1. Literature reviews of many such functions can be found in Marcus Castrén, RECREL: A Similarity Measure for Set-Classes (Helsinki: Sibelius Academy, 1994), Eric J. Isaacson, "Similarity of Interval-Class Content between Pitch-Class Sets: The IcVSIM Relation," Journal of Music Theory 34 (1990): 1-28, and Damon Scott and Eric J. Isaacson, "The Interval Angle: A Similarity Measure for Pitch-Class Sets," Perspectives of New Music 36(2) (1998): 107-142.

2. Marcus Castrén, "Joukkoluokitukseen perustuva sointuluokitus: perusperiaatteet ja esimerkkejä sovellusmahdollisuuksista" [Chord Classification Based on Set Classification: Basic Principles and Examples of Applicational Possibilities], Sävellys ja musiikinteoria 7 (1997): 6-25, and "Pairs of Chords as Objects Illuminating Set-Class Similarity: Some Viewpoints and a Computer-Assisted Procedure to Create Test Materials for Listener Experiments," Electronic Journal of Music Theory and Analysis vol.1, no.2 (2000), www.musictheoryresources.com; Olli Väisälä, "Concepts of Harmony and Prolongation in Schoenberg's Op. 19/2," Music Theory Spectrum 21 (1999): 230-59.

3. Eric J. Isaacson, "Issues in the Study of Similarity in Atonal Music," Music Theory Online vol. 2.7, 1996, parag. 11.

4. Diana Stammers, "Set Theory in the Perception of Atonal Pitch Relations" (Ph.D. diss., Cambridge University, 1994). Major and minor 3rds were treated by her listeners as being closely related and confusable, whereas the two types of 2nds were considered distinct. She attributed this to prior learning from exposure to classical tonality, in which the two types of thirds have similar structural roles but the two types of seconds do not.

5. Cheryl L. Bruner, "The Perception of Contemporary Pitch Structures," Music Perception 2 (1984): 25-39; Don B. Gibson, "The Aural Perception of Non-Traditional Chords in Selected Theoretical Relationships: A Computer-Generated Experiment," Journal of Research in Music Education 34 (1986): 5-23, "The Aural Perception of Similarity in Nontraditional Chords Related by Octave Equivalence," Journal of Research in Music Education 36 (1988): 5-17, and "The Effects of Pitch and Pitch-Class Content on the Aural Perception of Dissimilarity in Complementary Hexachords," Psychomusicology 12 (1993): 58-72; Tuire Kuusi, Set-Class and Chord: Examining Connection between Theoretical Resemblance and Perceived Closeness (Helsinki: Sibelius Academy, 2001), "Semantic Differential as a Method for Collecting Estimations of Chords," in Proceedings of the 5th Triennial ESCOM Conference (2003), and "The Role of Set-Class Identity in the Estimation of Chords," Music Theory Online vol. 9.3, 2003; Roger C. Lane, "A Multidimensional Scaling Study of Seven Theoretical Indices of Intervallic Similarity and Musicians' Perceptions among Twenty-One Pitch-Class Sets with Implications for Music Teaching and Learning" (Ph.D. diss, University of North Carolina, Greensboro, 1997); Panayotis Mavromatis and Virginia Williamson, "Towards a Perceptual Model for Categorizing Atonal Sonorities" (paper presented at the Society of Music Theory meeting, Atlanta, GA, 1999); Jana K. Millar, "The Aural Perception of Pitch-Class Relations: A Computer-Assisted Investigation" (Ph.D. diss., North Texas State University, 1984); Art Samplaski, "The Perceived Similarity of Some Non-Tonal Tetrachords Compared to Predictions of Selected Twentieth-Century Chord Classification Systems," presentation at the 2000 meeting of the Society for Music Perception and Cognition, Toronto, Ontario, and "The Relative Perceptual Salience of Tn and TnI," Music Perception 21 (2004): 545-59; and Virginia Williamson and Panayotis Mavromatis, "Similarity in Atonal Music Theory: A Perceptual Study" (paper presented at the Society for Music Perception and Cognition meeting, Cambridge, MA, 1997), and "Categorizing Atonal Sonorities: Multidimensional Scaling, Tree-Fitting, and Clustering Compared" (paper presented at the Society for Music Perception and Cognition meeting, Evanston, IL, 1999).

6. For other types of problems, other methods of data collection are also possible, e.g., consumer product preference ratings.

7. There is an entire family of MDS models for different situations, with differing limitations and requirements; a related set of techniques called cluster analysis also provides for visualization of similarity by generating tree diagrams instead of geometric configurations. See Joseph B. Kruskal and Myron Wish, Multidimensional Scaling, Sage University Paper Series on Quantitative Applications in the Social Sciences, Series no. 07-011 (Newbury Park, CA and London: Sage Publications, 1978), for a basic introduction. Their short monograph discusses both the underlying mathematics as well as various important methodological issues that will not be discussed in this essay; such issues include determining the appropriate number of dimensions for the derived configurations (i.e., are three dimensions needed to adequately describe the relationships between the objects under investigation, or will two suffice?), and how to deal with any underlying asymmetries in the data (e.g., in a tonal context I->V will almost certainly be rated differently than V->I). I am currently finishing preparation on an article illustrating a different application of MDS for music-theoretic analysis that will include a detailed but non-technical tutorial about such issues.

