Mathematics and Computation in Music

The present paper is concerned with the existence, meaning and use of the phases of the (complex) Fourier coefficients of pc-sets, viewed as maps from ℤ _c to . It explores a particular cross-section of the most general torus of phases, representing pc-sets by the phases of the third and fifth coefficients. On this 2D torus, triads take on the well-known configuration of the Tonnetz. Some other (sequences of) chords are viewed in this space as examples of its musical relevance. The end of the paper uses the model as a convenient universe for drawing gestures – continuous paths between pc-sets.

This article presents a first attempt at establishing a category-theoretical model of creative processes. The model, which is applied to musical creativity, discourse theory, and cognition, suggests the relevance of the notion of “colimit” as a unifying construction in the three domains as well as the central role played by the Yoneda Lemma in the categorical formalization of creative processes.

We represent chord collections by simplicial complexes. A temporal organization of the chords corresponds to a path in the complex. A set of n-note chords equivalent up to transposition and inversion is represented by a complex related by its 1-skeleton to a generalized Tonnetz. Complexes are computed with MGS, a spatial computing language, and analyzed and visualized in Hexachord, a computer-aided music analysis environment. We introduce the notion of compliance, a measure of the ability of a chord-based simplicial complex to represent a musical object compactly. Some examples illustrate the use of this notion to characterize musical pieces and styles.

While analysing large corpora of music, many of the questions that arise involve the proportion of some musical entity relative to one or more similar entities, for example, the relative proportions of tonic, dominant, and subdominant chords. Traditional statistical techniques, however, are fraught with problems when answering such questions. Compositional data analysis is a more suitable approach, based on sounder mathematical (and musicological) ground. This paper introduces some basic techniques of compositional data analysis and uses them to identify and illustrate changes in harmonic usage in American popular music as it evolved from the 1950s through the 1990s, based on the McGill Billboard data set of chord transcriptions.

Sturmian words are balanced, almost periodic, self-similar and hierarchical infinite sequences that have been studied in music theory in connection with diatonic scale theory and related subjects. Carey and Camplitt (1996) give a brief but suggestive rhythmic example in which these properties are made manifest in a particularly visceral manner. The present paper expands upon this example, considering the properties of canons based on Sturmian words, or Sturmian canons. In particular, a Sturmian word of irrational slope a with a hierarchical periodicity of p gives rise to p-tuple canons, the voices and relations of which are determined by the terms of the continued fraction expansion of a.

Tempo is an important parameter that is varied and analysed in music performance. We argue that it is important to consider both tempo and log(tempo) in score time as well as performance time in the analysis of performances; performance time mirrors listeners’ real time experience, and log(tempo) gauges proportional tempo changes. As demonstration, we revisit Chew’s (2012) score time tempo analysis of performances of Beethoven’s “Moonlight” Sonata, and generate new results using log(tempo) and performance time. We show that extreme differences in score time tempo are ameliorated by considering log(tempo) and performance time, that the performers employed similar log(tempo) ranges and phrase lengths (in performance time), and that long score time phrases do not necessarily map to lengthy performance time spans due to speedier phrase traversal times. The results suggest that log(tempo) range and maximum performance time phrase length may act as perceptual constraints on the shaping of a performance.

Representing musical notes as points in pitch-time space causes repeated motives and themes to appear as translationally related patterns that often correspond to maximal translatable patterns (MTPs) [1]. However, an MTP is also often the union of a salient pattern with one or two temporally isolated notes. This has been called the problem of isolated membership [2]. Examining the MTPs in musical works suggests that salient patterns may correspond more often to the intersections of MTPs than to the MTPs themselves. This paper makes a theoretical contribution, by exploring properties of patterns that are maximal with respect to their translational equivalence classes (MTEC). We prove that a pattern is MTEC if and only if it can be expressed as the intersection of MTPs. We also prove a relationship between MTECs and so-called conjugate patterns.

A familiar problem in neo-Riemannian theory is that the P, L, and R operations defined as contextual inversions on pitch-class segments do not produce parsimonious voice leading. We incorporate permutations into T/I–PLR duality to resolve this issue and simultaneously broaden the applicability of this duality. More precisely, we construct the dual group to the permutation group acting on n-tuples with distinct entries, and prove that the dual group to permutations adjoined with a group G of invertible affine maps ℤ₁₂ → ℤ₁₂ is the internal direct product of the dual to permutations and the dual to G. Musical examples include Liszt, R. W. Venezia, S. 201 and Schoenberg, String Quartet Number 1, Opus 7. We also prove that the Fiore–Noll construction of the dual group in the finite case works, and clarify the relationship of permutations with the RICH transformation.

This paper introduces the key-assertion method, a parsimonious analytic method for labelling key areas using pitch-class content. Sensitivity to key change is maximized, providing a detailed account of tonal areas, large and small. The method also produces a surprising heuristic for guessing the overall key of a piece, which performs well in comparison with other methods.

The paper presents some new results on Z-related sets obtained by computational methods. We give a complete enumeration of all Z-related sets in ℤ _N for small N. Furthermore, we establish that there is a reasonable permutation group action representing the Z-relation.

Mathematical performance theory [1] uses a model of performative unfolding that is based on “sexual propagation” of successive performance refinements. It is formally described by a tree-shaped diagram, the performance stemma, starting at the primary “mother” performance that ramifies to a series of “daughter” performances. This propagation mechanism is induced by a series of performance operators stemming from the composition’s music analysis. In this paper we refine such networks to performance hypergestures whose curves represent continuous transitions from mother to daughter performances. This level of description uses the theory of Lie-type performance operators and enables a detailed analysis of different performative transition strategies. We then calculate the singular performance hypergesture homology H ₁ and discuss its significance for the classification of transitional strategies.

