Perception & Psychophysics1974. Vol. 15, No.2, 238·242

Nonmetric scaling of loudness and pitchusing similarity and difference estimates"

SCOTT PARKER and BRUCE SCHNEIDERColumbia University. New York. New York 10027

In Experiment I. nonrnetric analyses of estimates of similarity and difference were used to generate a scale ofloudness for 1,200-Hz tones varying in intensity. For both similarity and difference estimates, loudness was found togrow approximately as the 0.26 power of sound pressure. In Experiment 2, nonrnetric analyses of estimates ofsimilarity and difference were used to generate a scale of pitch for 83.3-{jB pure tones varying in frequency. For bothsimilarity and difference estimates, pitch was found to vary with frequency in accordance with the mel scale.

To develop subjective scales of loudness and pitch,most investigators rely on the ratio and interval scalingprocedures developed by S. S. Stevens (1958). In theseprocedures, the S is required, in one fashion or another,to directly report ratios or intervals of sensorymagnitude. These judgments are used to construct scalesof the amount of a given attribute. They typically showthat sensory magnitude is a power function of soundpressure and that pitch increases nonlinearly with tonefrequency.

There is reason, however, to be somewhat concernedabout the validity of scales established by directestimation procedures. In a typical ratio estimationexperiment, Ss are asked to assign numbers to stimulivarying along a single dimension such that the ratiosamong the numbers reflect the ratios of magnitudesamong the stimuli. To argue from these experiments thatpsychological magnitude is a power function ofintensity, it is necessary to assume that these numbersare multiplicatively related to psychological magnitudes.Unequivocal determination of the function relatingsensory magnitude to stimulus intensity requires anexperiment designed to determine a scale of sensorymagnitude in which much weaker assumptions about thenature of the sensory judgments are employed. Recentwork by Shepard (1962a, b) on nonmetric scalingtechniques supplies the basis for such experiments.

In the experiments reported below, scales of loudnessand pitch were developed using nonmetric scalingtechniques. These techniques require that the S ordertonal pairs with respect to how much the elements of apair differ with respect to some attribute. InExperiment I, the elements of a tonal pair were identical

*These data are based on a dissertation by the first author,conducted under the supervision of the second author, andsubmitted to the Department of Psychology, ColumbiaUniversity. in partial fulfillment of the requirements for the PhDdegree. We would like to thank Donald Hood and EugeneGalanter for their helpful comments and suggestions. We wouldalso like to thank Dr. Galanter for making available to us thefacilities of the Psychophysics Laboratory. This research wassupported by NSF Grant GB 36211. Reprints may be obtainedfrom Scott Parker. Department of Psychology. ColumbiaUniversity, New York, New York 10027.

in frequency (l,200 Hz) and differed only in intensity.Ss therefore ordered loudness differences. InExperiment 2, the elements of a tonal pair were identicalin intensity (83.3 dB re 0.0002 dynes/em") and differedonly in frequency. Ss in Experiment 2 ordered pitchdifferences. Two different methods were employed toobtain a rank ordering of the loudness and pitchdifferences. One group of Ss in each experiment wasasked to estimate magnitudes of differences in loudness(Experiment 1) or pitch (Experiment 2). A second groupof Ss was asked to estimate magnitudes of similaritiesbetween the elements of a pair. Difference estimateswere interpreted as distances along a psychologicalcontinuum, and similarity estimates were interpreted asproximities (order-inverse with distance). The rankorders of the difference and similarity judgments wereused to determine interval scales of the sensoryattributes in question (loudness or pitch). Aconcordance of the results for difference and similarityjudgments would suggest that the geometricrepresentation for the stimuli is in fact correct, since itcan be obtained via these different experimental tasks.(A discordance of results might mean that theexperimental approach used here is unstable andunreliable, or it might mean that instructions to considerdifferences induce perceptual structures unlike thoseinduced by instructions to consider similarity.) Theinterval scale representations of loudness and pitchobtained from these judgments of difference andsimilarity were compared to tonal intensity andfrequency. In particular, two objectives of the studywere to determine (1) whether Stevens's power law orFechner's logarithmic law was a better description ofthe loudness function, and (2) whether Stevens andVolkmann's (1940) mel scale was appropriate for pitch.

