Music and Geometry - part I: pitch class

Introduction

In this series of 4 articles I’m going to introduce you to what for many will be the most exotic piece of music theory you’ve ever come across. Even if you think you’re not good at theory, you may want to try following along and – who knows – perhaps pick up an insight or two.

Music shows deep inner structure on multiple levels that appeals to mathematically inclined people. Math professor Guerino Mazzola has written a book called the “Topos of Music”. It’s safe to assume that only a handful of people in the world really understand the finer details of what it describes. You literally need a PhD in mathematics (in category theory to be more precise) to even begin making sense of it. Professor Mazzola argues that all this complexity is unavoidable because the music it models is just as complex.

Luckily also other books showing neat connections between music and mathematics have been written. And luckily these books are accessible to a wider audience. One of the more intriguing ones, in my humble opinion, is “A Geometry Of Music” by professor Dmitry Tymoczko of Princeton University. In these articles we will dip our toes in some of the very basics of his ideas. To be perfectly clear: I’m not affiliated to professor Tymoczko or the book in any way, and all awesome ideas are his, whereas all mistakes in the explanation are mine.

Pitch line

If you just consider notes in a chromatic scale, starting with the very lowest note you can imagine, and rising to the very highest note you can imagine, it’s not so difficult to depict these notes as points on a line. Although the picture below highlights only a few special notes on the line, actually all conceivable pitches, including microtonal ones or pitches so high they can only be heard by dogs, have their place on the pitch line which has no beginning and no end. It’s an infinite line. In geometry, such a line is considered a 1-dimensional space.


Some simple composition operations can be done by reasoning on a pitch line directly. Take transposition, e.g., and observe how it corresponds to sliding along the line. Or consider inversion of a melody and observe how it corresponds to mirroring pitches around some other pitch on the pitch line.

Pitch class

Yet this pitch line doesn’t model all existing intuitions about pitches. One intuition that is not visible on this line, is the feeling that notes one octave apart in some sense sound the same. Who doesn’t remember the lyrics to the Do-Re-Mi song from the sound of music? When reaching the end of the song, it’s made very clear that “this will lead us back to do”, as if all do’s (C notes) are equal in some sense.

A more difficult way to say the same thing is to say that in some contexts it makes sense to abstract away the exact pitch, and instead reason on pitch class. In other words we don’t speak of a concrete C4 or D3 note anymore, but of a more abstract concept of a “C” note, or a “D” note, with no indication of exactly which octave it belongs to.

The question arises what introducing this “pitch class” abstraction does to our pitch line. How does incorporating pitch class change the pitch line?

The solution is straightforward, but not necessarily trivial to come up with by yourself. To model the concept of pitch class, we need to transform our pitch line so that all differences between octaves disappear. This can be done by curling up the pitch line in three dimensions, so that all equivalent notes (e.g. all “C” notes) perfectly line up. After this is done, we will squash the curled line until it’s flat again. If you think deeply about it, the lining up and especially the squashing of equivalent pitches indeed is a kind of geometrical equivalent to replacing concrete notes with pitch classes. Just like with the pitch line, the pitch class circle has plenty of room to host all microtonal pitches you can come up with. In case you’ve been wondering: the pitch class circle as such has nothing to do with the so-called “circle of fifths”. Other than the fact that both are represented on a circle, there’s no special relation between the two circles.

Please watch the first video animation to better understand how replacing concrete pitch with pitch class has the effect of curling up the pitch line into a pitch class circle.

Before going on to our next topic, let’s sum up a few quick facts about the concept of pitch class and related subjects:

• “Pitch” indicates a specific point on a pitch line.  C4, for example, is a different pitch than C5.  The higher the number, the higher the sound.
• “Pitch class” indicates an entire group of pitches, related by their “octave equivalence”.  The pitch class of C, for example, includes C4, C5, C9 and so on.
• “Pitch class” includes notes with enharmonic spellings.  The pitch class of “C”, for example, would include such pitches as C3, B#5, D double flat 6, and so forth.

Pitch classes are “octave equivalent”, which means that pitches in different octaves are still in the same class.   Pitch classes are also “spelling equivalent” which means that notes spelled differently but sounding the same, which we call enharmonic spellings, are in the same pitch class.

Here is an example of a piece which is built entirely on one pitch class, until the final few measures.