Diatonic Modes can be modeled through automorphisms of the free group F ₂ stemming from special Sturmian morphisms. Following [1] and [2] we associate special Sturmian morphisms f with linear maps E(f) on a vector space of lattice paths. According to [2] the adjoint linear map E(f)^∗ is closely related to the linear map E(f ^∗), where f and f ^∗ are mutually related under Sturmian involution. The comparison of these maps is music-theoretically interesting, when an entire family of conjugates is considered. If one applies the linear maps E(f ₁), ..., E(f ₆) (for the six authentic modes) to a fixed path of length 2, one obtains six lattice paths, describing a family of authentic common finalis modes (tropes). The images of a certain path of length 2 under the application of the adjoint maps E(f ₁)^∗, ..., E(f ₆)^∗ properly matches the desired folding patterns as a family, which, on the meta-level, forms the folding of Guido’s hexachord. And dually, if one applies the linear maps \(E(f_1^\ast), ..., E(f_6^\ast)\) (for the foldings of the six authentic modes) to a fixed path of length 2, one obtains six lattice paths, describing a family of authentic common origin modes (“white note” modes). The images of a certain path of length 2 under the application of the adjoint maps \(E(f_1^\ast)^\ast, ..., E(f_6^\ast)^\ast\) properly match the desired step interval patterns as a family, which, on the meta-level, forms the step interval pattern of Guido’s hexachord. This result conforms to Zarlino’s re-ordering of Glarean’s dodecachordon.

Recent computer-aided studies of harmony in various corpora of music (e.g., Bach and Lutheran chorales, late-twentieth-century rock music, etc.) have demonstrated how the treatment of various harmonies differs among repertoires. These differences are most often represented through transitional probability matrices showing the likelihood of any recognized sonority following any other sonority within a defined state space of possible sonorities. While such models of tonality are useful for demonstrating differences among genres, they tend to downplay the impact of temporal ordering and metric position on harmonic treatment. A potential source of this deficit is the difficulty in making meaningful temporal comparisons without a precise definition of phrase beginnings and endings and without a large collection of phrases of the same length. This paper mitigates these challenges by identifying 799 phrases from the Bach chorale corpus that are identical in length and cadence. It then creates a small state space of chord roots and functional categories and, further, demonstrates how the treatment of harmonies is conditioned by their location within phrases. In so doing, it is hoped that the paper will contribute to more refined models of tonalities that recognize music’s essential temporality.

We investigate the representation of n-tone rows as paths on an n-dimensional hypercube graph with vertices labeled in the power set of the aggregate. These paths run from the vertex labeled by the null set to the one labeled by the full set, passing through vertices whose labels gradually accumulate members of the aggregate. Row relations are then given as hypercube symmetries. Such a model is more sensitive to the musical process of chromatic completion than those that deal more exclusively with n-tone rows and their relations as permutations of an underlying set. Our results lead to a graph-theoretical representation of the duality inherent in the pitch-class/order-number isomorphism of serial theory.

The article discusses the application of Formal Concept Analysis to the algebraic enumeration, classification and representation of musical structures. It focuses on the music-theoretical notion of the Tone System and its equivalent classes obtained either via an action of a given finite group on the collection of subsets of it or via an identification of Forte’s corresponding interval vector and Lewin’s interval function. The use of concept lattices, applied to a simple case such as the division of the octave into five equal parts and the associated Chroma System, clearly shows that these approaches are conceptually different. The same result is obtained for a given subsystem of the traditional Tone System, as we will show by analysing the case of the pentatonic system. This opens a window towards generic tone systems that can be used as starting point for the structural analysis of other finite chroma systems.

Computational music theorists have long been concerned with ways to parse musical surfaces into workable chords that conform to music-theoretical intuitions. This study proposes an algorithm that groups surface structures into relational networks that balance a chord’s contextual position and its scale-degree content. Applying the algorithm to a corpus of thousands of MIDI files that stretch throughout the common practice successfully derives an intuitive chord alphabet. The study raises issues concerning traditional harmonic-function theory, suggests a potential model of listeners’ learning of tonality’s basic cognitive elements, and proposes to a method of reducing surface complexity in corpus studies.

The paper deals with evaluation of various n-gram-based composer classification algorithms. Our analysis has a broad scope: We have analyzed three labelled corpora, five similarity measures, several feature extraction methods, the influence of forced balanced training and an extensive range of n-gram lengths. We found that most of the approaches we analyzed, when properly parametrized, can give very good results, on par with other state-of-the art data mining techniques and greatly outperforming humans in composer recognition.

Euler’s speculum musicum is a finite selection of tones from the two dimensional tone lattice known as the Tonnetz. The idea of representing larger or smaller collections of tones as finite subsets of the Tonnetz reappears in the scholarly discourse in various contexts. However, formal rules for such selections that would satisfactorily reflect musical reality are not known: those proposed in the past are either too restrictive (not allowing all musically relevant tone systems to enter the model) or too loose (not preventing musically irrelevant tone systems from entering the model). The paper offers a formal framework that yields selections satisfactorily reflecting the musical reality. The framework draws methods from the Minkowski geometry of numbers. It is shown that only selection bodies of very specific shapes called (skewed) selection polygons lead to relevant selections. Manifold music-theoretical examples include chromatic, superchromatic, and subchromatic tone systems.