Contribution Offered Towards a Discussion of the Paper “Pitch and the Missing Fundamental.”

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For several years we have used in this laboratory a set of steel bars, each mounted on a resonating box and respectively tuned to the frequencies 110, 220, 330, 440, 550, 660, 770, 880 and so on in the Fourier series, to and including 1760 c.p.s. This gives in effect it fundamental tone with the succeeding Fourier harmonics tip to the 16th inclusive. By repeated experiment with many students taking courses in the theory and practice of tuning pianos in equal temperament, during the last three years, we have found that, if we sound several of the bars simultaneously, choosing the members of the series in their regular succession, that is to say, such a series as 220, 330, 440, 550, 660, we always hear clearly and, as it were, beneath the simultaneous sounds, blended together into a chord, the fundamental tone 110. When again we take such a series as 440, 660, 880, 1100, we hear the fundamental 220. When we take 660, 990, 1320, 1560, we hear the fundamental 330. When again we have taken an irregular series, such as 330, 550, 660, 880 and 900, we have always heard a tone of pitch corresponding with the frequency of the smallest number of beats between any pair of the bars, namely 110. In the case above mentioned, the 110 is very distinct, with 220 existing simultaneously, and traces of 330 also. The most important fact however seems to be that the lowest frequency deducible, namely 110, is always, in the cases mentioned, definitely and distinctly heard. It should be understood that these bars are excited simply by blows delivered with felt-covered hammers, and that the beat-tones, as I shall call them, are therefore feebler than they would be if the bars were excited electrically at relatively large amplitudes, and were steadily sustained thereat. Phenomena such as these described have been explained by the Koenig theory of beat-tones, which states that when the number of beats between two frequency-generating units exceeds the minimum number of atmospheric pulses sufficing to arouse a sensation of sound (that is to say, when the number exceeds 16 or thereabouts), sensations of tone are aroused and identified, giving sensations of pitch corresponding to the beat-frequencies. Presumably such sensations may be described as having subjective origin, although it appears that the psycho-physiological foundation of any such theory has hardly yet been laid firmly. What seems more important for the moment is that our experiences corroborate the observations made, and conclusions reached, by Fletcher, as shown in his paper. We do not overlook the fact of Fletcher's experiments pointing to the fundamental tones being regarded as the actual pitch-sensation generators for the groups of simultaneous harmonic sounds used by him, while in our case we recognize the fundamental tones as forming weak but very distinct roots to the chords which are constituted by the simultaneous harmonic tones. The distinction however does not appear to be essential.