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

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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.

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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 chromatic 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 measures 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 context was followed on successive trials with each tone of the chromatic scale (in randomorder); the final tone is called a probe tone (Krumhansland Shepard 1979). (For example, the sequence consisting of F-, G-, and C-majortriads was followed by some order-

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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 distances between their tonic triads.2For example, to obtain the distance 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 correlated. 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 considered next is whetherthe KKcor and KKdistvalues are bettermodeled if

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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). Under the numberof transformationsfor each triad is an example of a shortestpath.5 Therecan be alternativeshortest-paths with the same num- ber of transformations.In most cases these have the same numberof each kind of transformation(for example, to go from the C-majortriad to the F-minortriad, there are two possibilities:RLP or PLR,both of which have one R, one L, and one P). In a few cases, two possible shortest-pathsexist with differentnumbers of each of the transformations(for example,mov- ing from the C-majortriad to the Db-majortriad can be accomplishedby PLRL, with one P, one R, and two Ls, or by RPLP,with two Ps, one R and one L). The values in parenthesesindicate alternativeshortest-path routes;the data analysis showed these producedslightly less clear results and will not be consideredfurther. All threemodels were tested using multipleregression which finds the best-fittinglinear function predicting the dependentvariable from a num- ber of independentvariables. For example, in testing how well KKcor can be predicted by Model 1, KKcor is the dependent variable that is modeled as a weighted sum of the three variablesD, P, and R (shown in the firstthree columns of Tables 1 and 2). The analysis returnsa multiple correlationvalue, R, indicatinghow well the dependentvariable is modeled, with 1 indicating that a perfect fit is obtained.R values are evalu- ated in terms of statistical significance, denoted p, an estimate of the probabilityof the result for randomdata. By convention, if p < .05, the result is consideredstatistically significant. Table 3 summarizesthe results of the multipleregression analyses for 269

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 Model 1: DPR Model 2: PR L Model 3: DPRL Triad D P R P R L D P R L Db 2 1 1 1(2) 1(1) 2(1) 1 1 0 1 DDPR PLRL (or RPLP) PLD D 2 0 0 0(1) 2(2) 2(1) 2 0 0 0 DD LRLR (or RPRL) DD Eb 0 1 1 1 1 0 0 1 1 0 PR PR PR E 1 1 1 1 0 1 0 1 0 1 DRP LP LP F 1 0 0 0 1 1 1 0 0 0 D RL D F# 0 2 2 2 2 0 0(2) 2(1) 2(0) 0(1) PRPR PRPR PRPR (or DDLP) G 1 0 0 0 1 1 1 0 0 0 D LR D Ab 1 1 1 1 0 1 0 1 DPR PL PL A 0 1 1 1 1 0 0 1 1 0 RP RP RP Bb 2 0 0 0(1) 2(2) 2(1) 2 0 0 0 DD RLRL (or LRPR) DD B 2 1 1 1(2) 1(1) 2(1) 1 1 0 1 DDPR LRLP (or PLPR) DLP Table 1 Triaddistances from the C majortriad to all other majortriads measuredby shortestpaths for Model 1: D P R, Model 2: P R L, and Model 3: D P R L. The numbersindicate the numberof each kind of transformation;numbers in parenthesesindicate alternativeshortest paths. Below these are examples of shortestpaths. the KKcor and KKdistvalues in the first two columns. All three models were statistically significant, so the focus will be on the magnitudesof the correlations.Model 1 produced a somewhat better fit to the KKcor values than Model 2, but the two models were equivalenton the KKdist values. Thus, substitutingthe L transformationfor the D transformation did not substantiallychange how well the model fits the data. However, higher correlationswere found for Model 3 than either Model 1 or 2.6 These results suggest that all four transformationsD, P, R, and L con- tributeto the patternin the KKcor and KKdistvalues. The advantageof Model 3 over Model 1 shows the L transformationhas psychological reality independentof the other transformations,D, P, and R. Similarly, 270

