About the SA Blog Network

Roots of Unity

Roots of Unity

Mathematics: learning it, doing it, celebrating it.
Roots of Unity Home

A Twelve-Tone Trip Down Memory Lane, Courtesy of Vi Hart

Email   PrintPrint

Last week Vi Hart posted a wonderful 30-minute video on twelve-tone music, and it really took me back!

(If the video doesn’t load for you, watch it on YouTube.)

Ten years ago, I hadn’t yet decided I wanted to do math. In fact, I was enjoying my music theory classes immensely, and I thought I might study it in graduate school. In the spring semester of 2003, I was taking a twentieth-century music theory class. As part of an assignment for that class, I wrote “ftc,” a twelve-tone serial setting of part of the fundamental theorem of calculus. You can look at the pdf of the composition here. It’s a scan from 2004, and it’s a bit chopped off in places, but I don’t have an original anymore. It has a lot of colored pencil marks on it that show what rows I was using at what times.

At the time I wrote “ftc,” I didn’t really understand the importance of the fundamental theorem of calculus, but I knew that it was important. If I were writing the piece today, I would think more carefully about the lyrics of the setting rather than just copying it out of my calculus textbook.

I loved writing twelve-tone serial compositions because it was fun to play with different properties of the tone rows, and it had just enough limitations to encourage my creativity.

Warning: the rest of this post assumes some familiarity with music notation and intervals.

To start writing a twelve-tone serial piece, you just arrange all twelve pitches from the chromatic scale in some order. The row I chose for “ftc” was F# A E D C# F B G G# A# D# C.

My tone row. Except for Vi Hart's video embedded above, all images on this page are by Evelyn Lamb.

You have to play the notes in this order (either one at a time or in chords) to “count” as playing the tone row. But in a composition you can also make some alterations to the original row. You can play it backwards, which is called “retrograde,” or do the same intervals but starting on a different pitch. You can also play the same intervals but going in the opposite direction, which is called “inversion.” In my case, the inverted row would start with a descending minor third, or ascending major sixth, rather than an ascending minor third. You can also combine transformations, so retrograde inversions are fair game too.

My row is really neat, if I do say so myself, because it contains all possible intervals exactly once. We have to be a little careful to define what we mean by “all possible intervals.” As you can see, it contains a minor third twice: F# to A and D# to C. (Yes, D#-C looks like an augmented second, but we’ll call it a minor third because we’re rebellious modern music people!) In twelve-tone music, we really don’t care what octave we’re playing in, we’re only concerned with the note name, also called pitch class. Middle C and the C an octave above it are the “same.” So we can just assume that all the intervals are ascending (or all are descending), and that will be our definition of “all possible intervals.” (There is also a way to phrase this using the language of arithmetic modulo 12, but I think that is a post for another day.) In my row, we have an ascending minor third at the beginning and an ascending major sixth at the end.

My tone row with ascending and descending intervals labeled. That's supposed to be a little devil head under the TT because the tritone/diminished fifth/augmented fourth used to be called "diabolus in musica." Now you know why I'm a mathematician and not a police sketch artist.

I decided to draw some pictures of my tone row using the clock face method Hart used starting near the 24:45 mark of her video.

Pitches from are arranged like numbers on a clock face. The numbers in the red squares indicate the order of the pitches in my tone row. The smudge in the bottom right represents my nascent awareness that my pen was leaking all over my fingers.

I am a string player, so I generally prefer sharps to flats. Hart used flats in her video. It doesn’t matter because we’re using equal temperament. I numbered the pitches in order from 0 to 11 instead of 1 to 12 because that’s how I did it when I wrote “ftc.” We must have used that convention in our class. There are good reasons to use either convention.

I decided to connect the dots. I started with the first six pitches of the row.

Pitches 0-5 connected.

Then I erased those and connected the last six.

Hey, that looks familiar!

It’s not an accident that the second picture looks like a 180 degree rotation of the first one. The first interval of my row is an ascending minor third, and the last interval is a descending minor third. The second interval is an ascending perfect fifth, and the second to last interval is a descending perfect fifth, and so on. If you play the row backwards, you get the row that you started with, transposed down a tritone. Isn’t that cool?

Put together, it looks like this.

Isn't that pretty?

I drew the tritone in the middle in orange because it doesn’t belong in either set of six pitches. It kind of connects these two symmetric pieces that make up the whole tone row.

If I wanted to connect the last note to the first, I’d draw another orange line, because the first and last notes are also a tritone away.

It looks so nice and symmetric like this.

I decided to start from scratch and visualize my row a different way, too. This time I wanted to highlight the different types of intervals I used. First I drew lines between the first and last intervals, which are both minor thirds.

The arrows show the order of the pitches in time. I colored in the rest of the intervals the same way.

Each different color represents a different interval size. My color choices were limited because I just moved and had trouble finding more markers.

Now I have a bunch of parallel lines with arrows on them! If we held the clock face still and rotated the lines by 180 degrees, the blue line between D and C# would end up being a blue line from G# to G with the arrow pointing the other direction. The same thing is true for all the other colored lines. Once again, this is not an accident. It highlights the fact that the retrograde version of the row has the same intervals as the original row, starting a tritone (or half a rotation) away.

I wish I could share a sound file of the piece I wrote using this tone row, but my cheapo midi file of it is lost in the sands of time. Perhaps when I’m not unpacking my apartment and getting ready for a big travel month, I’ll try to make another recording. I promise the row sounds nice, although the whole composition is a little wild at times. The first four notes of the row create a D major triad with an added note, which has a nice tonal sound. Because of the symmetry of the row, the last four notes do the same thing but with a G# major triad.

Thanks, Vi Hart, for reminding me of how fun serialism is!

Evelyn Lamb About the Author: Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.

The views expressed are those of the author and are not necessarily those of Scientific American.

Rights & Permissions

Add Comment

Add a Comment
You must sign in or register as a member to submit a comment.

More from Scientific American

Scientific American MIND iPad

Give a Gift & Get a Gift - Free!

Give a 1 year subscription as low as $14.99

Subscribe Now >>


Email this Article