ubjective similarity.

nisms of subjective similarity. situation has been simplified as far as possible, so t;lat variation is a&owed only with regard to one single subjective dimension. the subjective magnitu in the dimension under colls’i6~er~ti~~ are more or less simila arity is given by the estimate srJ. Then it is to be ex cted that subjective similarity is a fuuction of the subjective :magnitudes,

41 =_f‘(& J$)*

ur problem is to determine the form af t, dimension of pitch was chosen for

(1)

articular function.

Gcnerat Design

Im order to carry out the investigation outlined above, thz, experi- mental steps were necessary:

(a) Because of the restriction to ensional. variatiol.1 of pitch,

The basic scaling experiments of this report belong to a series of investigations made possible by the Wallenberg Foundation. The particular investigation of

similarity in a unidimensional case as supported by the Swedish State Institute of Psychological and Educational Research. The authors want to express their gratitude for the heip and advice in technical matters received by r. Bertil Jo-

hansson at the Royal Institute of Technology, Stockholm.

HANNES EISLER and <j&TA EKMAN

Fig. 1. Wiring diagram of the apparatus.

A , B Frequency generators (“Philips”). c Switch for connecting one of the gemerators with transformer D. D Transformer for impedance matching. E Transformer, fed from mains, giving 50 c/s of low voltage. F Switch connecting ouiput from transformer E with headphoue G G Dynamic headphone (“Telephonic TD 39”).

a preliminary experiment had to be carried out in order to make it. possible uce tones of equal loudness at various frequencies.

(b) A second experiment was set up in order to obtain estimates of subjective similarity between the tone stimuli chosen for the investigation.

(c) A Srd e~xperiment was arranged in order to obtain a magnitude function OS pit&, from which scale values could be found for thee tones used in the similarity experiment.

For obvious reasons, experiment (a) had to precede the other Iexperi- merits. In order to obtain estimates of similarity as perceived in a nai,ve and

hist.icated way, it was decided to carry out the similarity experiment fore the scaling experiment (c).

1 pparatrrs 4

e apparatus, which is common to aN three steps is shown in Figure 1. headphone G, connected over transformer D with switch C, a pure tone was

ted, when 0 moved1 the lever of s.witch C right or left and thus connected one two frequency generators A or B with the circuit. A tone of 50 c/s could be

uced in the headphone by pressing the button of switch F. ‘The duration of the ~,ones was determined by 0. As long as he kept the lever at one side or the: button pr e tcne sounded:.

Special equipment for Experimeitf (a): Logarithmic potentiometers between A and C, B and C, and F %and point H of the circlrit enabled E and G to vary the

ity of the tones produced by each generator anlj the 50 c/s tone fro:m the mains. e volt sge over the headphone when connected with one of the generators or trans-

: could be measured with a vacuum tube voltmeter connected in parallel with phone. Similarly, the voltage behind the two potentiometers injerted before could be measured.

A 3

ent for Experiment (b): e potentiometers between A, and C equalizing networks (ba ass filters), constructed in accordance

with the result from nt (a) to keep the subjective loudness at a constant level\, irrespective of frequency. horizontal and vertical plates of a c. r. Q were con-

allel with the output of e frequency generators so that by inspection ed Lissajous’ figures the generators could be adjusted yielding exact

octaves. Special equipment for perirnent (c): Equalizing networks as in Experiment (b).

was enabled to vary the frequency of one of the tones produced by the Tenerators by turning a knob.

IIn all experiments, 0 put the headphone on his right ear and inse.ted an earwad into his left (in order to avoid possible interfering noise). The two tones were coupled up by O’s movin the lever right and left (except ;he 50 c/s tone which was heard when 0 pressed the button).

this standard tone adjusted to 55,5 db (re 0.0 2 dyne/cm*). The variable was set in the other frequency generator and 0 was instructed to adjust its loudness to the same as the one of the standard tone by turning the knob of the potentiometer. The frequencies investigate were SO, 200, 500, 2500, 5000,7000, 10000 and 13000 c/s. Four adju ments were made for each frsquency with due rotation. The experiment per was preceded by pre- liminary trials. This was a preliminary expe ent with tiae purpose: (1) To obtain a group of OS fairly homogeneous in hearmg ability, and (2) to make up an equal loudness curve, serving as a basis for construction of an equalizing network (bzndpass filter) whit ivas to keep loudness con- stant irrespective of frequency.