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 Model 1: DPR Model 2: PRL Model 3: DPRL Triad D P R P R L D P R L c 0 1 0 1 0 0 0 1 0 0 P P P c 1 1 2 1 1 1 0 1 1 DRPR RPL RPL d 1 0 1 0 2 1 1 0 1 0 DR RLR DR d 0 2 1 2 1 0 0 2 1 0 PRP PRP PRP e 1 0 1 0 0 1 0 0 0 1 DR L L f 1 1 0 1 1 1 1 1 0 0 DP LPR DP ft 0 1 2 1 2 0 0 1 2 0 RPR RPR RPR g 1 1 0 1 1 1 1 1 0 0 DP LRP DP g# 2 1 2 1(2) 0(0) 2(1) 0(0) 1(2) 0(0) 2(1) DDRPR LPL (or PLP) LPL (or PLP) a 0 0 1 0 1 0 0 0 1 0 R R R bb 2 1 0 1(2) 2(2) 2(1) 2 2 0 0 DDP RLRLP(or RPRPL) DDP b 2 0 1 0 1 2 1 0 0 1 DDR LRL DL Table 2 Triaddistances from the C majortriad to all minor triadsmeasured by shortestpaths for Model 1: D P R, Model 2: P R L, and Model 3: D P R L. The numbersindicate the numberof each kind of transformation;numbers in parenthesesindicate alternativeshortest paths. Below these are examples of shortestpaths.

the advantageof Model 3 over Model 2 indicates that, even thoughcom- binations of L and R can substitutefor D, the D transformationhas an independentpsychological reality. The resultsof this analysisof the two sets of values, KKcorand KKdist, showed thatthe model incorporatingall four transformations,D, P, R, and L, fit the empirical values better than the models containing only three transformations,D, P, andR, or P, R, andL. These resultshave two important implications. First, they established the independentpsychological statusof the L transformation.Adding the L transformationimproved the fit of the empirical values over the D, P, and R model developed previ-

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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 KKcor KKdist BPLmus BPLnonmus Model 1: D P R R = .94, p < .0001 R = .91, p < .0001 R = .74,p = .0009 R = .77, p = .0003 Model2: PR L R = .90, p < .0001 R= .91, p < .0001 R= .74, p= .0011 R= .78, p= .0002 Model3: DPRL R = .97, p < .0001 R = .96, p < .0001 R = .83, p < .0001 R = .87, p < .0001 TonalPitch Space R =.90, p < .0001 R = .89, p < .0001 R =.82, p < .0001 R = .73, p < .0001 PitchCommonality R =.85, p < .0001 R = .88, p < .0001 R = .81, p < .0001 R = .83, p < .0001 HorizontalMotion R =.35, n.s. R = .41, p < .05 R = .47, p = .02 R = .78, p < .0001 Table 3 Statisticalresults of the models tested. The four sets of empiricalvalues from the studies by Krumhansland Kessler (1982) and Bigand, Parn- cutt, and Lerdahl(1996) were fit by Model 1: D P R, Model 2: P R L, and Model 3: D P R L as shown in the first three rows of statistics. Three additionalmodels were also tested: the tonal pitch space model (Lerdahl 1988), pitch commonality(Parncutt, 1989), and the actual pitch distancesin the chord tension study (Bigand et al. 1996). ously (Krumhansland Kessler 1982). Psychologically, this can be ex- plainedby the importanceof pitch proximityand the fact thatthe L trans- formationinvolves shifting a single tone by just one chromaticstep. Sec- ond, these resultsestablished the independentpsychological statusof the D transformation,indicating that combinationsof R and L transformations cannot substitutefor the D transformation.This is consistent with empirical results showing that the dominant relation is an important underlying principle in psychological representationsof pitch. Before examiningwhether these resultsextend to two othersets of empiricalvalues, some structureson the toroidalrepresentation will be described.

Structures on the toroidal representation

Examples 2a, b, and c show the subgroupsgenerated by P R, P L, and R L, respectively,on the toroidalrepresentation (Krumhansl and Kessler 1982). Cohn (1997) calls these 'binary-generated'subgroups. The transformationpair P R generatesthree subgroupsof order 8 that cut diago- nally across the toroidal representationfrom upper left to lower right; tonics are separatedby minor thirds.The shortest-pathdistance used in 272

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 Db f Db/C#

(2a) • PR Subgroups Order8 K eb d#

u..L.../C#

(2b) PL Subgroups Order6 G Lt.. L L. d# e d# . d#

S..F L

Db/C# D L... A Dt/C# LF.. L. A\LR \ L.E L- L..Ab (2c) RR RL Group R R R Order24 L8 G Eb L 9\ el/d# ebl: \R ,R L .D b

Example 2 Neo-Riemannianstructures shown on the toroidalrepresentation (Krumhansland Kessler 1982). Example 2a shows the three PR subgroupsof order8. Example 2b shows the four PL subgroupsof order6. Example2c shows the RL group of order24. 273

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 f#, d bb......

Db/ A EDb/#

" C#6 (• A#bAl c+• •. A•.K•f E T" E CAb "Ebb

"-..g#.,"-e "" 0..0

B G Eb eb/d 0 b "b "gD C, IV, \A

D D F*? 13bb

FLIP-LA. -. ,.X. Example 3 The P, R, and L transformationsare superimposedon the toroidal representation(Krumhansl and Kessler 1982). The tones indicatedin outline letters at each vertex are the tones that exchange between that vertex and neighboringvertices. Each hexagon has a common exchanging tone, indicatedin the center of the hexagon surrounded by the small hexagon.

