A multi-level tonal interval space for modelling pitch relatedness and musical consonance

Abstract

In this paper we present a 12-dimensional tonal space in the context of the Tonnetz, Chew’s Spiral Array, and Harte’s 6-dimensional Tonal Centroid Space. The proposed Tonal Interval Space is calculated as the weighted Discrete Fourier Transform of normalized 12-element chroma vectors, which we represent as six circles covering the set of all possible pitch intervals in the chroma space. By weighting the contribution of each circle (and hence pitch interval) independently, we can create a space in which angular and Euclidean distances among pitches, chords, and regions concur with music theory principles. Furthermore, the Euclidean distance of pitch configurations from the centre of the space acts as an indicator of consonance.

Keywords:

• tonal pitch space
• consonance
• tonal hierarchy

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Corrigendum

Notes

This article was originally published with errors. This version has been corrected. Please see Corrigendum (http://dx.doi.org/10.1080/09298215.2016.1198463).

Gilberto Bernardes, INESC TEC (Sound and Music Computing Group), FEUP campus, Rua Dr. Roberto Frias, 4200 - 465 Porto, Portugal. E-mail: [email protected].

1 The use of the DFT in the context of our work was inspired by Ueda, Uchiyama, Nishimoto, Ono and Sagayama (2010), who identified a correspondence between the DFT coefficients of a chroma vector and Harte et al.’s (2006) 6-D space.

2 Because the pitch configurations for major and minor triads contain identical relative intervals when represented as chroma vectors, the Tonal Interval Space cannot disambiguate them, hence we must consider them equally ranked.

3 While Roberts (1986) provides consonance ratings of triads, these were obtained from an experimental design that relied heavily on a preceding musical context and not listener judgements of isolated triads. Therefore we do not attempt to directly incorporate these absolute ratings when determining the weights w(k).

4 In order to illustrate distances among pitch configurations in the 12-D Tonal Interval Space, we use nonmetric multidimensional scaling (MDS) to plot it into a 2-dimensional plane. Shepard (1962) and Kruskal (1964) first used this method, which has been extensively applied to visualize representations of multidimensional pitch structures (Barlow, 2012; Krumhansl & Kessler, 1982; Lerdahl, 2001). Briefly, nonmetric MDS attempts to transform a set of n-dimensional vectors, expressed by their distance in the item-item matrix, into a spatial representation that exposes the interrelationships among a set of input cases. We use the smacof library from the statistical analysis package ‘R’ to compute dimensionality reduction using a nonmetric MDS algorithm. More specifically, we use the function smacofSym, with ‘ordinal’ type and ‘primary’ ties.

5 A major difference between the two empirical studies conducted relies on their population. While Roberts’ (1986) study was conducted among Western listeners, Cook et al.’s (2007) study involved East Asian listeners. The population of both studies included individuals with and without music training. The remaining psychoacoustic-based models aim at measuring auditory roughness, which largely equates with sensory dissonance (Sethares, 1999).

6 By guaranteeing uniqueness, our space avoids the overlap between relevant tonal pitch configurations as in the Harte et al.’s (2006) Tonal Centroid Space, such as the pair of dyads F#-B (P4) and D-G# (D5)—pitch classes [5, 11] and [2, 8])—and the D diminished seventh chord and the dyad D-A# (A4)—pitch classes [2, 5, 8, 11] and [2, 8].

Additional information

Funding

This research was supported by the Project “NORTE-01-0145-FEDER-000020”, financed by the North Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, and through the European Regional Development Fund (ERDF) and by the Portuguese Foundation for Science and Technology under the post-doctoral grants SFRH/BPD/109457/2015 and SFRH/BPD/88722/2012.