Metrics for Pitch Collections

Models of the perceived distance between pairs of pitch collections are a core component of broader models of the perception of tonality as a whole. Numerous different distance measures have been proposed, including voice-leading, psychoacoustic, and pitch and interval class distances; but, so far, there has been no attempt to bind these different measures into a single mathematical framework, nor to incorporate the uncertain or probabilistic nature of pitch perception (whereby tones with similar frequencies may, or may not, be heard as having the same pitch). To achieve these aims, we embed pitch collections in novel multi-way expectation arrays, and show how metrics between such arrays can model the perceived dissimilarity of the pitch collections they embed. By modeling the uncertainties of human pitch perception, expectation arrays indicate the expected number of tones, ordered pairs of tones, ordered triples of tones and so forth, that are heard as having any given pitch, dyad of pitches, triad of pitches, and so forth. The pitches can be either absolute or relative (in which case the arrays are invariant with respect to transposition). We provide a number of examples that show how the metrics accord well with musical intuition, and suggest some ways in which this work may be developed. 1. BACKGROUND AND AIMS A pitch collection may comprise the pitches of tones in a chord, a scale, a tuning, or the virtual and spectral pitches heard in response to complex tones or chords. Modeling the perceived distance (the similarity or dissimilarity) between pairs of pitch collections has a number of important applications in music analysis and composition, in modeling of musical cognition, and in the design of musical tunings. For example, voice-leading distances model the overall distance between two chords as a function of the pitch distance moved by each voice (see Tymoczko (2006) for a survey); musical set theory considers the similarities between the interval (or triad, tetrad, etc.) contents of pitch collections (see Castrén (1994) for a survey); psychoacoustic models of chordal distance (Parncutt, 1989; Milne, 2009) have treated tones or chords as collections of virtual and spectral pitches (Terhardt, Stoll, & Seewann, 1982; Zwicker & Fastl, 1999) to determine their affinity; tuning theory requires measures that can determine the distance between scale tunings and, notably, the extent to which different scale tunings can approximate privileged tunings of intervals or chords (e.g., just intonation intervals with frequency ratios such as 3/2 and 5/4, or chords with frequency ratios such as 4:5:6:7). 2. METHOD 2.1 Pitch (Class) Vectors A pitch (class) vector contains elements whose values indicate pitches (typically in semitones). Standard metrics between two such vectors are based only on the pitch distances between elements in matching positions in the two vectors. For this reason, such pitch metrics are only meaningful when each tone in one pitch collection has a privileged relationship with a unique tone in another pitch collection; for example, when each element represents a different voice (bass, tenor, alto, soprano), or scale degree, or even metrical or ordinal position in a melody. Such metrics are not suitable when the pitches cannot be uniquely categorized in the manner described above. This occurs when two collections have a different number of pitches, or when there are so many pitches that categorization is inappropriate. For example, when modeling the distance between the large sets of spectral or virtual pitches heard in response to complex tones or chords, there is no unique way to reasonably align each spectral pitch of one complex tone or chord with each spectral pitch of another (Sethares, Milne, Tiedje, Prechtl, & Plamondon, 2009) and, even if there were, it is not realistic to expect humans to track the “movements” of such a multitude of pitches. 2.2 Expectation Arrays We present a novel family of pitch embeddings (expectation arrays), and associated metrics, that can be applied to the above areas. Expectation arrays model the uncertainties of pitch perception by “smearing” each pitch over a range of possible values, and the width of the smearing can be derived directly from experimentally determined frequency difference limens (Moore, Glasberg, & Shailer, 1984; Roederer, 1994). The effect of this pitch smearing is significant whenever pitches in one collection are similar, but non-identical, to pitches in another; for example, when comparing scales with different microtonal tunings, or the collections of virtual and spectral pitches produced in response to different chords. The arrays can embed either absolute or relative pitches (denoted absolute and relative expectation arrays, respectively): in the latter case, embeddings of pitch collections that differ only by transposition have zero distance; a useful feature that relates similarity to structure. Depending on their number of dimensions, expectation arrays indicate the expected number of tones, ordered pairs of tones, ordered triples of tones, and so forth, that will be heard as having any given pitch, dyad of pitches, triad of pitches, and so forth. This enables different pitch collections to be compared according The Open University, UK University of Wisconsin-Madison, USA ISBN: 1 876346 62 0  2010 ICMPC11 Proceedings of the 11th International Conference on Music Perception and Cognition (ICMPC11). Seattle, Washington, USA. S.M. Demorest, S.J. Morrison, P.S. Campbell (Eds)