Letters to the Editor

Now one can easily demonstratethat a(a, 0, •b)= Y•m(0,•b)

as the condition satisfied by the characteristic

is an eigenfunction of the homogeneous integral equation. Proceeding as before one finds that the eigenvalue X is given by

numbers,in agreementwith previousauthorso x'•' 1L. G. Copley,"FundamentalResults Concerning Integral Representations

x =

+5=

>'

(9)

Using the Wronskian of the two spherical Bessel func-

tions and again requiring X = 1 in Eq. (9) yield the result

h(ka) =0

(lo)

wave

in Acoustic Radiation,"

J. Acoust. Soc. Am.

•44, 28-32 (1968). 2H. A. Schenck,"ImprovedIntegral Formulationfor Acoustic Radiation Problems,"

Jo Acoust. Soc. Am. 44, 41-58 (1968).

3C. J. GarrisonandP. Y. Chow,"WaveForcesonSubmerged Bodies," J. Waterways, Harbors Coastal Eng. Div., 98, 375-392 (1972).

ASCE

Noncombination of pitch and loudness in multidimensional scaling M. ^. Zagorski o

Memorial Universityof Newfoundland,Departmentof Psychology, Newfoundland,CanadaAIC 5S7 (Received 13 June 1977; revised26 September1977)

Carvellasand Schneider[J. Acoust.Soc.Am. 51, 1839-1848(1972)], usingmagnitudeestimatesof the dissimilarityof sinusoidaltones in a multidimensionalscalingprogram, found that their data fit a cityblock and a Euclideanmetric equallywell, but other researchhas indicatedthat subjectsdo not combine dimensionsof sinusoidaltonesto arrive at an overall estimateof stimulussimilarity. Yet multidimensional scalingseemsto requiresubjectsto combinedimensionsof similarity. In this letter a model is put forward to demonstratethat subjectsneed not combinedimensionsdirectly in order to generatedissimilarity judgmentsthat are very closeto either the city-blockor Euclideanmetrics.Thus it is possiblethat while subjectsdo not combinethe dimensionsperceptuallythey can adopt a cognitivestrategythat leadsone to believethey do. PACS numbers:43.66.Ba, 43.66.Fe, 43.66.Hj

Although Carvellas and Schneider (1972) had scaled sinusoidal tones using direct estimates of tonal dissimilarities and found atwo-dimensional solution correspond-

ing to loudnessand pitch, Zagorski (1969) had previously shown that for

a multidimensional

bisection

task

sub-

jects did not combine loudness and pitch to arrive at an overall judgment of similarity. This result is corrobo-

rated by Zagorski's (1975) finding that subjects did not combine the dimensions with frequency and intensity in a signal-detection task but rather combined decisions

made separately about these dimensions. This latter result is consistent with an analysis by Pollack (1961) of data of Harris et al. (1958). The apparent discrepancy between these two kinds of findings can be resolved by assuming that unidimensional dissimilarities are not combined according to a (Minkowski) metric rule of combination but rather that the subject adopts either a simple cognitive strategy or mix of strategies for determining similarities that yield

resultssimilar to thosepredictedby metricdistances. In other words, the metric does not reflect, in detail, the process by which subjects make judgments of similarity but somehow characterizes average responses. This problem is certainly not peculiar to judgments of pitch and loudness, but might apply to other dimensions

as well.

The purpose of this note is to exhibit a reason-

able process that might account for both kinds of re273

J. Acoust.Soc.Am. 63(1), Jan. 1978

suits.

In particular we address the problem of how a

subject can generate similarity (dissimilarity) judgments which are well fit with Minkowski metrics,

without per-

cePtually combining the dimensions. No claim is made that the process exhibited here represents reality

(thoughit might), the main purpose of this note is to show that there exists at least one process, or class of processes, which might account for these data. The process exhibited here is as follows: first

notices

that the stimuli

differ

The subject

on two dimensions.

Then he attends to the dimension with greatest perceptual difference; that is, the one given the largest dis-

similarity rating. (If the perceptual differences appear to be equal it makes no difference which one he chooses.) Next he adds to his dissimilarity rating for the dimension with the largest dissimilarity an amount to account for the dissimilarity of the other dimension. It is how he chooses this amount that makes interesting differences among the metric representation. If he adds the usual dissimilarity rating which would be associated with the dimension with the smaller perceptual difference in the absence of the other dimensions, then his overall dissimilarity rating would be following a rule that suggests the city-block metric. If he adds nothing, that is, ignores the dimension with the smaller perceptual difference, then his overall dissimilarity rating would suggest the suprememum metric. Let us

0001-4966/78/6301-0273500.80

¸1978 Acoustical Societyof America

273

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274

Lettersto the Editor

consider

the

case

in which

274

he adds

a certain

fraction

of

the usual dissimilarity rating of the dimension with the smaller perceptual difference.

