Musical scale rationalization – a graph-theoretic approach

While most Western music today uses the well-established equal-tempered 12 tone scale, there are many reasons why composers and music theorists are still interested in “just” scales in which the scale tones are simple integer ratios. Therefore the rationalization of musical scales is an important problem in music theory. The first general scale rationalization algorithm based on a complete enumeration approach was proposed by the influential contemporary composer Clarence Barlow. This paper gives a general formulation of the problem and proves that the problem is NP-complete. The paper also describes new algorithms for scale rationalization based on clique search in graphs, and a method to draw rational scales using multidimensional scaling. A scale rationalization and visualization software based on these algorithms is discussed as well.