Sunday, February 22, 2015


I love exploring the relationships between mathematics and music. My exploration tends to be more hands-on research than library research. So I am surely rediscovering many old ideas, and I almost never learn the proper terminology. But I do have a lot of fun!

Lately I have been exploring a pitch class set that is sometimes called Hanson[11]. Others have certainly explored this, e.g. and which even include some music composed in this system. I must confess that the folks that work in this area have developed an enormous collection of terms and ideas, very little of which I understand. I am going to explain here some of the ideas that have led me to this approach.

Most any musical pitch system is based on a collection of intervals, i.e. frequency ratios. The most fundamental intervals are simple rational numbers. E.g. an octave is the ratio 2:1 and a perfect fifth is a ratio of 3:2. There is a simple algebra of intervals: if pitches P and Q are separated by an interval X, and then Q and R are separated by an interval Y, then the interval from P to R will be X*Y, the multiplicative product of X and Y.

One can certainly make very good music by starting with a single pitch and a small collection of primitive rational intervals. The various combinations of these intervals will generate an infinite collection of more complex intervals. This system is known as Just Intonation.

The challenge with just intonation is that different combinations of intervals can result in complex intervals that are very close together. The standard tuning for a guitar illustrates this quirk. The intervals between successive pairs of strings on a guitar are a fourth from E to A, a fourth from A to D, a fourth from D to G, a major third from G to B, and a fourth again from B to E. From the first E to the last E should be two octaves. In just intonation a fourth is the ratio 4:3 and a major third is the interval 5:4. So the complex interval which is the product of the simple intervals between the strings is 320:81. But two octaves would be 4:1 = 320:80. The small interval between these two intervals is 81:80, known as the syntonic comma.

There are many such small intervals that arise in just intonation. In fact, the infinite collection of complex intervals will fill the entire pitch space densely. In notating music, in building musical instruments, and in performance, managing such an infinitely dense collection of pitches is practically impossible. Still, the underlying music concept can adhere to just intonation, while the notation and the instrument can simply collect pitches into clusters and use a single pitch to stand for the cluster. Analysis might still reveal the underlying intention. Perhaps our ears even interpret the sound relationships as the appropriate just intervals, at least when the musical sense is clear.

But Easley Blackwood's book The Structure of Recognizable Diatonic Tunings shows that the mainstream European tradition from Bach to Mahler does not work this way. A particular note to be played at a particular time in a piece of music will stand related to several other notes, e.g. other notes played at the same time, or immediately before or after, or in corresponding positions in musical phrases played not too far away in time. One set of relationships might imply one just intonation interpretation of a note, while another set of relationships implies a different interpretation. And this sort of ambiguity is pervasive in the European tradition. A lot of what is happening with the music is a kind of playing with ambiguities, with musical puns.

From this perspective, what makes a tuning useful or interesting is the set of ambiguities that it provides! Because different tuning systems work with different ambiguities, music won't in general translate from one system to another. It's like trying to translate puns between languages. But of course each language can support a wonderful collection of entertaining and insightful puns. So different tuning systems should support music, though different music.

My fascination for quite a while has been the microtonal tuning system that divides an octave into 53 equal parts. More recently I have been looking at ways to pick a subset of the 53 pitch classes with which to make music. In the conventional tuning system that divides an octave into 12 equal parts, music typically works with just 7 of the 12 pitch classes at a time, though there might be accidentals or key changes too in a piece of music. One way to think about these 7 pitch classes is as a contiguous region of the circle of fifths. I.e. the seven notes C D E F G A B can be arranged as F C G D A E B. While the primary structure of this set is governed by this sequence of intervals of a perfect fifth, there are other relationships among these, e.g. C to E is a major third. This ambiguity, is C E four perfect fifths or a major third, is again the blurring of the small interval of a syntonic comma.

What I am working with now is a set of 11 pitch classes out of the total palette of 53. This set is generated by 10 intervals of a minor third. When an octave is divided into 53 microsteps, a minor third is 14 microsteps. We can start this set of 11 pitch classes from any of the 53 pitch classes, just as a major scale can start from any of the 12 pitch classes of conventional tuning. Here is a diagram of one set:

In this diagram, each pitch class has six neighbors. The intervals between the neighbors are perfect fifths (31 microsteps), perfect fourths (22 microsteps), major thirds (17 microsteps), major sixths (36 microsteps), minor thirds (14 microsteps), and minor sixths (39 microsteps). The first important observation to make about this subset of 11 pitch classes is that, while the subset is generated by a sequence of minor thirds, the subset also includes pairs of pitch classes separated by fifths. This property is due to one of the ambiguities that arise when dividing an octave into 53 equal parts. The small gap between a fourth and 6 minor thirds is known as a klesma. The blurring or tempering of the kleisma in this system corresponds to the tempering of the syntonic comma in the convention 12 pitch class system.

Another nice feature of this set of 11 pitch classes is that when the pitches are arranged in order, e.g. 0 8 11 19 22 25 33 36 39 47 50 53, there are just two sizes of intervals that occur, 3 microsteps and 8 microsteps. This is analogous to the half step and whole step of a conventional major scale.

Here is what this pitch class set sounds like: