Golomb Rulers and Ugly Music

A Golomb ruler has marks on it for measuring distances, but unlike ordinary rulers it has a smaller number of irregularly spaced marks that still allow for measuring a large number of distances. The marks are at integer multiples of some arbitrary unit. A regular ruler of six units length will have marks at 0, 1, 2, 3, 4, 5 and 6, and it will be possible to measure each distance from 1 to 6 units. A Golomb ruler of length six could have marks at 0, 1, 4 and 6.

Each distance from 1 to 6 can be found between pairs of marks on this ruler. A Golomb ruler that has this nice property that each distance from 1 to the length of the ruler can be measured with it is called a perfect Golomb ruler. Unfortunately, there is a theorem that states that there are no perfect Golomb rulers with more than four marks.

Sidon sets are subsets of the natural numbers {1, 2, ..., n} such that the sums of any pair of the numbers in the set are all different. It turns out that Sidon sets are equivalent to Golomb rulers. The proof must have been one of the lowest hanging fruits ever of mathematics.

An interesting property of Golomb rulers is that, in a sense, they are maximally irregular. Toussaint used them to test a theory of rhythmic complexity precisely because of their irregularity, which is something that sets them apart from more commonly encountered musical rhythms.

There is a two-dimensional counterpart to Golomb rulers which was used to compose a piano piece that, allegedly, contains no repetition and is therefore the ugliest kind of music its creator could think of.

Contrary to what Scott Rickard says in this video, there are musical patterns in this piece. Evidently they did not consider octave equivalence, so there is a striking passage of ascending octaves and hence pitch class repetition.

At first hearing, the "ugly" piece may sound like a typical 1950's serialist piece, but it has some characteristic features such as its sequence of single notes and its sempre forte articulation. Successful serialist pieces would be much more varied in texture.

The (claimed) absence of patterns in the piece is more extreme than would be a random sequence of notes. If notes had been drawn randomly from a uniform distribution, there is some probability of immediate repetition of notes as well as of repeated sequences of intervals. When someone tries to improvise a sequence of random numbers, say, just the numbers 0, 1, they would typically exaggerate the occurrences of changes and generate too little repetition. True randomness is more orderly than our human conception of it. In that sense the "ugly" piece agrees with our idea of randomness more than would an actually random sequence of notes.

When using Golomb rulers for rhythm generation, it may be practical to repeat the pattern instead of extending a Golomb ruler to the length of the entire piece. In the case of repetition the pattern occurs cyclically, so the definition of the ruler should change accordingly. Now we have a circular Golomb ruler (perhaps better known as a cyclic difference set) where the marks are put on a circle, and distances are measured along the circumference of the circle.

Although the concept of a Golomb ruler is easy for anyone to grasp, some generalization and a little further digging leads into the frontiers of mathematic knowledge with unanswered questions still to solve. 

And, of course, the Golomb rulers make excellent raw material for quirky music.

Who cares if they listen?

No, it wasn't Milton Babbitt who coined the title of that notorious essay, but it stuck. And, by the way, it was "Who cares if you listen?". Serialism, as Babbitt alludes to, employs a tonal vocabulary "more efficient" than that of past tonal music. Each note-event is precisely determined by its pitch class, register, dynamic, duration and timbre. In that sense, the music has a higher information density and the slightest deviation from the prescribed values is structurally different, not just an expressive coloration. Such music inevitably poses high demands on the performer, as well as on the listener. (Critical remarks could be inserted here, but I'm not going to.) Similarly to how recent advances in mathematics or physics can be understood only by a handful of specialists, there is an advanced music that we should not expect to be immediately accessible to everyone. For such an elitist endeavor to have a chance of survival at all, Babbitt suggests that research in musical composition should take its refuge in universities. Indeed, in 20th century America that was where serialist composers were to be found.

The recent emergence of artistic research at universities and academies is really not much different from what Babbitt was pleading for. Artistic research is awkwardly situated between theorizing and plain artistic practice as it used to be before everyone had to write long manifestos. But more about that on another occasion, perhaps.

In a more dadaist spirit, there is this new release out now on bandcamp, also titled Who cares if they listen.

In these days there are enough reasons to care if they listen (yes they do, but not wittingly) and to worry about its consequences, as discussed in a previous post.