Model 1: D P R can be conceptualizedas distancealong the P R subgroup with the D transformation(the othertransformation in the model) used to move from the referenceC major to the appropriateP R subgroup.The transformationpair P L generatesfour subgroupsof order6 thatcut across the toroidal representationhorizontally. Tonics are separatedby major thirds.The shortest-pathdistance used in Model 2: P R L can be conceptualizedas distancealong the P L subgroupwith the R transformation(the other transformationin the model) used to move from the reference C

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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 majorto the appropriateP L subgroup.The transformationpair R L gen- eratesone complete groupof order24; it is the circle of fifths,with minor keys interspersedbetween their R and L relatedmajor keys. Example 3 shows the Model 2: P R L representation(see also Dou- thett and Steinbach 1998). The exchanging notes are added at each vertex in outline letters. For example, the vertex C majorhas C exchanging with B (E minor), E exchanging with Eb (C minor), and G exchanging with A (A minor) with the L, P, R transformations,respectively. As can be seen, the six verticesof each hexagonhave a commonexchanging tone. For example, the hexagon with the vertices C major,C minor,Ab major, F minor,F major,and A minor sharethe common exchangingtone of C. The common exchangingtone is indicatedat the center of each hexagon (outlined by the small hexagon). Going along the diagonals from lower left to upper right, the circle of fifths appearsin the exchanging tones. Going along the diagonalsfrom upperleft to lower right,three subgroups of order4 separatedby minorthirds appear in the exchangingtones. Thus, the representationderived from the exchanging tones replicates the D transformationand the P R subgroups(Example 2a) found in the toroidal representation.Moreover, the representationof exchanging tones suggests that in regions of the space certaintones might be psychologically primed as potentialexchanging tones. This would account, for example, for the readiness with which listeners assimilate modulationsto closely relatedkeys (Krumhansland Kessler 1982). The discussion so far has focused on only four transformations:D, P, R, and L. It might be of interestto consider the geometricrepresentation of some of the other Riemanniantransformations. Example 4 shows the six Riemannian Schritte described by Klumpenhouwer(1994). These take a majortriad to anothermajor triad (Example 4a), and a minor triad to another minor triad (Example 4b). The transformationsare: Quint- schritt,(Q), Gegenquintschritt(-Q), Terzschritt(T), Leittonschritt(L, to be distinguishedfrom the use of L above and below as meaningLeitton- wechsel), Ganztonschritt(G), and Kleinterzschritt(K). Comparingthe two figuresmakes clear the complementarynature of the transformations for majorand minor triads.In principle,it would be possible to ascertain using the above method whether any of these transformationshas an independentpsychological reality from the other transformationscon- sidered here.7

Judgments of chord tension

An independentset of empiricaldata on triaddistances comes from a studyof musical tension (Bigandet al. 1996).8The essentialdetails of the methodology are:

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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 RiemannianSchritts for MajorTriads (4a)

Db C# A F b # D ) c# f )C -Q T C Ab g# c

B GEb eb)d# b G eb d#

F# D Bb

RiemannianSchritts for MinorTriads f# d bb (4b)

Db C# A F Db C# G c# a f

E c Q Ab g# e-Q K

B G Eb eb'd# b g eb d#

F# D Bb f# dbb Example 4 Shows RiemannianSchritts (Klumpenhouwer 1994) superimposedon the toroidalrepresentation (Krumhansl and Kessler 1982). Example 4a shows the Schrittefor majortriads; Example 4b shows the Schrittefor minor triads.

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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 1. Two groups of subjects participated,fourteen music conservatory students and fourteen musically naive students.The data from the two groups of studentswill be denoted BPLmus and BPLnonmus, respectively. 2. Each trialbegan with a shortsequence in C major,followed by three chords:C major,X, C major.The X chordwas any of the twelve possible major,minor, major-minor seventh, or minor seventh chords. Only the data for the major and minor triads will be considered because these can be compareddirectly with the KKcorand KKdist values analyzed above. 3. The subject's task was to rate the tension producedby the second chord of each trial on a 12-point scale. The tension ratings will be taken as an indirect measure of psychological distance from C major.9 Before turningto other models consideredin Bigand et al. 1996, the BPLmusand BPLnonmusdata will be analyzedwith Models 1, 2, and 3. The correlationvalues for these threemodels are shown in the two right- hand columns of Table 2. As before, Models 1 and 2 achieved approxi- mately equally good fits to the data,but Model 3 provideda betterfit then either of the other models.' The advantageof Model 3 over Model 1 shows the independentpsychological statusof the L transformation,while the advantageof Model 3 over Model 2 shows the independentpsychological status of the D transformation.Compared with the model fits to KKcorand KKdist, the presentcorrelations were somewhatlower, andthe optimal weights for the linear model were somewhatdifferent. However, again, the best fittingmodel was Model 3: D P R L. In general,the musicians' data were the least well fit by any of the three models, suggesting that furtheranalysis is needed to understandmusicians' cognitive representationof triad distances, with an eye towardunderstanding the rela- tionships between the differentmodels.

