The pitch of tonal masses.

Originally published in American Journal of Psychology, 46, 452-5, 1934.

By Adolph G. Ekdahl, University of New Hampshire, and
Edwin G. Boring, Harvard University

Twenty years ago Baley reported that a number of tones, sounding together and differing from each other by small equal increments, tend to fuse (zusammenfliessen) into a single resultant tone, the pitch of which corresponds to the arithmetic mean of the frequencies of the stimuli [1]. Baley presented data for only four cases: (1) 6 tones in the range 610-625 cycles, (2) 6 tones in the range 600-620 cycles, (3) 8 tones in the range 600-628 cycles, and (4) 10 tones in the range 600-636. In the first case the increment between successive tones of the series was about 3 cycles, in the other cases it was about 4 cycles. The tone resulting from the mixture was described in the four cases as lying respectively in the ranges 616-619, 608-612, 612-618, and 616-620. It is obvious that this work is not quantitatively exact. There is no evidence of a psychophysical procedure and the subjective equivalents were not exactly determined. Baley's use of the phrase "arithmetic mean" does not signify that the equivalence is not for the geometric mean. The arithmetic mean of the six tones in his first case is 617.60 cycles, the geometric mean is 617.57 cycles, and he fixed the subjective equivalent merely as lying between 616 and 619 cycles. Baley, however, stressed more than the quantitative results the intimate fusion of the tones into a single resultant tone.

In 1925 the junior author, working in the Harvard Psychological Laboratory, undertook to determine the subjective equivalents in pitch of the tonal masses described in Table I. The points of subjective equality were computed by the method of constant stimuli with five comparison stimuli and (except for two cases, see legend to Table II) 100 series. The six sets of notes defined in Table I were given by an Appunn reed tonometer and were employed as the standard stimuli. The standard was given first. The stops were pulled out, the air was turned on and the tonal mass allowed to build up, then the stops were all pushed in at once by a hinged felt-covered board, and the single stop for the comparison stimulus was pulled out. The O made comparative judgments of the pitch of the single note in respect of the preceding tonal mass.

There was no evidence at all of the fusion of any of these tonal masses to yield a single resultant tone or note. The mixtures always sounded complex, and the psychophysical judgment was a judgment of equivalence of pitch but by no means a judgment of identity. This resultant is not a direct contradiction of Baley's finding. The minimal increment in the present experiment was 8 cycles in the region of 264 cycles, where 8 cycles is equivalent to about 52 musical cents (100 cents = 1 semitone in equal temperament). Baley used steps of about 4 cycles, but they were in the region of 620 cycles where every step is only about 11 cents. However, the minimal range in the present experiment ("4 by 8," Table I) was 157 cents, and for it there was no good fusion; whereas the maximal range for fusion reported by Baley-- case (4) cited above-- was 102 cents, a quarter-tone less. Nevertheless it seems doubtful that Baley's tones could have fused in the way that colors fuse in color mixture to give identity in perception. It is much more likely that his "resultant tone" was merely a tone judged as equivalent to a perceived tonal mass. It was, however, not the primary purpose of the present experiment to test Baley's results. Rather we sought to examine the nature of the pitch of the noises which are generated as tonal masses. The complexes were chosen to yield results for different numbers of notes combined (from two to eight) and different ranges of pitch (from about three-fourths of a whole tone to almost an octave). Thus in the stimulus "2 by 24" we have an ordinary interval that is something less than a major second, and in the stimulus "2 by 48" we have an interval a little more than a minor third. These pairs are simple fusions. However, the stimulus "8 by 8," extending over an interval slightly less than a major third, yields a confused mass that can readily be described as a noise, and the stimulus "8 by 24," covering nearly an octave, is also a noisy complex.

TABLE I.
Description of standard stimuli

The tonal masses defined in the table served as standards to which single notes were equated in pitch by the method of constant stimuli.  The range is given in musical cents (1200 cents = 1 octave).

The results are shown in Table II [2]. Here the small size and variability of the intervals of uncertainty show that the judgment was difficult and that the Os tended to avoid the category equal, as they might not have done had it been possible for equal to mean identical instead of merely equivalent. Hence we may dismiss the intervals of uncertainty as unimportant.

