The subject of this movement is the ‘Butterfly Effect’. The left and right hands begin by playing the same melody, but gradually these lines diverge until they are playing completely different notes. In other words, this piece is demonstrating ‘sensitive dependence on initial conditions’. Occasionally the melody lines seem to re-converge, and this is emphasised dynamically, but all anew the lines become once more vastly different. Such is the nature of Chaos.
View the sheet musicThis movement has a 7/4 time signature. The left hand repeats a simple motif throughout the piece. The composition was based on the fact that one dimensional maps such as the Logistic Equation can not only display chaotic behaviour, but also periodic behaviour of any period (in this case period seven). The right hand melody exhibits chaotic behaviour as in the first movement.
View the sheet musicThe final movement in the series explores the behaviour of the Logistic Equation as its parameter is varied across the complete range of non-divergent behaviour. Its title refers not only to the fact that notationally it is the most abstract of the movements, but also to the fact that the parameter is allowed to be less than zero, leading to negative ‘fish populations’. An important feature of the piece which should be realised in its performance is the appearance of period three ‘windows’ amongst the chaotic regions.
View the sheet music
