JournalPsychonomic Bulletin & Review 2005, ??12 (?),(6), ???-???1068-1075 The subjective size of melodic intervals over a two-octave range FRANK A. RUSSO and WILLIAM FORDE THOMPSON University of Toronto, Mississauga, Ontario, Canada Musically trained and untrained participants provided magnitude estimates of the size of melodic intervals. Each interval was formed by a sequence of two pitches that differed by between 50 cents (one half of a semitone) and 2,400 cents (two octaves) and was presented in a high or a low pitch register and in an ascending or a descending direction. Estimates were larger for intervals in the high pitch register than for those in the low pitch register and for descending intervals than for ascending intervals. Ascending intervals were perceived as larger than descending intervals when presented in a high pitch register, but descending intervals were perceived as larger than ascending intervals when presented in a low pitch register. For intervals up to an octave in size, differentiation of intervals was greater for trained listeners than for untrained listeners. We discuss the implications for psychophysi- cal pitch scales and models of music perception. Sensitivity to relations between pitches, called relative scale describes the most basic model of pitch. Category pitch, is fundamental to music experience and is the basis labels associated with intervals assume a logarithmic re- of the concept of a musical interval. A musical interval lation between pitch height and frequency; that is, inter- is created when two tones are sounded simultaneously or vals spanning the same log frequency are associated with sequentially. They are experienced as large or small and as equivalent category labels, regardless of transposition. Lis- consonant or dissonant, and for musically trained listen- teners with musical training can explicitly label intervals ers, they are associated with category labels, such as the (Killam, Lorton, & Schubert, 1975; Plomp, Wagenaar, & perfect fourth, the minor sixth, and the octave. Sequential Mimpen, 1973). More important, judgments of interval intervals form the basis of melody and may occur in as- size by musicians are resistant to the influence of context cending or descending pitch directions. (J. A. Siegel & W. Siegel, 1977a) and exhibit steplike func- Relative pitch allows us to perceive, appreciate, and tions characteristic of categorical perception (J. A. Siegel remember melodies. Large melodic intervals, or leaps, & W. Siegel, 1977b; see also Burns & Campbell, 1994; form the basis for gap-fill melodies (Meyer, 1973) and Burns & Ward, 1978). These findings imply that estimates are experienced as a point of accent (Boltz & Jones, of interval size may depend on musical training. 1986; Drake, Dowling, & Palmer, 1991; Jones, 1987). Drawing from early studies of tone perception by Conversely, melodies sound more coherent or cohesive Titchener (1905, pp. 232–248), Stevens, Volkmann, and when they consist of a sequence of small intervals (Huron, Newman (1937) questioned whether the psychophysical 2001; Russo, 2002). Interval size may also influence me- relation between pitch and frequency was logarithmic and lodic expectancy (Larson & McAdams, 2004; Margulis, used scaling techniques to derive a new scale, called the 2005; Narmour, 1990) and grouping (Deliège, 1987; Le- mel scale. A pure tone of 1000 Hz at 40 dB above thresh- rdahl, 1989; Lerdahl & Jackendoff, 1983). In short, there old was first defined as 1,000 mels, and the pitch in mels is a need for research whose aim is to identify factors that of other frequencies was determined by asking partici- influence the experience of interval size. The goal of this pants to adjust a comparison tone until it was perceived to study was to investigate relative pitch by obtaining magni- be one half of the pitch height of a standard tone (method tude estimates of interval size and assessing the influence of fractionation). Although the mel scale and the logarith- of three factors: musical training, pitch register, and pitch mic scale are roughly equivalent below 500 Hz, above direction (ascending or descending). 500 Hz, the mel scale increases at a slower rate. That is, Psychophysical models of pitch can be used to make perceptually equivalent pitch interval sizes (in mels) span predictions about judgments of interval size. A logarithmic progressively smaller frequency ratios with higher and higher transpositions (Stevens et al., 1937; see also Beck & Shaw, 1961). These results suggest that estimates of This research was supported by a discovery grant awarded to the sec- interval size may depend on pitch register. ond author from the Natural Sciences and Engineering Research Council Another psychophysical scaling method, called the of Canada. We thank Bruce Schneider for helpful comments and Jane method of equal sense distances, involved presenting lis- Campbell for research assistance. Correspondence should be sent to F. A. Russo or W. F. Thompson, Department of Psychology, University teners