METHOD

Experiments I and 2

Subjects

Nineteen of the 20 Ss were Columbia Universityundergraduates or graduate students in Columbia's Department

238

Table 1Stimuli Used in Experiments 1 and 2

Experiment 1(dB re 0002 dynes/em")

5056*6068728086*9498

104

Experiment 2(Hz)

460525645760*830920

10601130*12901370

NONMETRIC SCALINGOF LOUDNESS AND PITCH 239

* in Table 1) was presented and identified. When the S wassatisfied with that pair, the 45-tone pair sequence was presented.In the second and third sessions, no instructions were given andeach tone-pair sequence was preceded by an identifiedpresentation of the standard pair. Ss listened to each pair as longas they wished and then spoke a number.

The sequences in the second and third sessions were such thateach tone pair appeared twice before and twice after each othertone pair. Also, each tone appeared equally often on eachoperative position of the three-position switch. The secondsequence of the third session was identical to that used in thefirst session.

RESULTS

Note-All stimuli in Experiment 1 are at 1200 Hz. All stimuliin Experiment 2 are 83.3 dB re .0002 dyneslcm",

of Psychology. One S was a graduate student at the University ofPennsylvania Department of Linguistics. Their ages ranged from18 to 28 years. All Ss claimed to have normal hearing. Eight ofthe Ss had had some musical training. Seven had previouslyserved as Ss in magnitude estimation experiments. None was paidfor participation.

Apparatus

Calibrations and listening conditions were identical to thoseused by Carvellas and Schneider (1972), except that the Ss sat inan Industrial Acoustics sound-resistant booth, Model 300.

Procedure

The 10 tones used in each experiment are listed in Table 1.Each S served in three experimental sessions. In the first session,the Ss were presented with the 45 pairs of unequal tonesconstructible from the set of 10 tones. In the second and thirdsessions, the 45 tone pairs were presented twice, with a lo-minbreak separating the presentations. In Experiment 1, five Ssestimated the loudness difference of the tones in each pair, andfive Ss estimated the loudness similarity of the tones in eachpair. In Experiment 2, five Ss estimated pitch similarity and fiveSs estimated pitch difference.

Prior to hearing the first-session tone-pair sequence, the Ssestimating difference in Experiment I (loudness difference) wereinstructed as follows: "This is an experiment on your perceptionof difference. You will hear pairs of tones. The tones in a pairwill differ in loudness. Your task is to decide how different thetones in a pair are and to assign a number to that difference. Youwill first hear a pair of tones whose difference we will assign thenumber 60, to give us a starting point. For any subsequent pair,if the tones in that pair sound twice as different as did those inthe first pair, assign it the number 120; if in some pair, the tonessound half as different as did those in the first pair, assign it thenumber 30. You may use any positive number-integer, fraction,or decimal. You may not use negative numbers or zero. Arethere any questions?"

The instructions were similarly constructed for all Ss, exceptthat they concerned pitch rather than loudness in Experiment 2,and concerned similarity rather than difference for half the Ss ineach experiment. Also, Ss estimating similarity had the first pairdesignated as having a similarity of 20 rather than a difference of60.

Ss estimating similarity commonly claimed that they did notunderstand what they were to do. They were then told to assigna number to "how much the tones sound alike."

The S was then led into the booth and shown how to wear theearphones and how to operate the three-position switch and theintercom. Two minutes later, the standard pair (tones marked by

Experiment I-Loudness

For both similarity and difference estimation, each S'sfirst estimate was discarded. The geometric mean of theremaining four estimates was computed for each of the45 stimulus pairs. The geometric mean was chosen as ameasure of central tendency since the variance ofmagnitude estimates generally increases with the mean.The geometric means were then ranked, within Ss, from1 to 45. Kendall's coefftcient of concordance, W (Siegel,1956, pp. 229-238), for these rank orders was found tobe 0.93 for the ftve Ss estimating loudness similarity and0.94 for the five Ss estimating loudness difference. Thus,agreement among Ss as to rank order of loudnesssimilarity was good, and agreement on rank order ofloudness difference was equally good.

