A. M. Mayer — Researches in Acoustics. 25 



23*1, but it is just at this magnitude of interval that the supe- 

 rior beats begin to assert themselves, to produce with what 

 remains perceptible of the inferior beats, the confused rumb- 

 ling, which evidently would be but a slight roughness (disap- 

 pearing entirely at a further increase of the interval), if the 

 superior beats, whose intensity from this point increases with 

 the interval, did uot exist." 



3. The Durations of the Residual Sonorous Sensations as de- 

 duced from the Smallest Consonant Intervals among Simple 

 Tones. 



If we assume that two simple tones form the smallest conso- 

 uant interval because the beats produced by these conjoined 

 sounds have blended into a smooth continuous sensation, then 

 we may deduce the durations of the residual sonorous sensa- 

 tions from the observed smallest consonant intervals in the 

 following manner. The reciprocals of the numbers in column 

 C, Table II, are taken as expressing the durations of the sono- 

 rous sensations given by tones whose numbers of vibrations 

 are the mean of those of the lower and higher tones of the 

 corresponding consonant intervals, for, when two sounds of 

 different pitch blend, there is no reason why the duration of 

 their residual sensation, as given by the reciprocal in column 

 C, should refer to the lower sound more than to the higher. 

 Therefore we have taken these reciprocals from column C as 

 expressing the durations of sounds having the mean pitch of 

 the two associated sounds forming the interval. The residual 

 sensations thus found were projected in a curve, drawn to a 

 large scale. From this curve were obtained the durations of 

 the residual sensations of the tones of the musical scale. These 

 durations are given in column H of Table I. 



In column I of Table I are given these durations as com- 

 puted by the formula 



D 



/ 48000 \ 



(^ + 21 i-0001 



\N + 30 / 



In column K are the differences between the durations com- 

 puted by the formula and the durations given in column H. 

 These differences show that, the formula expresses closely the 

 durations of the residual sensations thus deduced from the 

 determinations of the smallest consonant intervals, except in 

 the case of UT 2 ; for which tone the computed number of 

 vibrations to be added to it to form the higher tone of its 

 smallest consonant interval, as shown in Table II, is 5*2 vibra- 

 tions less than the number of vibrations really required. 



In figure 11 these durations, as determined from the smallest 

 consonant intervals are plotted in the curve F so that the com- 



