because the property 1. deals with minimizing trajectories’ lenghts, while the property 2. refers only to such trajectories’ directions.

However, both properties happen to be fairly related, and even extremely related when we focus on ${180}^{°}$ rotations of chords, a situation where symmetry arises with respect to the possible voice leadings. But, such connection will be explored in the next post because that’s a deep subject as well.

Let’s get back to our example, rotating the C major by ${90}^{°}$.
Does it exist an unidirectional minimal voice leading in that case?

After checking by hand the $3!=6$ possible voice leadings, we actually find one unidirectional minimal voice leading, the one that I labeled as voice leading 3. Such voice leading is comprised of clockwise trajectories and it has a voice leading size of ${90}^{°}.$
Of course, there is some ambiguity with respect to the direction of the trajectory in voice leading 3 which has a size of $0^{°}$; but, we can ignore its direction (in the same way that the number zero has no sign) when we are checking if some voice leading is a unidirectional minimal voice leading. The other trajectories in voice leading 3 have the same direction, so that’s all that matters to us.
Notice that voice leading 1 and voice leading 2 fail to be minimal voice leadings because their respective voice leading sizes, ${90}^{°}\cdot 3={180}^{°}\text{ and }{150}^{°}+{180}^{°}+{120}^{°}={450}^{°}$, are greater than the voice leading size of the minimal voice leading that I already presented (voice leading 3).
Also note that voice leading 1 has no contrary motion but voice leading 2 does have contrary motion.

Putting it all together

We are 80% ready to understand the graphs shown previously.

The first step is to choose a chord size ($n$), that is, the amount of distinct points in a circle that we will be rotating afterwards.

The second step is to chose some rotation (‘rotation’), a value between $0^{°}$ and ${360}^{°}$, that we will apply to ‘all’ chords with the size already chosen.

In that sense, if we use the vector representation of chords via their angles, as we did with C major ‘=’ $(0^{°},{120}^{°},{210}^{°})$, the domain of the function, music_graph(…), function from which these graphs shown come from, would be $$\{(x_0,x_1,x_2,\dots ,x_{n-1})\in {\mathbb{R}}^n:0\le x_0<x_1<\cdots <x_{n-1}<360\}$$

However, we can be more efficient and work only with a special subset of such domain, which would be the new domain of music_graph(..), yet still achieve our goal of finding for which chords of size $n$ there exists an unidirectional minimal voice leading when they are rotated by ‘rotation’. Such efficiency is a direct result of the following:

Important property: If there exists a unidirectional minimal voice leading with respect to some rotation $\theta$ of some chord $X$, then there also exists a unidirectional minimal voice leading with respect to the rotation $\theta$ of any chord which is itself some rotation of $X$.

As a consequence, based on the fact that for the rotation ${90}^{°}$ of the C major chord, which we can label via its angles as $(0^{°},{120}^{°},{210}^{°})$, there exists a unidirectional minimal voice leading, if we were to rotate also by ${90}^{°}$ chords whose angles are of the form $((0+\alpha {)}^{°},(120+\alpha {)}^{°},(210+\alpha {)}^{°})$, where $\alpha$ can be any real number, then there would also exist at least one unidirectional minimal voice leading for those chords.

This means that the domain of music_graph(…) can be reduced in one dimension, after which it would take the following form as the new domain of music_graph(…):

$$\{(0,x_1,x_2,\dots ,x_{n-1})\in {\mathbb{R}}^n:0<x_1<\cdots <x_{n-1}<360\}\dots (\mathrm{△})$$

The third step is to check, for every chord $(0,x_1,x_2,\dots ,x_{n-1})$ in $(\mathrm{△})$ if there exists an unidirectional minimal voice leading after rotating such chord by ‘rotation’.
Practically, we can’t check each of those chords, because there’s an infinite amount of them, but the computer will check a sufficient amount so that we can accurately approximate how the graph of music_graph(…) actually looks like.
Let’s denote the set of chords in $(\mathrm{△})$ for which there exists an unidirectional minimal voice leading after rotating those chords by ‘rotation’, as ‘success’.

The fourth step is displaying, appropriately, on the 3-dimensional space ${\mathbb{R}}^3$ the elements of ‘success’. Allow me to explain the main 4 cases:

Case $n=2$:
The domain in $(\mathrm{△})$ becomes $\{(0,x)\in {\mathbb{R}}^{\mathbb{2}}:0<x<360\}.$
The elements of ‘success’ are special chords represented by $(0,\overline{x})$, with $0<\overline{x}<360$. Such chords will be displayed to be in ‘success’ via coloring the points $(\overline{x},0,0)\in {\mathbb{R}}^3.$ The last two zeros in $(\overline{x},0,0)$ I added them in for simplicity only, they aren’t important.