with a melodic interval and asking them to divide of Toronto, Mississauga, ON, L5L 1C6 Canada (e-mail: frusso@utm it into four equal pitch interval sizes (Greenwood, 1997; .utoronto.ca or [email protected]). Stevens & Volkmann, 1940). Results based on this method Copyright 2005 Psychonomic Society, Inc. 1068 THE SUBJECTIVE SIZE OF MELODIC INTERVALS 1069 were similar to those based on the method of fractionation, participants had 2 years or less of musical instruction (M ϭ 0.57 but an asymmetry was revealed. When the interval was years, SE ϭ 0.23) and no continued musical activity. They included ascending (i.e., low tone first), participants set dividing 10 females and 2 males ranging in age from 17 to 20 years, with a frequencies higher than when the interval was descending. mean age of 18.5 years. Trained participants had 7 years or more of musical instruction (M ϭ 11.07 years, SE ϭ 0.55) and continued This asymmetry suggests that estimates of interval size musical activity. They included 10 females and 2 males ranging in depend on pitch direction. age from 17 to 25 years, with a mean age of 18.9 years. All the par- Some researchers have questioned the relevance of psy- ticipants were given course credit. No participant reported having chophysical scales such as the mel scale (Burns, 1999; abnormal hearing. Krumhansl, 1990, 2000; Shepard, 1999). First, the tasks used to derive the mel scale have ambiguous interpreta- Stimuli Seventy-three tones were constructed by additive synthesis with tions (e.g., half as high), and responses to the tasks are not Praat software (Boersma & Weenink, 2004). The fundamental always reliable (Rasch & Plomp, 1999). It has been argued frequency of each tone was separated by 50 cents from its near- that the mel scale cannot be derived from other measures est neighbor, falling within the frequency range of 87.3 Hz (F2) to of pitch, such as difference limens, critical bands, or equal 698.5 Hz (F5). All the tones consisted of 12 equal-intensity compo- cochlear distances (Greenwood, 1997, p. 200), and differ- nents (the fundamental and the first 11 overtones). Tones were used ent strategies of deriving the mel scale yield somewhat dif- to create 48 melodic intervals (two-tone sequences) that varied in size (log frequency distance between tones) from 50 cents (one-half ferent functions (Lewis, 1942; but see Schneider, Parker, semitone) to 2,400 cents (two octaves). Each of the 48 intervals was & Upenieks, 1982). Second, the ease with which listeners presented in ascending and descending pitch directions and in low recognize and reproduce simple melodies independently and high pitch registers. The three factors of interval, pitch direc- of pitch register implies that frequency ratios are experi- tion, and pitch register were counterbalanced. Intervals in the low enced as invariant across transposition (Attneave & Olson, pitch register were centered on F3 (174.6 Hz), and intervals in the 1971). Third, the mel scale was derived from judgments of high pitch register were centered on F4 (349.2 Hz). For instance, the isolated pure tones, which may have limited relevance to largest ascending intervals (2,400 cents) spanned F2 to F4 in the low pitch register and F to F in the high pitch register. the way tones are perceived in a musical context. 3 5 Nonetheless, it is difficult to dismiss the data that were Procedure used to derive the mel scale. Specifically, they imply that Each participant completed two blocks each of ascending and there is a dimension of melody perception that is some- descending trials for a total of 384 trials (48 intervals ϫ 2 pitch what independent of the explicit labels associated with registers ϫ 2 directions ϫ 2 blocks). Block presentation adhered to musical intervals (Makeig, 1982). This implication moti- one of two orders: (1) ascending, descending, descending, ascend- ing or (2) descending, ascending, ascending, descending. The two vated us to obtain direct estimates of the size of melodic orders were counterbalanced across participants, and the order of intervals and to identify some influences on such judg- presentation of trials within blocks was randomly determined for ments. Trained and untrained listeners estimated the size each participant. of 48 different intervals ranging between 50 cents (one- The participants provided magnitude estimates of the size of each half semitone) to 2,400 cents (two octaves) presented in interval on a scale from 1 to 100. To discourage explicit mapping of a high or a low pitch register and in an ascending or a known interval categories, a time limit of 5 sec (with a 3-sec warn- ing) was imposed on each estimate. Before testing, the participants descending pitch direction. were familiarized with examples of the smallest interval (50 cents, We considered the possibility that the ability to dis- defined as 1) and the largest interval (2,400 cents, defined as 100). criminate intervals up to an octave might be somewhat different from the ability to discriminate intervals larger RESULTS than an octave.