Suppose that ZXxwas the magnitude of the difference on the dimension with the greater perceptual distance

and/xy was the magnitude of the difference on the other dimension. Now suppose the subject attenuates his estimate of the smaller dimension by a factor a; let us say ! a-•, so that his total estimate of dissimilarity D is

• D=ax+•ay

(Ax• Ay) .

One can consider other models for weighting the

smaller dimensionor weig,hting the dimensionsin general, but this one has the important property of giving similarities that lie between the city-block rule and the Euclidean rule. That is, given ZXx>•ZXy

ax +/Xy>•ax + •a ' y >•

+ (ay)•]•/• o

This first part of this inequality is easy to see. The second part, which compares this model to the Euclid-

ean rule can be seen by squaring both (non-negative) sides of the inequality

(•Xx+•ZXy)2 > ZXx 2+ zx

ax• + axay +¬aye>ax• + ay• axay + ¬ay•

Schneider's(1972) findingthat their data fit the cityblock and Euclidean metric equally well. They, in fact, stated a preference for the city-block conceptualization

becausethey did not believe that a metathetic (Stevens, 1957) dimension (pitch) and a prothetic dimension(ioudhess) could be combined according to a Euclidean rule. This model illustrates that it is possible that a simple cognitive strategy can account for similarities which are fit by Minkowski distances, without postulating a rule of combination which is an inviolable property of the perceptual system. Since the model allows the perceptual noncombination of pitch and loudness it is consistent with the findings of Pollack (1961), Harris et al.

(1958), and Zagorski (1969, 1973, 1975). No claim is made that the model put forward here is correct; there is no reason to suspect that this cognitive attentuation of one of the dimensions is a reality. Rather, it is put forward to illustrate the possibility that the fit of metric solutions to similarities data may be accounted for by simple cognitive processes. ACKNOWLEDGMENTS

>1

This work was supported in part by a National Research Council of Canada Grant, number A9804. I

/Xx

a-•+•>1.

would

Remembering that Ax)Ay equality holds.

we see that the latter in-

Thus the model that specifies an attenuation factor of • gives distances lying be[ween the Minkowski metrics

with r = 1 and r = 2.

In fact, all attenuation factors sat-

isfying

2a + a•'• 1 a•4•--1

also

like

to thank

Dr.

Donald

Robinson

and Dr.

Frank Restle of Indiana University for hearing me out on these problems, and would like to thank Fran Locke for helping with some simulation studies related to these problems.

CarveHas, T., and Schneider, B. (1972). of multidimensional tonal dissimilarity,"

"Direct estimation J. Acoust. Soc.

Am. 51, 1839-1848. Harris, J. D., Pickler,

H. S.,

A. G.,

Hoffman,

and Ehmer,

R. H. (1958), "The interaction of pitch and loudnessdis-

give distances which lie be[ween the ciW-block and Euclidean metrics even if they vary randomly from trial to trial. When a = 1, D is a city-block metric and as a gets smaller it gives distances approaching the Euclidean metric. And, as a approaches zero, D approaches the Minkowski

Euclidean distances or identical to city-block distances. If in fact, a subject was generating his similarities based on such a scheme his response then would be fit very well by Euclidean or city-block metrics. This model therefore, is not inconsistent with Carvellas and

metric

with

r

Whena =v•-- i this model gives "distances"which are identical to the Euclidean distanceswhen the ratio, Ay, is equal to zero, one, or infinity.

For other values

of Ax/Ay, the model gives distanceswhich are within 10%of the Euclideandistances. Thus this s,imple model can generate distances that lie bebveen Euclidean and city-block distances and depending on the cognitive attenuation factor a can generate distances very close to

criminations,"

J. Exp. Psychol. 56, 232-238.

Pollack, I. (1961). "On the combinationof intensity and frequency differences in auditory discrimination," Soc. Am.

J. Acoust.

33, 1141-1142.

Stevens, S.S. (1957). "On the psychophysicallaw," Psychol. Rev.

64, 153-181.



Zagorski, M. A. (1969). "Some necessary conditions for the spatial representations of stimulus similarity, "paper presented at Mathematical Psychology Meetings, University of Michigan, Ann Arbor, August, 1969 (unpublished). Zagorski, M. A. (1973). "A topological test of metric models of stimulus similarity," in Indiana Mathematical Psychology Report Series 1973 Report No. 73-10.

Zagorski, M. A. (1975). "Perceptual independenceof pitch and loudness in a signal detection experiment: A processing model for 2ATFC (2ITFC) experiments," Percept. Psychophys. 17, 525-531.

J. Acoust. Soc. Am., Vol. 63, No. 1, January 1978

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Noncombination of pitch and loudness in multidimensional scaling.

Letters to the Editor Now one can easily demonstratethat a(a, 0, •b)= Y•m(0,•b) as the condition satisfied by the characteristic is an eigenfunctio...
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