8. T. Jeffries, "The Effects of Order of Presentation and Knowledge of Results on the Aural Recognition of Melodic Intervals," Journal of Research in Music Education 15 (1967): 179-90; Catrina von Maltzew, "Das Erkennen sukzessiv gebener musikalischer Intervalle in den äusseren Tonregionen," Zeitschrift für Psychologie 64 (1913): 161-257; and Otto Ortmann, "Notes on Interval Discrimination," Peabody Bulletin vol. 28, no. 2 (1932): 45-46. Ortmann provides only qualitative bar graphs of confusion data for two intervals (i.e., he gives no indication as to whether the vertical axes should be interpreted as percentage of times the played interval was identified as this, that, or the other answer--and if so, what percentage--or as numbers of times it was so identified). Jeffries was concerned with how the presence of particular intervals before and after a target interval in specific melodic contexts affected how the target was identified; he presents his results in a couple of ways, but they again are qualitative, not quantitative, tables and diagrams.

9. Rosemary N. Killam, Paul V. Lorton, Jr., and Earl D. Schubert, "Interval Recognition: Identification of Harmonic and Melodic Intervals," Journal of Music Theory 18 (1975): 213-234; R[einer] Plomp, W. Wagenaar, and A. M. Mimpen, "Musical Interval Recognition with Simultaneous Tones," Acustica 29 (1973): 101-109. One slightly earlier study (W. J. M. Levelt, J. P. van der Geer, and R[einer] Plomp, "Triadic Comparisons of Musical Intervals," British Journal of Mathematical and Statistical Psychology 19 [1966]: 163-179) actually carried out MDS analysis of data obtained by the triadic comparison method for intervals sounded as pure tones and as complex tones; that analysis appeared to show some support for the hypothesis of interval-class as a psychological construct. In my opinion, however, their paper contains significant methodological problems, which would be beyond the scope of the present article to critique in detail; the citation is given for those readers wishing to consult the work for themselves.

10. Thirty additional duplicate trials were inserted into the generated random orders, subject to the same constraints between consecutive trials, as a check for subjects' self-consistency. There were thus 360 trials (18 blocks of 20) in total. One additional subject was eliminated from consideration for failing this check.

11. Art Samplaski, "Interval and Interval-Class Similarity: Results from a Confusion Study," poster presentation at the 2003 meeting of the Society for Music Perception and Cognition, Las Vegas, NV; manuscript submitted for publication.

12. The subsidiary arc drawn between the imperfect consonances on the schematic diagram in Figure 2 is to aid visualization. It is parallel to the arc along which the dissonances lie.

13. The configurations derived from Plomp et al.'s data are not shown because of the minimal time for which their stimuli sounded--they did four experimental runs of stimuli with durations of 120, 60, 30, and 15(!) milliseconds respectively, only the longest of which is relevant to actual musical situations. Plomp et al. were concerned with how interval recognition degraded as stimulus duration was reduced, rather than any possible implications for theories of musical structure. It is thus all the more remarkable that the configuration for their data bears any resemblance at all to the other two.

14. Killam et al. used a sample of 15 undergraduate music majors; they tested at two transposition levels rather than the ten used in the present study, using an organ timbre; the stimuli were presented in two durations, of 100 and 200 milliseconds. As in the present study, intervals were presented in all three presentation modes; while the authors provide some analysis separated by presentation mode, they only presented an aggregate confusion matrix. Plomp et al. used a sample of 15 music students; they used sine tones and sawtooth waves as timbres; the intervals were presented such that either the high or low notes of the intervals were always middle C or the C above, or else centered within that octave to the extent possible. The various intervals were thus presented different numbers of times (octaves could only be presented as C4 to C5 and major 7ths could only be presented with C as one component, while the other intervals were presented in three different forms); while they detail some efforts to reduce any bias caused by such lopsidedness, their paper remains problematic.

15. Indeed, given that in a number of undergraduate aural skills programs music majors receive little or no training on the recognition of compound intervals, it is entirely possible that such MDS configurations could be chaotic jumbles if population samples of only undergraduates are used. This is a worst-case scenario, of course, but we have no way a priori to know what results might occur.

16. Again, we cannot be certain that this would be the case, although it may seem like a perfectly plausible scenario. In making pronouncements about empirical situations, there is no substitute for collecting actual data. Because the number of factors that might influence the results, and their possible interactions, proliferate rapidly in imbedded-context situations, it is more difficult to develop well-designed studies of such types--context-free studies must first be carried out to provide a baseline for follow-up research.

17. Scott and Isaacson, op. cit., pp. 136-39.

18. The most recent such statement is Tonal Pitch Space (New York: Oxford University Press, 2001), p. 351.  

End of footnotes