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Perceived Triad Distance: Evidence Supporting the Psychological Reality of Neo-Riemannian Transformations Author(S): Carol L

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Perceived Triad Distance: Evidence Supporting the Psychological Reality of Neo-Riemannian Transformations Author(S): Carol L Yale University Department of Music Perceived Triad Distance: Evidence Supporting the Psychological Reality of Neo-Riemannian Transformations Author(s): Carol L. Krumhansl Source: Journal of Music Theory, Vol. 42, No. 2, Neo-Riemannian Theory (Autumn, 1998), pp. 265-281 Published by: Duke University Press on behalf of the Yale University Department of Music Stable URL: http://www.jstor.org/stable/843878 . Accessed: 03/04/2013 14:34 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Duke University Press and Yale University Department of Music are collaborating with JSTOR to digitize, preserve and extend access to Journal of Music Theory. http://www.jstor.org This content downloaded from 128.84.127.82 on Wed, 3 Apr 2013 14:34:27 PM All use subject to JSTOR Terms and Conditions PERCEIVED TRIAD DISTANCE: EVIDENCE SUPPORTING THE PSYCHOLOGICAL REALITY OF NEO-RIEMANNIAN TRANSFORMATIONS CarolL. Krumhansl This articleexamines two sets of empiricaldata for the psychological reality of neo-Riemanniantransformations. Previous research (summa- rized, for example, in Krumhansl1990) has establishedthe influence of parallel, P, relative, R, and dominant, D, transformationson cognitive representationsof musical pitch. The present article considers whether empirical data also support the psychological reality of the Leitton- weschsel, L, transformation.Lewin (1982, 1987) began workingwith the D P R L family to which were added a few other diatonic operations. Subsequently,Hyer (1989) reduced the transformationsto the family consisting of only the D P R L transformations.Cohn (1996, 1997) has provided an extensive theoreticalanalysis of neo-Riemanniantransfor- mations excluding the dominant, D, transformation.Thus, one of the issues that arises out of this literatureis whetherthe D transformationis needed given that, as will be shown next, it is equivalentto a combina- tion of R and L transformations. 265 This content downloaded from 128.84.127.82 on Wed, 3 Apr 2013 14:34:27 PM All use subject to JSTOR Terms and Conditions The parallel, P, relative, R, and Leittonwechsel, L, transformations move between majorand minor triads with a minimalchange of one tone. The P transformationshifts the thirdof a majoror minortriad by one chro- matic step holding constantthe tones relatedby a perfect fifth (e.g., C E G to C Eb G). The R transformationshifts the fifth of a major triad (or root of a minor triad)by two chromaticsteps holding constantthe tones relatedby a majorthird (e.g., C E G to C E A). The L transformationshifts the root of the majortriad (or fifth of a minortriad) by one chromaticstep holding constantthe tones relatedby a minorthird (e.g., C E G to B E G). The dominant,D, transformation,shifts a majoror minortriad up or down a fifth (e.g., C E G to G B D or FA C, and C Eb G to G Bb D or FAb C); in this sense it is ambiguous.The D transformationcan also be writtenas a combinationof R and L transformations(e.g. applying the L transfor- mationto C E G producesB E G, thatis, E G B; then applyingthe R trans- formationto E G B producesD G B, that is, G B D). Thus, the D trans- formationmight be redundant. Psychologicalconsiderations, namely numerouspsychological results showing the effect of pitch proximity (see summariesin Bigand, Parn- cutt, and Lerdahl1996; Krumhansl1990), also suggest that the P, R, and L transformationsmight have priorityover the D transformationbecause the latterrequires more than a minimal change of one tone. Given this, the firstthree of these transformations,P, R, and L, might provide a good model of triad distance as measuredin empirical studies. The empirical measuresto be examinedcome from an experimentderiving indirect mea- sures of triad distance from probe-toneratings (Krumhansland Kessler 1982). That study also produceda geometricrepresentation, in the form of a torus, that will be used to representvarious structuresrelated to the neo-Riemanniantransformations. The second set of empiricalmeasures come from an experimentthat collected judgments of chord tension in a fixed tonal context (Bigand, Parncutt,and Lerdahl 1996). Derivation of the torus from probe-tone ratings The toroidalconfiguration was derivedfrom probe-tone ratings (Krum- hansl and Kessler 1982) throughthe following steps:1 1. Ten subjects (average years musical instruction 10.9) were pre- sented with the following contexts in both majorand minor keys: I triad,IV V I cadence,VI V I cadence, and II V I cadence. Each con- text was followed on successive trials with