Aline Honingh

Convexity and the well-formedness of musical objects

Abstract It is well known that subsets of the two-dimensional space ℤ2 can represent prominent musical and music-theoretical objects such as scales, chords and chord vocabularies. It has been noted that the major and minor diatonic scale form convex subsets in this space. This triggers the question whether convexity is a more widespread concept in music. This article systematically investigates the convexity for a number of musical phenomena including scales, chords and (harmonic) reduction. It is hypothesised that the notion of convexity may be a covering concept of musical phenomena and possibly reflects other mathematical properties of these musical structures. Furthermore, convexity can be used in a pitch-spelling model.

Mathematical and computational approaches to music: challenges in an interdisciplinary enterprise

This Special Issue is dedicated to explicate and discuss methodological issues in the interdisciplinary research field of mathematical and computational approaches to music. It arose from a lively panel discussion at the third International Conference on Mathematics and Computation in Music 2011 in Paris. We have organized this panel in order to initiate the much needed interdisciplinary dialogue on the How, Why, and What of our modelling of and theorizing about music in the wide field of science, humanities and cognitive approaches to music research. From the contributions of the three panelists to this Special Issue, we extract key topics that the interdisciplinary scientific community needs to address in order to enable the different disciplines to productively complement one another in achieving a comprehensive approach to music as a complex, yet fundamental human trait.

Clustering and classification of music by interval categories

We present a novel approach to clustering and classification of music, based on the concept of interval categories. Six interval categories exist, each with its own musical character. A piece of music can be represented by six numbers, reflecting the percentages of occurrences of each interval category. A piece of music can, in this way, be visualized as a point in a six dimensional space. The three most significant dimensions are chosen from these six. Using this approach, a successful visual clustering of music is possible for 1) composers through various musical time periods, and 2) the three periods of Beethoven, which illustrates the use of our approach on both a general and a specific level. Furthermore, we will see that automatic classification between tonal and atonal music can be achieved.

In search of universal properties of musical scales

Abstract Musical scales have both general and culture-specific properties. While most common scales use octave equivalence and discrete pitch relationships, there seem to be no other universal properties. This paper presents an additional property across the worlds musical scales that may qualify for universality. When the intervals of 998 (just intonation) scales from the Scala Archive are represented on an Euler lattice, 96.7% of them form star-convex structures. For the subset of traditional scales this percentage is even 100%. We present an attempted explanation for the star-convexity feature, suggesting that the mathematical search for universal musical properties has not yet reached its limits.

A model for rhythm and timbre similarity in electronic dance music

Music similarity is a multidimensional concept to which so-called “sub-similarities”, such as timbre and rhythm similarity, contribute. In this study, two models are presented: one for timbre similarity, and one for rhythm similarity. The musical domain for which the models were established is Electronic Dance Music (EDM). The models extract feature values from segments of audio and calculate a distance between two segments based on their feature vectors. The models are evaluated on perceptual data using linear regression. The accuracy of the rhythm similarity model reaches an empirically established upper bound to model performance. The accuracy of the timbre model is moderate, possibly due to insufficient data. From the selection of features and their weights resulting from the regression analysis, periodicity of rhythmic elements turned out to be the most important feature group for rhythm similarity in EDM.

Perception of timbre and rhythm similarity in electronic dance music

Music similarity is known to be a multi-dimensional concept, depending among others on rhythm similarity and timbre similarity. The present study aims to investigate whether such sub-dimensions of similarity can be assessed independently and how they relate to general similarity. To this end, we performed a series of web-based perceptual experiments on timbre, rhythm and general similarity in electronic dance music. Participants were asked to rate similarities of music pairs on a 4-point Likert scale. The results indicated that the ratings in the three types of similarity did not completely overlap and that participants showed slight to fair agreement in their ratings in all conditions. Together, the results suggest that it is possible to assess sub-dimensions of similarities independently to some extent. Interestingly, general music similarity was not completely explained by the summation of timbre and rhythm similarity. Based on this, a novel hypothesis of how general music similarity follows from its contributing sub-similarities is proposed.

Compactness in the Euler-lattice: A parsimonious pitch spelling model

Compactness and convexity have been shown to represent important principles in music, reflecting a notion of consonance in scales and chords, and have been successfully applied to well-known problems from music research. In this paper, the notion of compactness is applied to the problem of pitch spelling. Pitch spelling addresses the question of how to derive traditional score notation from 12-tone pitch classes or MIDI. This paper proposes a pitch spelling algorithm that is based on only one principle: compactness in the Euler-lattice. Generally, the goodness of a pitch spelling model is measured in terms of its spelling accuracy. In this paper, we concentrate on the parsimony, cognitive plausibility and generalizability of the model as well. The spelling accuracy of the algorithm was evaluated on the first book of Bachs Well-tempered Clavier and had a success rate of 99.21%. A qualitative discussion of the models cognitive plausibility, its parsimony and its generalizability is given.