As I have shown in previous posts, there is a very close relationship between music and mathematics. Whether we are talking about the ratios between polyrhythms, how to count in irrational time signatures, or the similarities between rhythm and pitch, math always has its place in music theory. In this week’s topic, the two subjects are more related than ever. Today we will be calculating tone rows.
By definition, a tone row is a series of non-repeating notes. The most common type is called a 12 tone row, which is created using all 12 notes from a single octave. There are four types of tone rows: prime, retrograde, inversion, and retrograde inversion.
Notationally, tone rows are classified by a letter and a number. The letter refers to the type (P=prime, R=retrograde, I=inversion, and RI=retrograde inversion). The number refers to the transposition, which is simply a shift. 0 means no shift, 1 means a shift of one half step, and so on. For example, P0 refers to a prime order tone row with no transposition. Let’s start with the simplest type, which is prime.
A prime order tone row (P0) is the foundation for all of the other types. It is simply a series of twelve different pitches, in any particular order. Here is an example:
P0: C F# D B A# F C# D# G A E G#
Note that there are no repeating notes.
To find P1, we just shift all of the notes up one half step:
P1: C# G D# C B F# D E G# A# F A
Another type is retrograde. This is just a fancy word for “backwards”. R0 is just P0 backwards:
R0: G# E A G D# C# F A# B D F# C
Similarly, R1 is P1 backwards.
The third type is inversion. To invert a tone row, there are two things to consider: distance and direction. We start with P0. The distance from C to F# is 6 half steps, and the direction is up. To invert this, we once again use a distance of 6 half steps, but this time we go down instead of up. This processes is repeated for all twelve notes, and the resulting I0 looks like this:
I0: C F# A# C# D G B A F D# G# E
You can calculate I1 by simply inverting P1.
Finally, RI, or retrograde inversion, is just the inverted tone row ordered backwards.
RI0: E G# D# F A B G D C# A# F# C
Now that we can calculate them, what do we even use tone rows for? Tone rows are a key device in jazz improvisation. They allow for soloists to create interesting phrases due to the seemingly random nature that they provide.
One of my favorite examples is Cory Henry’s keyboard solo in the song Lingus by Snarky Puppy:
There are multiple instances within his solo that he uses tone rows to structure the phrases and development of the melody.
Overall, all four types of tone rows can be created using fairly simple calculations, and the applications of these devices are very useful for creating crazy jazz solos.
But that’s just a theory. A music theory.
-gwilliams 5
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