# How A Little Mathematics Can Help Create Some Beautiful Music

A standard 2:3:6 polyrhythm. For clarity, the hexagonal level is not shown.

Well-Formed Polygons

Well-formedness elegantly generalises three properties commonly found in real-world multilevel rhythms:

• each rhythmic level comprises only a small number of distinct beat lengths, often only one or two;

• each level’s beats are fairly evenly spaced in time – there aren’t sudden clusters of events followed by long gaps; and

• the rhythmic levels are hierarchical – there is a slow and metrically dominant level; above this is a faster and weaker level that splits the previous level’s beats; above this is an even faster and weaker level that splits the previous level’s beats; and so on.

Every level of a well-formed rhythm has two beat lengths: a long beat and a short beat. The length of a given beat is the time between its onset and the onset of the following beat.

A multilevel well-formed rhythm can then be fully defined by three numerical parameters: the numbers of long and short beats in the lowest level rhythm, and the ratio of the sizes of its long and short beats.

From these three numbers, an entire rhythmic hierarchy can be calculated, such that each level has no more than two beat lengths, each level arranges these beat lengths to make them as evenly spaced as possible and each successive level in the hierarchy is created by splitting the long beats of the level below.

Using XronoMorph, the above three parameters can be freely manipulated. The rhythmic hierarchy emerging from them often has great aesthetic appeal. Every level is related to every other level and is also intrinsically well-formed. Together, they create a somewhat self-similar and interwoven structure reminiscent of fractals.

A demonstration of how a well-formed hierarchy is built, and the effect of the lowest level’s beat-size ratio.

Perfectly Balanced Polygons

Perfect balance is a mathematical principle that can generalise polyrhythms, a type of rhythm commonly used in sub-Saharan African music.

Unlike well-formed rhythms and most Western rhythms, polyrhythms are not hierarchical. They are more like an alliance of different rhythmic levels, each of equal status.

In a polyrhythm, two or more levels, each comprising evenly spaced beats, are superimposed.