Permutations as Proposed Solution
Our interest in sonifying basketball scores minute-by-minute was more than showing which team is ahead by how much. We were looking for a way to show which combinations of five players are more effective (in terms of increasing their team's score relative to the other team's).
We tried to represent each of the twelve players by a different musical instrument. We also tried to represent each player by a different note in a chord, and each minute by a measure, in which the number of notes played indicated roughly the difference in score between the two teams. Neither of these generated music which was pleasant to listen to or easy to understand.
To avoid producing dissonant chords we decided to represent the players on the court in a given minute not by a chord but by a permutation of the notes (from a major scale) representing those players. In this way we do not have to be concerned with generating chords or sequences of chords which are pleasing to the ear but rather with generating short melodies which change over time as function values change.
The permutation varies according to any of several patterns. For example, if we present the permutations in lexicographic order (defined by pitch) we would play the series of notes (representing three players):
Any sequence of permutations of these notes can indicate that the team of interest is ahead in points. To indicate that this team is tied or behind, we sharpen or flatten one note to make the sequence of notes be from a minor scale.
This work is inspired by work in comma-free codes (Ridley, 1993; Buchanan, 1992). In constructing a comma-free code, an n element alphabet and a set of code words each of length k over that alphabet are chosen in such a way that if any sufficiently long fragment of a series of code words is selected the code words can be recognized uniquely. For example when n=4 and k=3 there are potentially 43 = 64 possible code words but it can be shown that the largest possible set of code words contains only 20 in order to maintain this "comma-free" property.
However, in our work we are not looking for comma-free sequences which can be recognized logically but rather for melodies which are recognized more subtly and intuitively by a listener.
There are two main kinds of permutations we are currently studying: permutations in lexicographic order and in bell change ringing order. We are also beginning to look at a subset of permutations, those forming a comma-free code.