Music and Geometry - part III: Intervals, Counterpoint and Voice Leading

In the first and second article we’ve studied single pitches. This is great for analyzing melodies or scales, but its usefulness is limited when considering multiple pitches sounding simultaneously. In this article we will turn our attention to intervals (2-note chords).

Representing intervals

A straightforward idea would be to represent intervals as line segments on the pitch line, or as circle segments or chords on the pitch class circle. The following could represent an interval made of pitch classes C and E.

While this representation has been used as the basis for other exotic music theories (e.g. Peter Schat’s tone clock theory), as far as the geometrical structure of the space of intervals is concerned this approach won’t bring us many new insights.

To represent intervals we will instead represent groups of 2 notes as one point in a 2-dimensional space. Representing an interval as a point in a 2-dimensional space sounds abstract, so let’s examine first what this means.

Probably the simplest way to represent an interval as a point in a two-dimensional space, is to consider two pitch lines, each of which contribute a note to the interval. A horizontal pitch line contributes a note, and a vertical pitch line contributes another note. Both together determine a point in the 2-dimensional plane. Any point on the plane can be analyzed as consisting of a combination of 2 pitches.

As an example, we look at the representation of the interval (E4,G4). The fact that each point consists of the contribution of two pitch lines is what makes this a two-dimensional space (you need two coordinates to unambiguously define a point in this space). This is a 2-dimensional extension of the pitch line we discussed in the first article. Because every point in this space represents an interval, we call it the interval space.

The first note for every interval is contributed by the horizontal pitch line, and the second note in the interval is contributed by the vertical pitch line. In the above drawing each note in each note pair has an octave number associated to it. Just as with the pitch line, also microtonal intervals can be drawn in the interval plane.

A composition in 2 voices can be represented as a path in the two dimensional interval space. Consider e.g. a counterpoint exercise in the D dorian mode taken from Johann Joseph Fux’ book “Gradus ad Parnassum”. Suffice to say there’s a lot more to counterpoint than just drawing lines in 2d interval space but in this article we won’t go any deeper.

When moving from interval to interval, the jumps (thick black lines) are kept relatively small, especially if you mentally cross out intervals (points) that cannot be used because they contain notes that are not available in the D dorian mode. This is no coincidence: one of the (many) rules in counterpoint is to limit the size of the jumps in melodies, to keep the intervals easy to sing.

Voice leading

Voice leading is a compositional principle governing many styles of music. It is used by classical, pop, folk and jazz musicians alike. The basic idea is that when you go from one chord (or interval) to the next, you do so by moving the different voices the shortest distance possible, so as to gently lead the listener from one chord to the next. Consider the following examples of good and bad voice leading. Both measures move between the same chords. Play them on the piano to hear the difference! In the smooth version adjacent notes take small leaps or remain the same as much as possible.

Because of how the two-dimensional plane of intervals is constructed, points that lie closer together in the plane have a shorter voice-leading distance. Another way to say the same is that a jump between two points in the plane (i.e. a move from one interval to another interval) will have better voice leading if the distance you jump is smaller. This follows from the fact that adjacent points in the 2d interval space are made from adjacent notes on the contributing pitch lines.

Note that our 2d interval space contains all chromatic notes. In many pieces, composers restrict themselves to a subset of the chromatic notes (e.g. only the white keys on a piano). In such cases the interval space can redrawn showing only the relevant intervals.

Voice leading is not only used to write choir pieces! Listen, e.g. how smoothly J.S. Bach moves from chord to chord in his prelude #1 from the well-tempered clavier:

or the dazzling tribute to Bach’s prelude made by Frederic Chopin in which he guides us through a wild ride from c major through the most distant keys and back to c major with ease (well… for the listener at least!)

Motion

Now that we’ve introduced the concept of voice leading, it’s useful to mention the concept of motion, and to study how it translates to our 2d interval space. When moving from one interval to the next, one has a choice between four types of motion.

In parallel motion, both voices move together, keeping the same interval between them. In similar motion, both voices move up or down together, but not necessarily keeping the same interval. In contrary motion, when one voice moves up, the other moves down and vice versa. In oblique motion, one voice remains stationary while the other voice moves.

In many music styles certain parallel motions are forbidden: parallel fifths and parallel octaves are commonly forbidden. This is not just because music teachers like to be pedantic, but also because they sound really bad in a piece that for the rest uses more traditional rules. People who didn’t study music composition can still hear that something “strange” happens when hearing a parallel fifth, although they will struggle to define exactly what causes this sensation of “strange”. It’s also advised not to keep using parallel motion for too long because that starts to tie the two voices together, threatening their melodic independence.

In contrapuntal writing at the same time contrary motion is applauded, as it helps suggesting that the melodies in the two voices are independent.

Parallel and contrary motion in 2d interval space

How is parallel and contrary motion visible in our 2d interval space? Look at the diagram and verify for yourself: any two points in the 2d interval space that are connected by a straight line under 45 degrees represents parallel motion. And any two points in the 2d interval space that are connected by a straight line under -45 degrees represents perfect contrary motion (i.e. contrary motion whereby each voice moves the same number of scale degrees).

If you check the 2d path again that we drew in the counterpoint exercise, you can immediately see from the path segment angles that the composer switches between parallel motion and contrary motion after every chord.

Because voice leading and motions play such a central role in classical music theory, professor Tymoczko decided to rotate the 2d interval pitch space a bit, to make parallel and contrary motion happen along horizontal and vertical directions in the 2d space. This new space contains all the same information as the original space, just rotated a bit. The new space is called the “voice leading space”.

This article was mostly about explaining the preliminaries for the next article where we will examine what happens to the voice leading space if we replace pitch with pitch class again. Remember from the previous article that the pitch line was bent into a pitch class circle.

What do you expect will happen to the 2d voice leading space? Will it simply bend into a ring? Or can we expect something more twisted? Stay tuned for the next episode!