Empirically testing Tonnetz, voice-leading, and spectral models of perceived triadic distance

Abstract

We compare three contrasting models of the perceived distance between root-position major and minor chords and test them against new empirical data. The models include a recent psychoacoustic model called spectral pitch-class distance, and two well-established music theoretical models – Tonnetz distance and voice-leading distance. To allow a principled challenge, in the context of these data, of the assumptions behind each of the models, we compare them with a simple “benchmark” model that simply counts the number of common tones between chords. Spectral pitch-class distance and Tonnetz have the highest correlations with the experimental data and each other, and perform significantly better than the benchmark. The voice-leading model performs worse than the benchmark. We suggest that spectral pitch-class distance provides a psychoacoustic explanation for perceived triadic distance and its music theory representation, the Tonnetz. The experimental data and the computational models are available in the Online Supplement (http://dx.doi.org/10.1080/17459737.2016.1152517).

Keywords:

• spectral pitch-class distance
• voice-leading distance
• Tonnetz
• similarity
• fit
• harmony

2010 Mathematics Subject Classification:

• 15A04
• 15A06
• 65C20

2012 Computing Classification Scheme:

• applied computing
• arts and humanities
• sound and music computing

Acknowledgments

We would like to thank the two anonymous reviewers for helpful comments and suggestions. The first author would also like to thank Tuomas Eerola and Petri Toiviainen for supporting this project as part of a Master's degree for the MMT programme at the University of Jyväskylä (the models used in the resulting thesis, Milne 2009, were quite different from those described in this paper), as well as the students, staff, and other volunteers who participated in the experiment.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental online material

Supplemental online material for this article can be accessed at http://dx.doi.org/10.1080/17459737.2016.1152517 and http://www.dynamictonality.com/harmonic_distance_files/.

Notes

1 A question also raised by Krumhansl (1990).

2 Although not relevant to the more abstract argument being made here, the spectral pitch-class model does indeed predict that the C maj chord is closer to the C major scale than is the G maj chord (Milne, Laney, and Sharp 2015).

3 The two types of Tonnetz can also be characterized as geometrical duals (Cohn 1998; Tymoczko 2012).

4 In this paper, enharmonic equivalence refers to all Tonnetz locations with the same pitch in 12-tone equal temperament, which is the tuning of the experimental stimuli (in just intonation, their pitches would differ). This includes standard enharmonic equivalences such as …, B$\mathrm{♯♯♯}$, C$\mathrm{♯♯}$, D, E$\mathrm{♭♭}$, F$\mathrm{♭♭♭}$, …, but it also includes Tonnetz locations with the same pitch name. More precisely, all notes separated by some integer combination of the syntonic comma (up four Tonnetz fifths and down one Tonnetz major third) and the major diesis (up four Tonnetz fifths and down four Tonnetz major thirds) are termed enharmonically equivalent because they have the same pitch in 12-tet. They have the same pitch because these two commas are a basis of the null space of the linear map from 5-limit just intonation to 12-tet (Milne, Sethares, and Plamondon 2008).

5 The triangle inequality requires that $d(x,z)\le d(x,y)+d(y,z)$; that is, the distance between two objects is shorter than or equal to the distance when passing through a third object.

6 Due to the abstraction (i.e. permutations) over voices, the ordering of elements in this vector has no meaning; hence this vector is most usefully thought of as just one out of all possible orderings of a single (unordered) multiset – the latter corresponding to the mathematical formalization used by Tymoczko (2006, 2011). Vectors may, however, be a more appropriate formalization whenever voices are not abstracted over.

7 It is worth noting that the model described here is quite different from a voice-leading type model – even if each harmonic were to be represented as an individual “voice.” For discussion of the differences between the category domain pitch embeddings used in voice-leading models and the pitch domain embeddings used in expectation tensors, and their implications when used to obtain distances, see Milne et al. (2011).

8 For concision here, and in future examples, we use uppercase for major triads and lowercase for minor triads.

9 A harmonic complex tone comprises frequency components at integer multiples of a fundamental frequency, an octave complex tone comprises only those frequency components at $2^n$ multiples of the fundamental – the first, second, fourth, eighth, sixteenth, etc., harmonics. Most pitched Western musical instruments, and the sung human voice, produce harmonic complex tones. Octave complex tones do not occur in nature and must be artificially synthesized.

10 Cronbach's α is equivalent to the mean split-half correlation of a data set and is used as an estimate of reliability, consistency, or homogeneity. A split-half correlation is given by splitting the data into two equally-sized halves, summing across each half, and then correlating these two summed halves. The mean split-half correlation is the mean correlation over all possible equal splits. It is equivalent to Cronbach's α, though the latter is calculated in a computationally simpler – though intuitively less understandable – manner.

11 Bootstrapping is a method for estimating the variance of a statistical estimate – in this case the variances of the means. It makes no assumptions about the underlying distribution of the data.