Sounding of two Notes. 435 



and consequently 3 n + x and 3 n — x would give the beats 

 2 x. Nothing, however, can be discovered of all these inter- 

 mediate notes ; and we may well suppose therefore that, if with 

 such extraordinarily strong notes as I have used there is hardly 

 any probability of secondary beats being produced by combi- 

 nation notes, then with the use of weaker simple notes (such 

 as, for example, those produced by organ-pipes) the supposition 

 seems to be deprived of all probability whatever. But if, on 

 the other hand, we succeeded in producing such powerful 

 simple primary notes that the combination notes necessary for 

 the formation of the secondary beats according to the old view 

 could be formed with sufficient intensity, even in this case the 

 two beat-notes (m and mf) and their beats might have attained 

 to such strength that the combination notes of the higher order, 

 falling together with the latter beats, might still form only an 

 extremely small part of the intensity of the beats heard. 



In order to allow an easy review of all my observations on 

 primary and secondary notes and beat-notes, I have drawn up 

 the following Table. The column A contains the primary 

 notes with their vibrations, B the ratio of these two notes, 

 C the number of lower beats m, c the ratio they form with the 

 fundamental note of the interval, D the number of the upper 

 beats m / , and d the ratio they form with the fundamental note. 

 Under E is stated how the lower beats m, and under F how 

 the upper beats mf are to be heard. Finally, the column G 

 contains the secondary beats and secondary beat-notes arising 

 from the combined action of m and mf. In this Table I have 

 only given such results as may be perceived by any ordinarily 

 good ear from the use of the notes which I have employed 

 in these experiments ; and I have noticed especially the cases 

 in which notes cannot be directly distinctly heard, whose 

 undoubted existence is proved not only by secondary beats, 

 but also by the help of auxiliary forks, as is, for example, the 

 case in the beat-notes of the intervals d : d and d :f . An 

 " ordinarily good ear," and u notes such as I have employed," 

 are certainly, in spite of the dimensions of the tuning-forks 

 and sounding-boxes stated above, premises much wanting in 

 precision ; but it stands to reason that even the phenomena 

 resulting from the simultaneous sound of two simple notes 

 can only be stated with perfect accuracy in relation to their 

 intensity when it becomes possible to express the intensity of 

 notes of different pitches by a common ratio, with the same 

 precision with which we can now state the pitch of their 

 vibrations. 



A few apparent anomalies which are shown by this Table, 

 as, e. </., that the system of secondary beats can be less perfectly 



2 G2 