For the similarity estimates and the differenceestimates, the arithmetic mean of the ranks across thefive Ss was computed for each stimulus pair. These meanranks were then themselves ranked from 1 to 45,providing an ordinal index of each group's similarity ordifference estimates. These ranks were reversed (R' =46 - R) for the similarity group, but not for thedifference group, and the ranks were used as input to anonmetric scaling computer program (Carvellas &Schneider, 1972). The use of these averaged ranksassumes that geometric distance is monotone increasingwith increasing loudness difference estimates, andmonotone decreasing with increasing similarity estimates(since these ranks were reversed). Since 1,200-Hz tonesvary minimally in pitch over the range of intensities used(Stevens, 1935), the basis for all judgments waspresumed to be loudness variation among the tones.Hence, the analysis was one-dimensional, and the toneintensities in decibels were used as the startingconftguration in both cases.

Stress, Kruskal's (1964) measure of goodness of fit,was computed for the outputs of the nonmetricprogram. Stress measures the discordance between thepredicted distances, ds, and a set of distances, d, that are(1) monotonically related to the original distances (i.e.,preserve the rank-ordering) and (2) are as much like theds as they can be within the restrictions imposed by (1).Stress is given by [~(d - d)2/r.d2] '12, often expressed asa percentage. Notice that perfect ordinal agreement

240 PARKER AND SCHNEIDER

produces d :=: d, and, in this case, stress> O. The stressvalues were 4.7% for difference estimates and 3.1% forsimilarity estimates. Kruskal (1964) states that stressvalues of 5% or less indicate "good" agreement betweenthe ds and ds.

The index of metric determinacy (M), which wasoriginally developed by Shepard, was estimated fromYoung's (1970) nomogram. M is the squared Pearsoncorrelation coefficient between the true distances(whose rank ordering serves as the input to thealgorithm) and the ds produced by the algorithm. Hence,M varies between 0 and 1, and M:=: I means that the truedistances have been perfectly reconstructed. In noempirical investigation using these techniques are thetrue distances known, but Young provides a nomogramfor estimating M from the number of points, number ofdimensions, and stress-all of which are available. Theresult is that, in nonmetric scaling analysis, if M issufficiently high (above .98, say), thepoint coordinatesproduced by the algorithm are properly regarded as aninterval scale representation of the original points. Inthese experiments, M was, conservatively, 0.98 for thedifference estimates and 0.99 for the similarityestimates. Thus, for both experiments, the projectionvalues achieved from the nonmetric program may betaken as an interval scale representation of stimulusloudness.'

One objective of this study was to determine whetherloudness was related to sound pressure as described byFechner's law, Stevens's law, or something elseentirely. Since the projection values, P;, provide intervalscale representation for loudness, Li :=: aPi + b, where L,is the loudness of Stimulus i, Pi is the projection valuefor Stimulus i (from the nonmetric scaling program), anda and b are constants. If loudness is as described byFechner's law, L is linear with log I, where I is stimulussound pressure. Since decibels are a logarithmictransform of sound pressures, Fechner's law states L islinear with stimulus intensity in decibels. Hence, P islinearly related to stimulus intensity in decibels, ifFechner's law is correct.

Stevens's law is L, :=: kIf. Thus, the present datafollow Stevens's law if aPi + b :=: kl]'. Equivalently, Pi +(b/a):=: (k/a)If, or Pi + b' :=: k'If, where b' is unknown.Taking logarithms on both sides of the equation,Stevens's law describes thepresent data if there is a b'such that log (Pi +b') :=: nlog(Ii) + log k', i.e., if for someb', 10g(Pi+b') is linear with stimulus intensity indecibels. A value of b' was therefore sought that wouldincrease the squared correlation coefficient (r 2

) between10g(Pi +b) and decibels. The values of r2 for numerouschoices of b' were computed, and b* designates that b'which maximized r2

• The numbers Pi' :=: 100(Pi + b *) arereferred to as adjusted loudness projections.

