Like impressionist art – such as Monet’s work Sunset – impressionist music does not have fixed structures. Both artforms use the art of abstraction to give a sense of the theme of the work.

On the other hand, classical music, such as sonatas, flows with a rhythmic beat with a clear beginning, middle, and end to the work.

Since there is little theoretical study on the compositional patterns of the contemporary style of music, Duke senior Rowena Gan finds the mathematical exploration of impressionist music quite exciting, as she expressed in her senior thesis presentation April 17.

Sunset: Impressionist art by Claude Monet

Classical music is well known for its characteristic chord progressions, which can be geometrically represented with a torus – or a product of circles – as shown in the figure below.

Torus depicting C major in orange highlight and D minor in blue highlight

By numbering each note, the Neo-Riemannian theory can be used to explain chord progressions in classical music by finding mathematical operations to describe the transitions between the chords.

Expressing chord progressions as mathematical operations

Basic transformations between chords described by the Neo-Riemannian theory.

Similar to a chord, a scale is also a collection of notes. In classical music, scales typically played have seven notes, such as the C major scale below:

C Major Scale.

Impressionist music, however, is marked by the use of exotic scales with different numbers of notes that tend to start at notes off the key center. In that case, how do we represent scales in Impressionist music? One of the ways of representation that Gan explored is by determining the distance between the scales – called interscalar distance – by depicting each scale as a point, and comparing this value to the modulation frequency.

Essentially, the modulation frequency is determined by varying the frequency of the audio wave in order to carry information; a wider range of frequencies corresponds to a higher modulation frequency. For example, the modulation frequency is the same for the pair of notes of D and E as well as F and G, which both have lower modulation frequencies than between notes D and G.

Gan calculated the correlation between modulation frequency and interscalar distance for various musical pieces and found the value to be higher for classical music than for impressionist music. This means that impressionist music is less homogenous and contains a greater variety of non-traditional scale forms.

Gan explores more detailed findings in her paper, which will be completed this year.

Rowena Gan is a senior at Duke in Mathematics. She conducted her research under Professor Ezra Miller, who can be contacted via email here.