A Multi-Level Tonal Interval Space for Modelling Pitch Relatedness and Musical Consonance
Journal of New Music Research, 2016 Vol. 45, No. 4, 281–294, http://dx.doi.org/10.1080/09298215.2016.1182192 A multi-level tonal interval space for modelling pitch relatedness and musical consonance Gilberto Bernardes1 ,DiogoCocharro1,MarceloCaetano1 ,CarlosGuedes1,2 and Matthew E.P. Davies1 1INESC TEC, Portugal; 2New York University Abu Dhabi, United Arab Emirates (Received 12 March 2015; accepted 18 April 2016) Abstract The intelligibility and high explanatory power of tonal pitch spaces usually hide complex theories, which need to account In this paper we present a 12-dimensional tonal space in the for a variety of subjective and contextual factors. To a certain context of the ,Chew’sSpiralArray,andHarte’s Tonnetz extent, the large number of different, and sometimes con- 6-dimensional Tonal Centroid Space. The proposed Tonal tradictory, tonal pitch spaces presented in the literature help Interval Space is calculated as the weighted Discrete Fourier us understand the complexity of such representations. Exist- Transform of normalized 12-element chroma vectors, which ing tonal spaces can be roughly divided into two categories, we represent as six circles covering the set of all possible pitch each anchored to a specific discipline and applied methods. intervals in the chroma space. By weighting the contribution On the one hand we have models grounded in music theory of each circle (and hence pitch interval) independently, we (Cohn, 1997, 1998;Lewin,1987;Tymoczko,2011; Weber, can create a space in which angular and Euclidean distances 1817–1821), and on the other hand, models based on cogni- among pitches, chords, and regions concur with music theory tive psychology (Krumhansl, 1990;Longuet-Higgins,1962; principles.
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