From 15 OS 4 were discarded because of t large dtxkhm of their equal loudness settings from majority. The bandpass filters were constructed in accordance with the values obtained for the remaining subjects. The intensity deviations from the desired values were kept within k 5 db, except over a range between 2 and 6 kc/s, the intensiity of which exceeded the desired one by somewhat more than 5 db.

Experiment (b)

The OS were instructed to rate the “similarity” (without further expla-

nations) of the two tones presented by E on a scale with 10 steps (0 deno- ting no similarity at all and 10 denoting identity), to one place if desired. The presented tones were 200 c/s or exact octaves of 200 c/s up to 12800 C/C. This restriction was introduced in order to farce the C’s tc make their

ratings only regarding pitch, i. e. to avoid any 1 ‘nbirfc ‘C1,W M;1Lh t

Table 1:. %nilarity satig data.

kcj!~ 0.4 0.8 1.6 3.“r! 6.4 12.8

0.2 6.78 5.07 3.71 2.96 2.57 1.60

0.4 7.28 5.71 5.50 3.85 2.61

0.8 7.50 6.64 5.10 4.08

1.6 7.47 6.72 5.40

3.2 7.94 6.49

6.4 8.7

similarity of two t0nes constituting a musical interval wi tave. As the dials of &e two frequency generators were not exact, ttings were corrected by means of the 6. r. 0. e czorrection

t was .,wrformc;J with 9 OS of the 11 Imentioned in experi- ach pair of t:$nes was rated 4 times with due rotation. The ex-

ent proper was prX2ded b_/ preliminary trials. The means of’ these ratbgs are bound in Table 1.

The OS were instructed to adjust the pitch of a v e tone t.0 one of the pit.ch of standard tone by turning a kno Y SQ wished,

they CBUM listen to the 50 c/s tone as a kind of reference tone with a pitch a~~r~~~ately zero ( except for the two lowest frequencies). (The intensity

Table 2. Ratio setting data. ._I

S Sua c/s c/s

387 203.5

685 282.6

987 363.9 14911 443.9 199.; 551.8

249C 677.3

301c 722.7

41o(E 1028.8

4850 1060.6

6900 1.396.5

8810 1797.6

11cuO 2247.2

12980 2541.9

Smke

c/s and 13COO c/s were inwstigated.

correct’ed values are shown

a method previously descri ‘ie equations for the least

mean square solution of the straight lines are:

s 1,1= 0.2174 S + 132.37,

&I = 0.1860 S + 153.61.

The: values of Sl, 2 calculated by these e~uat~o~~ ; never deviate y more

than t’k standard erro:; of the mean from the measured ones Tht; corresponding equations for the magnitude function of

(21 (31

The magnitude function is plotted in by computing R (=: 11.54) for S - - 387 and plott

S 111 = 203.5.

s in Fig. 3 illustrate the goodness of fit. computed from the experiment

mental S,, :!=-values.

6 IiANNES EISLER and GC)STA EK

R S 01 2 3 15 6 7 6 9 $0 11 12 13 kc/r

Fig. 3. Pitch as a function of frequency.

As to the discontinuity in lthe range 3-4 kc/s we refrain from any expla- natlion, as the problem proper did not consist in constructing a correct magnitude function of pitch, but in co paring the mcgnitude functions of

itch obtained by different methods. If, e. g., there were any shortcomings in. the apparatus, they q#rt to affect both experimental series equally. It

, howver, be point~zd out: (a) that a linear plot of the values of Stevens ‘VoUcmann (8) likewise reveals a discontinuity, if not of the function

i&e&, en of its deriviitive, ;nd (b) that the experience of tone chrema diiappears in about that range (5) The remainh~g difference between our

Table 3. Scale values of pitch computed from ratio setting data (R,) and from similarity rating dabi ( Rs),,

.-

S R, R, -.-

0.2 5.63 0.269 0.4 11.84 0.494 0.8 18.70 0.771 1.6 27.12 1.134 3.2 35.98 1 . m 6.4 49.60 2.142

12.8 68.16 3.056

from the similarit;r rating

Table 3.

I, RJ, could be found, sU could be compute against the experimentally ol~tainetl

vahes su. This p’o+ th-- .

