1. Surveys and critiques of most similarity measures can be found in Castrén (1994), Isaacson (1990, 1992), and Scott and Isaacson (1998).

2. Isaacson (1996) raises some basic conceptual questions. Quinn (2001) is a well-argued critique of several fundamental issues.

3. Rahn (1989) relates an early attempt to apply computation to the analysis of similarity relations; he was defeated by the speeds of then-available processors. Computational complexity has not completely disappeared since then, in that commercially-available software still has limits that prevent, e.g., a single analysis of all ratings for trichords through nonachords (a 208 x 208 matrix); but excessive length of computer runs is no longer an obstacle to such analyses.

4. Correlations (denoted by "r") can vary from a perfect negative of -1.0 (as quantity A goes up, quantity B goes down by the same proportion) to a perfect positive of +1.0; for these functions over the integers 1 to 50, a domain size comparable to the number of pcsets studied here, y = x/SQRT(x) have r = .983, y = x/x2 have r = .969, and y = x2/SQRT(x) have r = .911--all close to perfect positive.

5. Narmour's (1990, 1992) implication-realization model for melodic expectation is a case in point. He proposes five interacting factors to account for listeners' melodic expectancies; studies by Krumhansl (1995) and Schellenberg (1996, 1997) indicate that the model can be simplified to just two factors without significant loss of explanatory power.

6. In doing so, I fulfill the promise in my previous article in this journal (Samplaski, 2004, fn. 7) to provide "a detailed but non-technical tutorial" about MDS and some of the issues in its use. Nonetheless, this overview must perforce still be very superficial and exclude the theoretical underpinnings necessary to use the techniques appropriately; I can but hope that it will stimulate some readers to investigate possible applications to their own areas of music-theoretic research. The best initial pointer into the MDS literature remains Kruskal and Wish (1978), even though there have been a number of developments in the field since then.

7. Quinn (2004) has recently completed a dissertation that analyzes in depth the mathematics behind pcset genera, and by extension, similarity functions. Since he considers not only twelve-fold division of the octave but other equal temperaments, his results clarify issues that remain obscured if one examines only the 12-ET universe. Interested readers should consult this important work.

8. The ratings can be obtained in any of several ways: as direct estimates or impressions of similarity or distance; as same-different confusion rates (non-identical but similar stimuli are more likely to be mistaken for each other than dissimilar ones, so high confusion rates correlate with high similarity); etc.

9. Without going deeply into topology, a torus can be embedded in variously-dimensioned spaces. In a three-dimensional space an observer on a torus' surface would notice that the surface was non-Euclidean: the angle sum of a triangle would not be 180 degrees, etc. If a torus is embedded in a four-dimensional space, its surface will be "flat" in the sense of Euclidean geometry. This was the nature of the configuration found by Krumhansl and Kessler.

10. Other distance metrics are possible. One, intuitively used daily by millions, is the "city-block" or "taxi-cab" metric, where to get from point A to point B you go J units in the first dimension, then K units in the next, etc. The general formula for such distance metrics is called the "Minkowski distance metric."

11. Even this assumption fails in the real world depending on the precision needed: the builders of the Verrazano Narrows Bridge, between the NYC boroughs of Staten Island and Brooklyn, had to take the earth's curvature into account.

12. Note that models such as ASCAL and INDSCAL are not invariant with respect to rescaling, or to rotation or reflection about axes. They also have a significantly larger number of free parameters to be estimated than simpler models. Moreover for INDSCAL, because individual biases are not canceled out by averaging responses into a single matrix, it can produce a much poorer fit to the data. A researcher should thus not automatically use one of these models in hopes of obtaining the most general result.

13. In technical terms, as the number of free parameters being calculated increases relative to the number of data points, there become too few constraints on the possible configuration.

14. If questions remain, the researcher should examine several different dimensionalities and be prepared to choose on the bases of clarity and logical interpretation. There are times where a solution one dimension above or below "optimal" as indicated by the stress/r2 values might be better: 1) if there is a clear interpretation given an added dimension; or 2) if one configuration is easier to visualize (e.g., a 3-D vs. a 4-D solution), especially in a situation where it is unclear what can be gained in explanatory power by using the extra dimension.

15. In an MDS analysis of N objects, one of which is an exemplar, the only way to minimize distortion (i.e., stress) is to place the exemplar at the center of the configuration and arrange the other objects around it, along the rim of a circle/sphere/equivalent higher surface. One can always draw a circle through three non-colinear points, a sphere through four non-coplanar points, etc.; so, adding in the exemplar, an N-dimensional MDS solution can in general accommodate N+2 objects (where one is an exemplar) without problems. Of course, for some datasets one will be able to fit more objects than this along the surface's rim; and in other cases the limit can be somewhat liberalized by locating the exemplar off-center in the circle/sphere/etc., or by using an ellipse/oblate spheroid/etc. as the surface. This will, though, only go so far: if you try to fit a set of 23 objects including an exemplar in two or three dimensions, you must expect a fair amount of stress in the solution.

16. An example dating back to the great nineteenth century psychologist William James is: the moon is like a ball because they are both round; the moon is also like a gas lantern because they both illuminate; but we do not think of a ball as being like a gas lantern.

17. This study considers only results from MDS, so no further discussion of CA is necessary; it is mentioned to raise awareness of the issues that render it appropriate for various situations. Some of Quinn's (1997, 2001) analyses of pcset similarity measures use CA; as discussed in paragraphs 61-63, he obtains results compatible with those reported here.