Other models of triad distance

The originalarticle (Bigand et al. 1996) tested a numberof othermod- els. The firstmodel was Lerdahl's(1988) tonal pitch space theory which consists of three components.The firstcomponent is pitch-classproxim- ity, which measuresthe numberof distinctiveelements in the basic pitch space. The basic pitch space consists of five levels: chromatic,diatonic, triadic,fifth, androot. In a given key region, the root is representedat five levels, the fifth above the root at four levels, the third above the root at three levels, other scale tones at two levels, and nonscale tones at one level. Pitch-class proximityis the numberof distinctive elements of the second triadcompared with the first (in this case, C major).The second

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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 componentis triadproximity within a key, measuredas distance around the diatonic circle of fifths (C-G-D-A-E-B-F-C).The third space is the distances between keys or regions, measuredon the chromaticcircle of fifths (C-G-D-A-E-B-F#-Db-Ab-Eb-Bb-F-C).The distances measuredin all three spaces are summedto give the predicteddistances from C major to each triad.Any triadcan, of course, be representedin a numberof different key regions, so Lerdahluses a shortest-pathcriterion. For example, the shortestpath from C majorto A majoris V/d. Table 3 shows the correlationsbetween the tonal pitch space predictions and the four data sets considered here. For all except BPLmus, Model 3: D P R L gives a better account of the data;the two models are essentially equivalent for the BPLmus data. This result suggests that, comparedto nonmusicians,musicians more strongly weigh a hierarchi- cal model of embeddedtone, triad,and key (region) distancesin judging chordtension. In summary,of the variousmusic-theoretic models of triad distance, the strongestempirical supportis for Model 3: D P R L, with the exceptionjust noted of thejudgments of chordtension madeby musicians. A psychoacoustically-basedmodel of chord tension was also tested (Bigand et al. 1996). It is based on known propertiesof sensory process- ing, independentof cognitive representationsthat are acquiredthrough learning.The model, called pitch commonality,was developedfrom previous work on virtualpitch (Terhardt1974; Parncutt1989). Accordingto this approach,the perceivedpitches of a triadare not limitedto the notated pitches and their harmonics. Virtual pitches are also perceived to the extent that the spectral (physically present) pitches approximatea har- monic series above the virtualpitch. So, for example, an Eb-majortriad weakly implies the pitch C. The pitch commonalitymodel computes the predictedpitch salience of each X chord and comparesthese (by correla- tion) with the predictedpitch salience of the referenceC-major triad. The resultingpitch commonality predictions were correlatedwith the four sets of empiricalvalues, as shown in the next line of Table3. The resultswere very similarto those for the tonal pitch space model (Lerdahl1988). That is, for all except BPLmus,Model 3: D P R L gave a betteraccount of the data; the two models were essentially equivalentfor the BPLmus data. Overall,Model 3: D P R L providedthe most accuratepredictions for the data. The final analysis consideredthe voicing of the chords in the Bigand et al. (1996) experiment.The independentvariable used in this model, horizontalmotion, is the sum of the sizes of the intervalscovered by each voice when passing from the C-majorchord to the X chord.This is com- puted from actual pitches, not abstract pitch classes. The horizontal motionvalues, when correlatedwith the fourempirical sets of values, produced the results shown in the last row of Table 3. As would be expected,

