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.

Decimals of π in 10TET

The digits of π have been translated into music a number of times. Sometimes the digits are translated to the pitches of a diatonic scale, perhaps accompanied by chords. The random appearance of the sequence of digits is reflected in the aimless meandering of such melodies. But wouldn't it be more appropriate, in some sense, to represent π in base 12 and map the numbers to the chromatic scale? After all, there is nothing in π that indicates that it should be sung or played in a major or minor tonality. Of course the mapping of integers to the twelve chromatic pitches is just about as arbitrary as any other mapping, it is a decision one has to take. However, it is easier to use the usual base 10 representation and to map it to a 10-TET tuning with 10 chromatic pitch steps in one octave.

Here is an etude that does precisely that, with two voices in tempo relation 1 : π. The sounds are synthesized with algorithms that also incorporate the number π. In the fast voice, the sounds are made with FM synthesis where two modulators with ratio 1 : π modulate a carrier. The slow voice is a waveshaped mixture of three partials in ratio 1 : π : π².

Despite the random appearance of the digits of π, it is not even known whether π is a normal number or not. Let us recall the definition: a normal number in base b has an equal proportion of all the digits 0, 1, ..., b-1 occuring in it, and equal probability of any possible sequence of two digits, three digits and so on. ("Digit" is usually reserved for the base 10 number system, so you may prefer to call them "letters" or "symbols".) A number that is normal in any base is simply called normal.

Some specific normal numbers have been constructed, but even though it is known that almost all numbers are normal, the proof that a number is normal is often elusive. Rational numbers are not normal in any base since they all end in a periodic sequence, such as 22/7 = 3.142857. However, there are irrational, non-normal numbers, some of which are quite exotic in the way they are constructed.