18. This model underlies MacKay and Zinnes' (1999) PROSCAL program. A different model underlies an older PMDS program called MULTISCALE (Ramsay, 1977), but we need not worry about the distinctions.

19. The present article is a case in point: analyses of the datasets herein took a few seconds on a PC of moderate speed, using the statistical package SPSS; the equivalent analyses on the same machine using PROSCAL took up to forty minutes.

20. AMEMB2 is a modification of Rahn's (1979-80) MEMB2 function by Isaacson for inclusion in the latter's Winsims calculator, available at <http://theory.music.indiana.edu/isaacso/winsims.html>; Isaacson applies a normalization factor equivalent to that used by Rahn to derive ATMEMB from his TMEMB function. For narrative simplicity, it seemed preferable to refer to it as Rahn's function. Note that while AMEMB2 is concerned with cardinality-two sets, i.e., interval-classes, it is not an icv-based function in the sense of ANGLE et al., since it only counts how many instances of each dyad are mutually imbedded in two pcsets--for example, in returning a comparison for [0156] and [0167], only one ic6 in the latter is counted. I thus treat it as a subset-based measure.

21. That is, the icv of any nonachord is a function of the icv of its trichord complement, and similarly for octachords/tetrachords and heptachords/pentachords; ratings produced by icv-based similarity functions are therefore also related. The precise functions involved differ for each pair of cardinalities, but they are systematic for those pairs.

22. That value is actually a severe understatement; it is the limit of accuracy reported by the statistical package used. Obtaining an r of better than .96 (between ANGLE and RECREL) over 3160 observations is so close to perfection as to be basically unheard-of for a "real-life" dataset.

23. Various additional types of cross-check analyses were carried out using PROSCAL to the extent possible given the version of the program available--the 4x5 and 3x5 datasets were too large. These are omitted for considerations of space, minimization of technical detail, and reader patience.

24. In particular, the 3x3 dataset, with only 12 objects, is far too sketchy to understand what is going on for most of its configuration. The same holds true to lesser extents for the other smaller datasets.

25. If one is determined to try to visualize a four-dimensional object, the best starting place is Edwin Abbott's classic story Flatland (Abbott, 1885/1952). With modern computer graphics, it is now possible to get a direct visceral appreciation of such objects via programs that manipulate their projections onto the screen.

26. Since the Procrustes rotation deals with the derived configurations, which involve the final relative set-class locations in abstract space, we need not worry that the latter two functions rate similarity while RECREL rates dissimilarity.

27. A p-value of .05 is the typical cutoff value for empirical studies, in that researchers are usually willing to risk a 1 in 20 chance of reporting a false positive result. (Certain types of studies, e.g., clinical drug trials, obviously must set far more stringent standards.)

28. There is no possibility of problems due to axial reflection, since the Procrustes rotation would have handled that.

29. The most surprising of these cases is the disappearance of the ic1/ic5 dimension for cardinality-three set-classes in the 3x4 dataset. Given the coordinates of the rotated subconfiguration, that dimension appears to involve an "ic3/anti-ic6 vs. ic6/anti-ic3" opposition, as seen on the second section of Table 5.

30. Forte only lists hexachords up to 6-35 in his table on pp. 264-266 that summarizes the makeup of the genera. From the inclusion rules on p.192 one can deduce that the remaining 15 hexachords belong to the same genera as their Z-related counterparts, but it would have been better just to list them explicitly, since he lists all Z-related tetrachords and pentachords.

31. Harris (1989) follows this same procedure, except that he draws from a wide body of musical literature, and he is explicitly not working from a pcset-influenced background. His system of chord families is much more complicated than Parks', but it has a great deal of solid and thoughtful musicianship behind it. His proposal deserves closer attention by the music-theoretic community.

32. If we adjust cutoff values, we can re-include 5-2 and 5-3, although 5-4 and 5-8 ([01236] and [02346]) will also come along.

33. The term refers to a probabilistic process that is repeatedly carried out (as in, "repeated throws of the dice at Monte Carlo") to determine a result. The technique is often used to simulate physical processes; it has some relation to the stochastic algorithms used to generate a number of Xenakis' compositions.

34. The speculation two paragraphs ago about Quinn's hexachord group B, whose members all had fairly high connections to other clusters (unlike members of the remaining six groups), applies: might an MDS analysis of hexachords show all his group B hexachords to be "garbage" set-classes?

35. As an example, if I alternately play 0137/0146 in closest spacing with the same bass note I feel a definite sense of tonic/dominant, presumably due to the imbedded minor triad in 0137 and the semitone neighbor in the soprano in 0146 acting like a leading tone. I could easily envision exploiting this type of perceived relationship in a composition.

36. Using Tn-equivalence raises a number of questions; in particular, would the dimensionality of the configurations change? Since a major priority for this essay was to compare RECREL with several other functions, all of which use Tn/I-equivalence, it was necessary to eliminate the B-forms of asymmetrical set-classes from consideration, obviating all such issues. Morris (1995) lists several other possible levels of abstraction. Most of these have as yet received little or no attention by music theorists, a situation which sorely needs correction.

37. To cite one example, Samplaski (2004) found support for grouping interval-classes by category of acoustical dissonance, rather than treating them as separate isolated entities. If other studies confirm this result, that would strongly imply a need for substantial modification of existing similarity measures.

End of footnotes