Figure 1 shows plots of adjusted loudness projectionsvs stimulus intensity in decibels for both loudnesssimilarity estimates and loudness difference estimates.The ordinate is spaced logarithmically in Panels a and c

32 ISIMI~ARITY OIFFEI'lENCE

16

!

~....

8u

iIII 4

I@ 3S

i 30

2S

20

IS

10

80 eo 10 10 10 100 110 eo eo 10 .0 10 100 110

SOUNO ""EStUllE ~VEL. INOECIIlEL.t

Fig. 1. Adjusted loudness projections (see text) as a functionof stimulus intensity in decibels. Notice that the ordinate isspaced arithmetically in Panels b and d, and logarithmically inPanels a and c.

and arithmetically in Panels band d. Best-fitting straightlines, determined by the method of least squares, aredrawn in each panel. It can be seen that there are nosystematic deviations from linearity in the upper panels,whereas the point configurations in the lower panels,ppear to be concave upwards. Linearity in the upperpanels indicates conformity of the data with Stevens'slaw; linearity in the lower panels would indicateconformity of the data with Fechner's law. Values of r2

are 0.998 and 0.997 for Panels a and c, and 0.949 and0.963 for Panels band d. These data, then, are not inaccord with Fechner's law, but do conform to Stevens'slaw. The best estimated for n in the formulation P; :=:kIf are 0.27 for similarity estimation and 0.24 fordifference estimation.

Experiment 2-Pitch

As in Experiment 1, each S's first estimate wasdiscarded. The geometric mean of the remaining fourestimates was computed for each of the 45 stimuluspairs. The geometric means were ranked within Ss from1 to 45. Kendall's coefficient of concordance for theserank orderings was 0.92 for pitch similarity estimatesand 0.95 for difference estimates. Here again, the Ssagreed both on the rank order of pitch similarities andon the rank order of pitch differences.

For the similarity estimates and for the differenceestimates, the arithmetic mean of the ranks across thefive Ss was computed for each stimulus pair. These mean

NONMETRIC SCALING OF LOUDNESS AND PITCH 241

Fig. 2. Pitch prqjections (see text) u a function of stimuluspitch in melll/lOO.

ranks were then themselves ranked from 1 to 45. Theranks were reversed for the similarity estimates but notfor the difference estimates, and these ranks were usedas input to the nonmetric analysis. Since 83.3-dB tonesvary minimally in loudness over the range of frequenciesused (Fletcher & Munson, 1933), the basis for alljudgments was presumed to be pitch variation among thetones. Hence, the analysis was one-dimensional, and thefrequencies of the tones in hertz were used as thestarting configuration. The projections, Ph obtainedfrom the pro~ram were multiplied by 100, pt =Pi X 100. The Pi are referred to as pitch projections.

Stress was computed for the output of the program.The value of stress for the similarity estimates was 6.0%;that for the difference estimates was 3.5%. Thus,agreement between the ds and cis is good for thedifference estimates, and fair for the similarity estimates[Kruskal (1964) states that a "fair" fit is indicated by astress between 5%and 10%]. Estimates of M, the indexof metric determinacy, were made from Young's (1970)nomograms. M was, conservatively, .98 for the similarityestimates and .99 for the difference estimates. Thus, thePi in both cases may be taken as an interval scalerepresentation of stimulus pitch.2

Figure 2 shows a plot of pitch projections vs stimuluspitch in mels/IOO (Stevens & Volkmann, 1940, p. 336)for both the similarity and difference estimates.Best-fitting straight lines, determined by the method ofleast squares, are drawn in each panel. There are nosystematic deviations from linearity in either panel.Values of r2 for these plots are 0.992 for the similarityestimates and 0.996 for the difference estimates. Thus,agreement is quite good between the pitch projectionsand the mel scale.