The first line of the table shows that Ds with the "8 by 8" stimulus gave the point of subjective equality at 261.46 cycles. This value is quite close to the mean of the stimulus (range 236-292 cycles) which is 264 cycles. It will be observed from the values of h that the point of subjective equality, 261.46, is quite precisely established. The interquartile range in this case is about 6 cycles, i.e. with a change of 6 cycles one passes from the stimulus that yields 25% judgments of higher to a stimulus that yields 75% of higher (or conversely for lower).

TABLE II.
Subjective equality of single notes to tonal masses.

The first two columns show the number of notes and the interval (cycles) between successive notes in the tonal masses making up the standard (cf. Table I). The body of the table gives, in cycles, the lower and upper limens, the values of h for each of the limens, the interval of uncertainty, and the point of subjective equality (intersection of the two psychometric functions). A value of h = 0.2 corresponds to an interquartile range on the psychometric function of about 4.8 cycles, since P.E.=0.4769 h. All frequencies in the psychometric functions are based on 100 observations, except that there were only 75 observations for Zr and 85 for Ka.

Examination of Table II shows that, with the exception of the case of "8 by 8" for Ka and the two cases of "2 by 48," all the points of subjective equality are near the mean, 264, and that nearly all of them are placed within interquartile ranges of less than 6 cycles. This degree of precision is high, although it is not great enough to give significance to the question as to whether it is the arithmetic or the geometric mean that is approximated (cf. Table I). What the Table II shows is that an accurate judgment of pitch can be made for tonal masses where as many as eight tones, distributed within a musical third or over an entire octave, are sounded together. In such cases the pitch of the mass is taken by O as equivalent to a mid-pitch of the mass. The same principle held in this experiment for masses of four tones, and for a two-tone combination when the two tones were separated by an interval less than a major second ("2 by 24"). It did not hold for an interval approximating a minor third ("2 by 48").

There were three exceptions to the general rule. In the case of "2 by 48" both DJ and Br equated the pitch of the pair to the pitch of the lower member of the pair. We probably have here the limiting case where the number of components is so few (two) and the separation is so great (minor third) that the O, in making a judgment of pitch, could not avoid discriminating the two pitches and selecting one as the equivalent. It is apparent that a complex tonal mass or one within a small range of pitches is localized as to pitch at its mid-pitch, but that simpler complexes may be characterized by one of the actual pitches in the complex. Thus we know that clangs are located as to pitch by their fundamentals, the lowest pitch of the complex, but that in musical harmony the upper note of a chord often characterizes the fusion as to pitch.

The other exception is the case of "8 by 8" with Ka. He found the judgment very difficult and his subjective assurance was low. His psychometric functions show erratic frequencies, as the negative interval of uncertainty and the low values of b indicate. The point of subjective equality (280.13) is well above the average, but not as high as upper note of the stimulus (292). Presumably this case can be ignored, as dependent upon an instable attitude.

In brief we may say that, if we start with a single tone and add successively to it other tones that bear no particular musical relation to one another, we pass through musical fusions to confused tonal masses which are noisy and which may be regarded as noises approximately located as to pitch. The nature of these complexes as noises is most obvious when the masses are given briefly and explosively by an abrupt quick pressure on the tonometer bellows. The fact that they have location in the scale of pitches is shown by the capacity of Os to equate them to simple pitches, usually a pitch approximating the pitch that corresponds to the mean of the frequencies of the tones in the mass. However, the mass is still perceived as a mass, and its equation to a single pitch is only a rough localization that resembles the manner in which a large cutaneous impression is localized, when its exact position is required, approximately at its midpoint.

In this manner we come to see the relation of noises to tone. Complexes of tones are noises, and a noise is tonal when pitches within it are continued and when the range of pitches allows of some relatively definite location of the complex noise in the scale of pitches.

Footnotes

1.   S. Baley, Uber den Zusammenklang einer grosseren Zahl wenig verschiedener Tone, Zsch. f. Psychol., 67, 1913, 261-276.

2.  The observers in this experiment were H. R. DeSilva, A. M. Brues, R. A. McFarland, K. E. Zener, and T. F. Karwoski, then research students in the Harvard Psychological Laboratory.