Case $n=3$:
The domain in $(\mathrm{△})$ becomes $\{(0,x,y)\in {\mathbb{R}}^{\mathbb{3}}:0<x<y<360\}.$
The elements of ‘success’ are special chords represented by $(0,\overline{x},\overline{y})$, with $0<\overline{x}<\overline{y}<360$. Such chords will be displayed to be in ‘success’ via coloring the points $(\overline{x},\overline{y},0)\in {\mathbb{R}}^3.$ The last zero in $(\overline{x},\overline{y},0)$ I added in for simplicity only, it’s not important.

Case $n=4$:
The domain in $(\mathrm{△})$ becomes $\{(0,x,y,z)\in {\mathbb{R}}^{\mathbb{4}}:0<x<y<z<360\}.$
The elements of ‘success’ are special chords represented by $(0,\overline{x},\overline{y},\overline{z})$, with $0<\overline{x}<\overline{y}<\overline{z}<360$. Such chords will be displayed to be in ‘success’ via coloring the points $(\overline{x},\overline{y},\overline{z})\in {\mathbb{R}}^3.$

Case $4<n\le 6$:
We’ve runned out of dimensions, because now the graphs are either 4 dimensional or 5 dimensional. For that reason, in order to be able to visualize the graph in a 3 dimensional space, $n-4$ of the variables in $(0,x_1,x_2,\dots ,x_{n-1})$ will need to be chosen in advance and be kept fixed during the third step.
Which angles will remain fixed is what the last two numbers in the command music_graph(…) refer to.

How to interpret the function music_graph(chord_size, rotation, $m_1$, $m_2$) :
We already know what chord_size and rotation refer to.
The extra caveat to consider is that the chords represented in $(\mathrm{△})$ must contain the angles $\frac{m_1}{\text{chord\_size}}\cdot {360}^{°}$ and $\frac{m_2}{\text{chord\_size}}\cdot {360}^{°}$ .

Now you can see why in the case $m_1=m_2=0$ we didn’t need to fix any angle in $(\mathrm{△})$, because the angle $0^{°}$ is already present in those chords which we are considering.

The fifth step, the final one, is to color appropriately the points displayed in the fourth step.
After having chosen the parameters in music_graph(chord_size, rotation, $m_1$, $m_2$) and doing all the first 4 steps just mentioned, we established a finite set of points to represent/approximate $(\mathrm{△})$, after which we selected a subset of it, let’s call it ‘success_approximation’, which represents/approximates ‘success’. Each element of the finite set ‘success_approximation’ has a certain minimal voice leading size, so, I’ll define a coloring based on such minimal voice leading sizes.
Blue will be the color of the displayed point (fourth step) for which its associated chord had the smallest minimal voice leading size (min_VL) with respect to the minimal voice leading sizes of all the chords in ‘success_approximation’.
Red will be the color of the displayed point (fourth step) for which its associated chord had the biggest minimal voice leading size (max_VL) with respect to the minimal voice leading sizes of all the chords in ‘success_approximation’.
The other points displayed in the fourth step will be colored with a color between blue and red, depending on the position of their minimal voice leading size in the interval $[\text{ min\_VL };\text{ max\_VL }].$

Notice that there may be more than one chord in ‘success_approximation’ with the smallest (or biggest) minimal voice leading size, out of the minimal voice leading sizes of all the chords in ‘success_approximation’.
Besides this, it may also be the case that, for certain parameters in music_graph(chord_size, rotation, $m_1$, $m_2$), all the chords in ‘success_approximation’ have precisely the same minimal voice leading size. In that case, all the displayed points (fourth step) would be colored in purple.

Of course, we don’t have to stop at the case of chords with size 6, we could consider, for example, chords of size 666 and so on. However, I only coded the function music_graph(…) till chord size 6, so that’s the limit for now.

I encourage you to play around with the graphics in this post. For example, notice the bizarre fact that each of the graphs shown contains regions where the minimal voice leading size remains constant, that is, where the coloring applied doesn’t change.

In conclusion, I hope that after reading this post, you’ve gained some appreciation for music. We only focused on rotating points along a circle, while checking for certain properties of the trajectories of the points, a very simple problem indeed; yet, the graphs shown displayed no simplicity whatsoever. Do we even know which mathematical equation determines the shape of the graphs? From what I’ve researched online, it’s not known publicly.
I do know that precise mathematical equation, but I’d rather not share it till I present it in my undergraduate thesis.

Thanks for reading!
This post took a huge amount of work, so I am glad you made it till here.
I’ve also implemented an online graph calculator so that you can play around changing the inputs of the function music_graph(…) and see what bizarre shapes come into existence.

Finally, a couple more graph examples in case you are curious about how other cases look like:

music_graph(2,90,0,0)

music_graph(2,180,0,0)

music_graph(3,65,0,0)

music_graph(3,180,0,0)

music_graph(4,155,0,0)

music_graph(4,180,0,0)

music_graph(5,55,2,0)

music_graph(6,35,2,3)

music_graph(6,180,3,4)