DISCUSSION

The high inter-S agreement, low stress values, and high

(estimated) M values indicate that interval-scalerepresentations for loudness and pitch can be obtainedfrom perceptual-interval scaling experiments in which Ss'responses are treated as ordinal proximity indices. Inboth experiments (loudness and pitch), the interval scalerepresentations were essentially identical for two distinctproximity indices (similarity and difference).

The representation found for the tones varying inintensity (Experiment I) is a validation of the notionthat psychological loudness is a power function ofstimulus intensity rather than a logarithmic one; i.e.,that loudness is given by Stevens's rather than byFechner's law. Rule et al (1970), using the rank-orderproperties of judgments of area difference for pairs ofcircles, also found that their scale values were a powerfunction of the actual areas of the circles. Similar resultswere obtained from lifted weights in the sameexperiment. The Rule et al study and the loudness dataof the present one indicate that for judgments ofperceptual difference, a power function representationof the stimuli is often appropriate. Furthermore, toobtain this representation, one need only assume that Ss'magnitude estimates are monotonic with perceptualdifferences. This is much weaker than the assumptionemployed in direct estimation techniques: namely, thatthe numbers generated by the Ss are proportional tosensory magnitudes. The fact, however, that thenonmetric techniques also result in a power functionrepresentation of sensory magnitude lends support tothe assumptions underlying the direct estimationtechniques.

In the present experiment, both the similarity and thedifference estimates produced spatial representationsthat were essentially identical power functions ofstimulus intensity (exponents of 0.27 and 0.24 forsimilarity and difference, respectively). This suggeststhat the perceptual structures underlying judgments ofloudness difference, loudness similarity, and directestimates of loudness are all power functions of stimulusintensity. And, with respect to pitch, the perceptualrepresentation is identical for all three kinds ofjudgments. A study by Markley et al (1969), however,suggests that this convergence to a single representationfor these three kinds of judgments may not hold for allsensory continua. They had Ss rate the similarity of pairsof lines, and used the rank-order properties of thesesimilarity judgments to determine a scale of line length.They found that their scale values were a logarithmicrather than a power function of actual line length. Thus,in this instance, a different perceptual structure appearsto underlie judgments of line length similarity ascompared to direct estimations of line length where apower function representation holds (exponent close to1.0). The reasons why similarity, difference, and directestimates converge on the same representation forloudness and pitch, but apparently not for line length,remain to be determined.

The exponents of the best-fitting power functions(0.27 and 0.24 for similarity and difference,

242 PARKER AND SCHNEIDER

respectively) are approximately equal to each other, butare at variance with previously reported values obtainedfrom straightforward magnitude estimation experiments,where the exponent for monaural loudness isapproximately 0.54 (e.g., Reynolds & Stevens, 1960).The sources of the discrepancy in exponent between thepresent results and those of the more traditional metricscalingmethods are not clear.

However, there are some results on loudness that arein accord with those of the present experiment. Garner(1954), combining equisection and fractionation datafor 1,000-Hz tones ranging from 50 to 110 dB re .0002dynes/em? developed the lambda scale for monauralloudness. Loudness in lambda units grows as the 0.26power of sound pressure. Garner shows, in addition, thatthe lambda scale provides a better account of previousbisection and equisection experiments than doesStevens's (1956) sone scale. The data of Beck and Shaw(1961), who also worked with loudness differences,agreed more closely with the lambda scale than with thesone scale.

The results of Experiment 2 (pitch) are in accord withStevens and Volkmann's (1940) revised mel scale. Therecovery of the mel scale is also in agreement with theresults of Carvellas and Schneider (1972). Theseexperiments all involve the Ss' responding on the basis ofpitch intervals.

