# Introduction

In the first article we introduced the pitch line. The second article examined what happens to the pitch line if you remove all octave information and replace pitch with pitch class. As we found out the pitch line bends into a pitch circle. In the third article we left the one-dimensional spaces of pitch and pitch circle, and we started looking at intervals, represented as points in a 2d-plane. In this fourth (and for now final) article in this series about geometry and music, we will examine the effect of removing octave information from the voice leading plane and replacing pitch with pitch class.

First, let’s recall what we ended up doing last time.

## Live improvisation in the 2d voice leading space

If you are not afraid of computers and trying new things, I’ve created a supercollider program that allows you to explore the 2d voice leading space auditively by navigating it using the computer mouse.

If you want to play with the program yourself, you will first need to install the free and open source supercollider system. You can download it from the supercollider download site.

Before you worry: this is free of cost and completely legal.

The second step is to download the voiceleadingplane.sc code I wrote from my github page. If you have no idea how to use github, the easiest is to click the green “Clone or download” button on my github page, and then the “download ZIP” link that appears. This will download a .zip file to your computer that contains the voiceleadingplane.sc program as well as a LICENSE agreement that basically grants you the right to use this software for any purpose you see fit, including modifying it and calling it your own, as long as you distribute your derived work under the same license for others to learn from and to modify.

Finally you will need to start the “scide” editor that you installed as part of supercollider, and paste the contents of the voiceleadingplane.sc file in it, put the cursor just before the first bracket and press ctrl-enter (or command-enter on a Mac). You can change the sounding interval by clicking the different points in the voice leading space.

The next video gives a demonstration of me exploring my newly created musical instrument. Navigating the voice leading space like this to actually create music is new to me as well. As with any new instrument this one too would benefit from some practice to make (better) music!

I kept the program as simple as possible. Did you like what you see? Did you encounter problems? Do you wish you had additional features to play with? Consider logging a bug or a suggestion on github then.

## Removing octave information

Recall from last article that every point in this plane consists of a pair of notes with associated octave number. Now what happens if we remove the octave information?

If we no longer consider the octave numbers, the space contains a lot of points that represent duplicate information, e.g. the points (E3,G3), (E3,G4), (E4, G3), (E4, G4), (G3, E3), (G4,E3), (G3, E4) and (G4, E4) are all equivalent (as well as all other intervals consisting of an E-note and a G-note). In the following figure, these (E,G) and (G,E) intervals are colored.

As there’s no need to keep all those equivalent points around, we will select some that, taken together, represent all possible intervals (discarding octave information).

### Selecting each possible interval

In the next figure each point is colored according to the size of the interval it represents. We select a subset of the points that covers all distinct intervals ignoring octave information. Only the points that fall completely in the black rectangle are part of the selection. Given that horizontal movement represents parallel motion, it should not come as a surprise that the colors form horizontal bands. (The colors represent the size of the interval.) The fact that the colored bands are distributed symmetrically around the horizontal axis should not come as a surprise either: this is the consequence of the fact that (because of ignoring octave information) an interval like (E,G) is equal to an interval (G,E). When you move along the 45deg axes, you’re moving in parallel with one of the contributing pitch lines (and therefore perpendicular to the other one), and therefore you expect to hear oblique motion.

Let us look at this selection of points in more detail. In the following image, the selection of points has been stretched horizontally, and squeezed vertically, but it is still the exact same selection of points as in the previous image.

Each ellipse shape in the ribbon corresponds to a point on the 2d voice leading plane. Moving vertically in this plane still represents perfect contrary motion, and moving horizontally represents parallel motion.

## Bending the voice leading plane

To fully understand the beauty of what we’ve created here, we need to take a second, very close look at the colorful ribbon. We’re especially interested in what happens at the borders of the ribbon. It will become apparent that the ribbon does not represent all our intuitions about the 2d voice leading plane. Recall that moving horizontally causes the 2 voices to move in parallel motion. Look at the upper side of the ribbon. It starts with (C,C), and as we move horizontally, it goes to (C#,C#), …, all the way to the right where it says (F#,F#).

Now what happens if we try to go even further horizontally? Well, at first sight we just drop off the ribbon into empty space. But look closely… what we should be doing, taking into account that moving horizontally implies parallel motion, is jumping to the lower left corner and continuing our journey going horizontal until we reach the lower right corner of the ribbon, where it says … (B,B), (C,C) again. And what happens if we try to go even further horizontal? Indeed! We jump back to the upper left corner and continue our journey horizontally.

It is as if the upper left corner and the lower right corner are one and

the same point in space. And in the same way the lower left corner and the

upper right corner are the same point in our interval space. Our ribbon does

not represent these intuitions explicitly.

Similarly, you can verify that travelling vertically on the ribbon represents perfect contrary motion of the voices, and that again the same corners seem to be connected to each other as in the previous paragraph. Now also verify how every point on the right side of the ribbon matches up with the upside-down points on the left side of the ribbon.

Our 2d ribbon representation does not explicitly show these connections between corners and left and right side of the ribbon. Can we do something, similar to what we did with bending the pitch line into a pitch circle, to get a graphical representation that makes all these connections explicitly visible?

The answer is: yes! But it’s not as easy as when we bent the pitch line into the

pitch circle. I’ve created a video to illustrate how it can be done.

This time we use 3 dimensions to represent the effect of ignoring octave information and order between notes in the intervals on our 2d plane. Despite using 3 dimensions to draw our space, the resulting space is really still two-dimensional. Only points glued to the surface have a physical interpretation as an interval. The resulting figure is a so-called Moebius strip. It is a fascinating geometrical object for many reasons.

Amusing videos illustrating some of the wonders of Moebius strips can be found on Youtube. Here’s one (not made my be) that illustrates the unexpected things that can happen if you try to cut Moebius strips in half.

Here’s another one (also not made by me) that demonstrates how Moebius strips, despite first appearance, have no top or bottom side, and no left or right side

## Conclusion

Congratulations if you made it until here! We’ve come a far way. We started with pitches and pitch lines in the first article, then discarded octave information and saw how this bent the pitch line into a pitch class circle in the second article. Next we turned our attention to intervals and discussed how to represent these intervals as points on a plane in the third article. Finally we discarded octave information again in the fourth article and ended up with a Moebius strip.

Geometry doesn’t stop at 2 or 3 dimensions. Mathematics is not limited to things we can easily imagine and represent. You can create a voice leading cube with three-note chords, and keep on adding dimensions to analyze four- and more-note chords, and study the geometrical properties of these spaces. But as the number of dimensions in these spaces increases, things get harder to represent with movies and drawings. That’s where you need to start using more rigorous maths.