Interval class

Abstract: Models of the perceived distance between pairs of pitch collections are a core component of broader models of music cognition Numerous 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 or conceptual framework or to incorporate the uncertain or probabilistic nature of pitch perception This paper embeds pitch collections in expectation tensors and shows how metrics between such tensors can model their perceived dissimilarity Expectation tensors indicate the expected number of tones, ordered pairs of tones, ordered triples of tones, etc, that are heard as having any given pitch, dyad of pitches, triad of pitches, etc The pitches can be either absolute or relative (in which case the tensors are invariant with respect to transposition) Examples are given to show how the metrics accord with musical intuition

Abstract: Undergraduate music majors (N = 27) identified simple musical intervals (m2 through M7), presented at 10 different pitch levels and in three different presentation modes (ascending, descending, harmonic). Resulting error matrices were analyzed by direct inspection, repeated measures ANOVA, and multidimensional scaling (MDS), Minor 6ths were the most difficult to identify; sizes of larger intervals were systematically underestimated; and an interaction of several factors including interval type and acoustical dissonance appeared to shape error rates. ANOVA found no effect for pitch level but a significant effect for presentation mode, with ascending intervals easiest and harmonic intervals the most difficult to identify. A three-dimensional MDS configuration was obtained, indicating an interaction of interval size, interval type, and class of acoustical dissonance. The "classical" interval classes of pitch class set theory can be derived from a particular planar projection of the configuration. The ability to identify intervals is a basic skill for music majors. College music programs devote substantial instruction time to developing and refining this skill. This suggests several research questions that could help teachers optimize classroom time spent on interval identification: Which intervals are most difficult to idenlify? With which other intervals is any given interval more likely to be confused? How is this confusion affected by presentation mode (melodic vs. harmonic) and/or pitch level at which intervals are played? Despite the obvious pedagogical importance of the skill, classroom teachers still rely primarily on "folk wisdom" (e.g., "of course, harmonic intervals are harder to identify ...") for their class preparations. While a number of empirical studies have involved interval identification, almost all of them have dealt with issues of categorical perception (e.g., Burns & Ward, 1978) or thresholds for detecting mistiming (e.g., Vos, 1982). Very few studies have investigated the ways that interval types confuse with each other. In particular, the question of whether interval identification ability is affected by the pitch level at which trials are played never has been considered systematically, even though perceived ioudness varies according to frequency (Fletchcr and Munson, 1933) and thus might possibly interfere with interval identification. Of the few existing studies, the earlier ones either provide no usable quantitative data (von Malt/ew, 1913; Ortmann, 1932), or else any useful data must be mined from within various tables (Jeffries, 1967). (Orlmann only provided qualitative error figures for two interval types, while von Malthew only studied interval recognition at the uppermost extremes of human hearing-thus her results arc-not relevant to typical musical experience.) Two later studies (Killam, Lorton, and Schubert, 1975; Plomp, Wagenaar, and Mimpen. 1973) do give detailed matrices of confusion data generated by somewhat different methodologies. Their results are somewhat similar, but both have problematic or limiting aspects. First, Plomp, Wagcnaar.and Mimpcn studied only harmonic intervals. Furthermore, their set of stimuli could have skewed their results-the stimuli either had one tone fixed at middle C or the octave above, or were at frequencies centered around the middle of that range so that not all interval types were presented an equal number of times. Finally, their stimuli durations were completely unrealistic from a pedagogical perspective (four series of durations at 15, 30, 60, and 120ms). In fairness, however, they were investigating different models for acoustical dissonance and were not interested in pedagogical implications of their work. Ki llam, Lorton, and Schubert used stimuli of reasonable pedagogical duration and studied melodic as well as harmonic intervals, but they only tested at two pitch levels. Thus basic work remains to be done in this area, and the present study attempts to address such need. …

Abstract: Selecting an appropriate representation for chords is important for encoding pertinent harmonic aspects of the musical surface, and, at the same time, is crucial for building effective computational models for music analysis. This chapter, initially, addresses musicological, perceptual and computational aspects of the harmonic musical surface. Then, two novel general chord representations are presented: the first, the General Chord Type (GCT) representation, is inspired by the standard Roman numeral chord type labelling, but is more general and flexible so as to be applicable to any idiom; the second, the Directed Interval Class (DIC) vector, captures the intervallic content of a transition between two chords in a transposition-invariant idiom-independent manner. Musical examples and preliminary evaluations of both encoding schemes are given, illustrating their potential to form a basis for harmonic processing in the domain of computational musicology.

Abstract: This study examined the connection between the concept interval-class (IC) and aural estimations of intervals. Sequences of five piano-sound or Shepard-tone intervals were used. In each sequence four intervals represented one IC (the context) and one interval represented another IC (the deviant). The participants (N = 36) were asked to select one interval deviating from the other intervals of the sequence. The participants responded according to IC more often when the intervals were played with Shepard tones (average 57.2%) than with piano sounds (30.5%). With piano-sound intervals they were also guided by the actual interval size. Additionally, the participants responded according to IC more often when the context was more consonant than the deviant compared with the opposite order of presentation.