An interesting aspect of the results of Experiment 2(pitch) is the absence of any phenomenon like octavegeneralization. Two pairs of tones in the stimulus arrayhad a frequency ratio of2:1 (460 and 920 Hz: 645 and1,290 Hz), and three of the Ss (two estimatingdifference and one estimating similarity) affirmed uponinquiry that they had heard tone pairs with octaveseparation. Yet, in neither the similarity nor thedifference portion of the experiment was there anysuggestion in the data that octaves sounded alike forthese Ss. Octave generalization has been found inconditioning studies with rats (Blackwell & Schlosberg,1943), pigeons (Gerry, 1971), and humans (Humphreys,1939; Bersh, Notterman, & Schoenfeld, 1956). Theremay well be some set of instructions for experiments ofthe present type that would produce evidence of octavesimilarity, but it is disheartening that "how much thetones sound alike" is inadequate to the task.

REFERENCES

Beck, J., & Shaw, W. A. Ratio-estimations of loudness intervals.American Journal of Psychology, 1967, 80, 59-65.

Bersh, P., Notterman, J., & Schoenfeld, W. N. Generalization tovarying tone frequencies as a function of intensity ofunconditional stimulus. Publication 56-79, School of AviationMedicine, USAF, Randolph AFB, Texas, 1956.

Blackwell, H. R., & Schlosberg, H. Octave generalization, pitch

discrimination, and loudness thresholds in the white rat.Journal of Experimental Psychology, 1943,33,407-419.

Carve lla s, T., & Schneider, B. Direct estimation ofmultidimensional tonal dissimilarity. Journal of the AcousticalSociety of America, 1972.51,1839-1848.

Fletcher, H., & Munson. W. A. Loudness, its definition,measurement, and calculation. Journal of the AcousticalSociety of America, 1933.5,82-108.

Garner, W. R. A technique and a scale for loudnessmeasurement. Journal of the Acoustical Society of America,1954, 26, 73-88.

Gerry, J. E. Peak shift on the tonal frequency continuum: Theeffects of extinction and punishment. Psychonomic Science,1971, 23, 33-34.

Humphreys, L. G. Generalization as a function of method ofreinforcement. Journal of Experimental Psychology, 1939,25,361-372.

Kruskal, J. B. Multidimensional scaling by optimizing goodnessof fit to a non-metric hypothesis. Psychornetrika, 1964, 29,1-27.

Markley, R. P., Ayers, D., & Rule, S. J. Similarity judgments ofline length. Perception & Psychophysics, 1969,6,58-60.

Reynolds, G. S., & Stevens, S. S. Binaural summation ofloudness. Journal of the Acoustical Society of America, 1960,32, 1337-1344.

Rule, S. J., Curtis, D. W., s: Markley, R. P. Input and outputtransformations from magnitude estimation. Journal ofExperimental Psychology, 1970,86,343-349.

Shepard, R. N. The analysis of proximities: Multidimensionalscaling with an unknown distance function. l. Psychometrika,1962a, 27,125-140.

Shepard, R. N. The analysis of proximities: Multidimensionalscaling with an unknown distance function. II. Psychornetrika,1962b, 27, 219-246.

Siegel, S. Non-parametric statistics for the behavioral sciences.New York: McGraw-Hill, 1956. Pp. 229-238.

Stevens, S. S. The relation of pitch to intensity. Journal of theAcoustical Society of America, 1935,6,150-154.

Stevens, S. S. The direct estimation of sensorymagnitude-loudness. American Journal of Psychology, 1956,69, 1-25.

Stevens, S. S. Problems and methods of psychophysics.Psychological Bulletin, 1958, 54, 177-196.

Stevens, S. S., & Volkmann, J. The relation of pitch tofrequency: A revised scale. American Journal of Psychology,1940, 53, 329-353.

Young, F. W. Nonmetric multidimensional scaling: Recovery ofmetric information. Psychornetrika, 1970,35,455-473.

NOTES

1. The data on difference estimation were reanalyzed usingYoung's (1968) Torsca 9, nonmetric scaling procedure, whicharrives at a set of point coordinates via a different algorithmfrom that used in the present analysis. The value of r2 betweenTorsca's projections and those reported here was0.999-essentially perfect interval scale agreement.

2. Pitch difference estimation data were reanalyzed usingTorsca 9. Again, r2 for the projections produced by the twononmetric analyses was 0.999.

(Received for publication July 26, 1973;revision received September 20, 1973.)