each tone of the chro- matic scale (in randomorder); the final tone is called a probe tone (Krumhansland Shepard 1979). (For example, the sequence con- sisting of F-, G-, and C-majortriads was followed by some order- 266 This content downloaded from 128.84.127.82 on Wed, 3 Apr 2013 14:34:27 PM All use subject to JSTOR Terms and Conditions ing of the tones C, CO,D,.....B.) The subjectsrated on a scale from 1 to 7 how well each probe tone "fit into or went with" the musical contextjust heard.The ratings were shifted to a common tonic for the different major keys and the different minor keys used in the study.Because the four context types producedsimilar results, they were averaged.This yielded a probe-toneprofile for majorcontexts and for minor contexts (the numericalvalues appearin Krumhansl 1990, 37). 2. The probe-toneprofiles were then shifted and correlatedwith one another.This yields an indirect measure of the distances between keys, which will be takenhere to be approximatelyequal to the dis- tances between their tonic triads.2For example, to obtain the dis- tance from C major to F minor, the major profile was shifted to tonic of C, the minor profile was shifted to tonic of F, and the two sets of twelve ratings (for the chromatic scale tones) were corre- lated. A correlationhas a value between -1 and 1, where 1 is the maximum degree of similaritybetween the probe-toneprofiles. In the present context, the correlationsconsidered are those between C major and all other major and minor triads,denoted KKcor (the numericalvalues appearin Krumhansl1990, 38). 3. The correlationsbetween probe-toneprofiles were enteredinto an analysis programcalled multidimensionalscaling (MDS, Shepard 1962, availablein statisticalpackages such as SYSTATand SPSS). This analysis producesa configurationof points such thatdistances are inversely related to the KKcor values. In other words, probe- tone profiles that are similar (highly correlated)are representedby points that are close together in the space. The analysis found an excellent fit to the KKcorvalues with a 4-dimensionalconfiguration in which the points fall on the surface of a torus (the coordinates appearin Krumhansl1990, 42). The interpointdistances, measured by a Euclidean4-dimensional metric, will be denotedKKdist.3 4. Because a torus is the cross-productof a circle with anothercircle, S' X S1, it can be depictedin two dimensionsas shown in Example 1. The horizontalaxis representsthe angulardistance around one of the circles and the vertical axis represents the angular distance aroundthe other circle. In this representation,it is understoodthat the right edge is the same as the left edge, and the top edge is the same as the bottom edge. The configurationwas originallyinterpreted in termsof the threemore familiar transformationsshown in Example la: dominant, D, parallel major/minor,P, and relativemajor/minor, R.4 The question to be consid- ered next is whetherthe KKcor and KKdistvalues are bettermodeled if 267 This content downloaded from 128.84.127.82 on Wed, 3 Apr 2013 14:34:27 PM All use subject to JSTOR Terms and Conditions 9 ' AllP o !D/I DI/ bi• D DI/C / D/ D/ (la) D/ . '. Model 1:DPR D/ D D/ et /id# D/ /d# K d D ... #••...LC#•. ,L.? (lb) E.. L.. L... Model 2: PRL eb (d# ,C P -1 e•,•# D,,•.? - 'd t \ Oc/ .I J C D• f. ... L..." . .. L. \R / I\ (Ic) LLL. J' / Model 3: DPRL L.../b,4 , / // eb'd# IOW /tl 'L?R / D.C?_ . Example 1 The toroidalrepresentation (Krumhansl and Kessler 1982) is depicted in two dimensions, where it is understoodthat the left and right edges are the same and the top and bottom edges are the same. Example 1a shows the toroidalrepresentation of Model 1: DPR super- imposed. Example lb superimposesModel 2: PRL. Example Ic super- imposes Model 3: DPRL This content downloaded from 128.84.127.82 on Wed, 3 Apr 2013 14:34:27 PM All use subject to JSTOR Terms and Conditions the Leittonwechseltransformation, L, is added to the model or is substi- tutedfor the dominanttransformation, D (because D can be expressedin terms of L and R). Modeling triad distances with D, P, R, and L transformations All models to be considereduse the shortest-pathdistances. In other words, the distancebetween one triadand anotherwill be takento be the smallestnumber of transformationsneeded to move from one triadto the other.Three models will be compared: Model 1. The model with D, P, and R transformations,that is, the orig- inal model, shown in Example 1a. Model 2. The model with P, R, and L transformations,shown in Example lb. Model 3. The model with D, P, R, and L transformations,shown in Example Ic. Tables 1 and 2 show the numberof each type of transformationfor each model. The values shown are the transformationsfrom C major to all major triads (Table 1) and from C major to all minor triads (Table 2).
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