1 L, UICII, O*@i+c t0 be IiiXXii, tith the SlOpt2 = 1 iifid the y-intercept = 0. Likewise, a plot of R-vabres out of sir-Clarity ratirlg data

R,) against R-values out of ratio setting data ( ,) should be linear wi the y-intercept = 0 if the zero-points of the two scales coincide.

In order to ~v~stigate the for e similarity function (I), the simi- larity rating data were transforme fractions of identity (with iden-

1) by dividing by !O. he fist specific hypothesis about the fo

(1) which we tested, was the simple ratio hyp but it was not confirmed by the experiment hypotheses had to be rejected, too, before the fo g one was set up:

Sfj = 4

m, + q/2’ i. e.

(4)

w-9

where R,, is the arithmetic mean of R, and All sI, were computed from R,-vahcs according to equation (4); they

are shown! in In Fig. Cl experimentah y obtained s,l-Bvalues of Table 1 (divide

are plotted against the theoretical values of Table 4. The 21 points cluster rather well round the straight line which represents perfect agreement.

computed horn R, accMling to formula Wn .-

kc/s 0.4 0.8 1.6 3.2 6.4 12.8

0.2 .6 .463 .3 .270 .204 .152 0.4 .776 .608 ,495 .386 .296 0.8 .8ff .684 S48 .43 1 1.6 .860 .707 .‘i’69 3.2 .1141 A91 6.4 .842

Far further investigations of relation (41, a new ratio matrix was made up with the elements

%j = % u:-

RI 2-SPj

and qj* = 1 /q jj _ The scale u&es RI wme computed from aczo+di~g to a procedure described by one of the au

~ww111 !i the last column of Table 3. Fig. 5 shows the graph RI vs. R,. The y-intercept of the least mean square

Worn ks - 0.04, which implies a rather close agreement between the zero- ES c)f the two scales. The Ipints seem, however, to lie on a slightly

curved Ike, ‘but as the deviatkns from the calculated points do not e 3%,exceptinS= 0.2 kc/s, a straight lime is a very good approximation.

0 02 Oh El6

~~~e~t~ estimates of similarity plotted against theoretical values of ty wrnpted from scale values of pitch according to Equation (4).

@tch si;ale comput from experimental estimates 06 similarity according uation (4) plotted against the pitch scale o tained by ratio sel:ting.

values for S = 0.2 an

bandpass filters. lated value and as s h. somewhat uncertain.

Relation (4) is the main result of this investiga as an empirical equation without e~piri~Q1 constants.

Ttis su sts the possibility of a al derivation of the equation.

rform~d on the basis of

judged as the smaller stimulus rela arithmetic meu of the subject

comparison, seemed to

10 IIAXNES EISLER and GUSI’A EKMAN

SUMMARY

The mtxhanism of perception of similarity in the dimension of pitch was investigated. Stiti@ estimates were obtained and a pitch scale was constructed. It was shown

subjective similarity between two tones of equal loudness is equal to the ratio ecn the lower pitch and the average of the two pitch values. Thi!; relation is in neral agreement with Helson’s concept of adaption level.

REFERENCES

. Eisler, II., A note on treatment of ratio setting data for constructing phychological scales. Rep. Psychol. Lab. Univ. Stockholm, 1958, Nr. 5;4.

2. n, G., Two generalized ratio scaling methods. 1. Psychol., 1958, 45, 287-295. 3. II., Adaptation level as a frame of reference fr,l, prediction of psycho-

phyiical data. Amer. J. Psych&., 1947, 60, 1-29. 4. - Adaptation level as a basis for a quantitative theory of frame of rieference.

Psychor”. Rev., 1948, 55, 297-313. I. C. R., Basic correlates of the auditory stimulus. In S. S. Stevens (Ed.),

001: of experimental psychology. Ne:w York: Wiley, 1951, Pp. 985-1039. .S. & Davis, I-I., Hearing: its psychology and physiology. New York:

yens, S. S., Volkmann, J. & Newman, E. B., A scale for the measurement of the chological magnitude pitch. J. acoust. SQC. Amer., 1937, 8, 1%5-19(.X

ens, S. S. & Volkmana, I., The relation of pitch to frequency: a revised scale. Amer. /. Psychoi., 1940, 53, 329-353.