Dynamical and topological tools for (modern) music analysis

Is it possible to represent the horizontal motions of the melodic strands of a contrapuntal composition, or the main ideas of a jazz standard as mathematical entities? In this work, we suggest a collection of novel models for the representation of music that are endowed with two main features. First, they originate from a topological and geometrical inspiration; second, their low dimensionality allows to build simple and informative visualisations. Here, we tackle the problem of music representation following three non-orthogonal directions. We suggest a formalisation of the concept of voice leading (the assignment of an instrument to each voice in a sequence of chords) suggesting a horizontal viewpoint on music, constituted by the simultaneous motions of superposed melodies. This formalisation naturally leads to the interpretation of counterpoint as a multivariate time series of partial permutation matrices, whose observations are characterised by a degree of complexity. After providing both a static and a dynamic representation of counterpoint, voice leadings are reinterpreted as a special class of partial singular braids (paths in the Euclidean space), and their main features are visualised as geometric configurations of collections of 3-dimensional strands. Thereafter, we neglect this time-related information, in order to reduce the problem to the study of vertical musical entities. The model we propose is derived from a topological interpretation of the Tonnetz (a graph commonly used in computational musicology) and the deformation of its vertices induced by a harmonic and a consonance-oriented function, respectively. The 3-dimensional shapes derived from these deformations are classified using the formalism of persistent homology. This powerful topological technique allows to compute a fingerprint of a shape, that reflects its persistent geometrical and topological properties. Furthermore, it is possible to compute a distance between these fingerprints and hence study their hierarchical organisation. This particular feature allows us to tackle the problem of automatic classification of music in an innovative way. Thus, this novel representation of music is evaluated on a collection of heterogenous musical datasets. Finally, a combination of the two aforementioned approaches is proposed. A model at the crossroad between the signal and symbolic analysis of music uses multiple sequences alignment to provide an encompassing, novel viewpoint on the musical inspiration transfer among compositions belonging to different artists, genres and time. To conclude, we shall represent music as a time series of topological fingerprints, whose metric nature allows to compare pairs of time-varying shapes in both topological and in musical terms. In particular the dissimilarity scores computed by aligning such sequences shall be applied both to the analysis and classification of music

Musicians and Machines: Bridging the Semantic Gap In Live Performance

PhDThis thesis explores the automatic extraction of musical information from live performances – with the intention of using that information to create novel, responsive and adaptive performance tools for musicians. We focus specifically on two forms of musical analysis – harmonic analysis and beat tracking. We present two harmonic analysis algorithms – specifically we present a novel chroma vector analysis technique which we later use as the input for a chord recognition algorithm. We also present a real-time beat tracker, based upon an extension of state of the art non-causal models, that is computationally efficient and capable of strong performance compared to other models. Furthermore, through a modular study of several beat tracking algorithms we attempt to establish methods to improve beat tracking and apply these lessons to our model. Building upon this work, we show that these analyses can be combined to create a beat-synchronous musical representation, with harmonic information segmented at the level of the beat. We present a number of ways of calculating these representations and discuss their relative merits. We proceed by introducing a technique, which we call Performance Following, for recognising repeated patterns in live musical performances. Through examining the real-time beat-synchronous musical representation, this technique makes predictions of future harmonic content in musical performances with no prior knowledge in the form of a score. Finally, we present a number of potential applications for live performances that incorporate the real-time musical analysis techniques outlined previously. The applications presented include audio effects informed by beat tracking, a technique for synchronising video to a live performance, the use of harmonic information to control visual displays and an automatic accompaniment system based upon our performance following technique.EPSR