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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 given that these values are based on the particularsof the Bigand et al. (1996) experiment,the correlationswere low for the KKcor and KKdist values. The correlationbetween horizontalmotion and the BPLnonmus data was relativelyhigh, althoughlower than Model 3: D P R L. This is interestingin the presentcontext because it shows a direct effect of tone distance on the chord tension judgments. Horizontal motion added to Model 3: D P R L resultedin a very good fit of the BPLnonmusdata,11 suggestingthat tone distancemeasured both in termsof actualpitches and abstractpitch classes (as embodied in Model 3: D P R L ) are psychologically highly salient for nonmusicians. The analysis of the datafrom the Bigand et al. (1996) studyfound different results for musicians and the nonmusicians. For the musicians, threemodels performedapproximately equally: the D, P, R, and L model, the tonal pitch space model, and the pitch commonalitymodel. That the tonal pitch space model performedrelatively well for these datasuggests thatmusicians employ a hierarchicalmodel of embeddedtone, triad,and key (region) distancesin judging chordtension. That the pitch commonality model also performedwell reinforces Parncutt's(1989) claim that music-theoreticconstructs such as those tested in the D, P, R, and L and tonal pitch space models have a psychoacousticbasis.12 For the nonmusicians, the model using the four transformations,D, P, R, andL, provided the best fit of the empiricalvalues, as it had for the Krumhansland Kes- sler (1982) values. In addition, the nonmusicians'data correlatedquite stronglywith the Bigand et al. (1996) measureof horizontalmotion. This suggests thatnonmusicians are especially sensitiveto pitch proximity,the principle underlyingthe neo-Riemanniantransformations, measured in both actual and abstractpitch class distances.13

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1. Onlythe directlyrelevant aspects of the methodswill be described. 2. This is justifiedgiven the similarratings for the tonic triadsand the cadences, althoughit mightbe objectedthat this confuses tonic triads with the keys schemat- ically representedby the cadences.The similaritybetween the probe-toneprofiles for triadsand cadencescan be understood,however, because the latterall ended on the tonic triad. 3. The differencesbetween KKcor and KKdist,although minor, may be important becausethe latterare distances in a geometricrepresentation that abstracts the pat- ternsunderlying the correlations. 4. Thedominant transformation, D, is shownonly for majortriads; it appliesequally to minortriads. 5. The conventionwill be used thatthe left-mosttransformation is the firsttransfor- mationapplied. 6. The followinganalysis was doneto demonstratethat the improvementof Model3 was not simplya matterof addingone moreindependent variable (and thus reducing the degreesof freedomby one). ForModel 1, 2, and3, the unweightedsum of the independentvariables was correlatedwith the KKcorand KKdistvalues. All thesecorrelations have the samenumber of degreesof freedom.This produced the followingresults: Model 1, correlationof SUM(D,P,R)= .91 and .88 for KKcor and KKdist,respectively. Model 2, correlationof SUM(P,R,L)= .73 and .78 for KKcorand KKdist, respectively. Model 3, correlationof SUM(D,P,R,L)= .95 and .95 for KKcorand KKdist, respectively. Again, by thismeasure Model 3 fit the values consistentlybetter than the othertwo models. 7. Kopp(1995) criticizedLewin for privilegingthe P, R, andL transformations,and emphasizedinstead the third-relatedtriads of the samemode (shownin Example 4). 8. I am gratefulto EmmanuelBigand for providingthe datafor the musiciansand nonmusiciansin the study(Bigand et al. 1996). 9. This is supportedby the significantcorrelations between BPLmus and BPLnon- mus data,on the one hand,and the KKcorand KKdistvalues, on the other.The correlationsfor the BPLmuswere r = .81 and .83 for KKcorand KKdistvalues, respectively;the correspondingcorrelations for the BPLnonmuswere r = .73 and .77. Thesesignificant correlations show that these different empirical measures tap into the sameor similarcognitive representations of triaddistance. 10. To demonstratethat the improvementof Model 3 was not simply a matterof addingone moreindependent variable (and thus reducing the degreesof freedom by one), the unweightedsum of the independentvariables was correlatedwith the BPLmusand BPLnonmus values for Model 1, 2, and3. Thisproduced the following results:Model 1, correlationof SUM(D,P,R)= .74 and .75 for BPLmusand BPLnonmus,respectively. Model 2, correlationof SUM(P,R,L)= .68 and .78 for BPLmusand BPLnonmus, respectively. Model 3, correlationof SUM(D,P,R,L)= .82 and .78 for BPLmusand BPLnonmus,respectively. Again, by this measure Model 3 fit the valuesbetter than the othertwo modelsexcept for the equalfits of Model2 andModel 3 for the BPLnonmusdata. 11. The multiplecorrelation predicting BPLnonmus using Model 3 D P R L andhor- izontal motion was R = .95, p < .0001

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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 12. Thatthese three models have strong underlying commonalities is supportedby the fact thatthe tonalpitch space and the pitchcommonality models can be quitewell accountedfor by Model3 D P R L, with multiplecorrelations of R = .94 and .91, respectively. 13. I am gratefulto RichardCohn, Fred Lerdahl, and JohnClough for commentson an earlierversion of this manuscript.

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