Siren and an Organ Pipe. 91 



nary. Without wishing to enter into the memorable Helm- 

 holtz-Konig controversy, it is certain that the difference tones 

 vanish with the failure of the ear completely, before any beats 

 are heard as such. These introduce an entirely different sensa- 

 tion (by contrast a very commonplace sensation), the sibilant 

 rattling being at first rapid, then slowing up into complete uni- 

 son, finally to reappear again with excess of frequency in the 

 pitch of the siren over that of the organ pipe. The beats 

 gradually gain in rapidity, finally to vanish in the profound dif- 

 ference tones of the ascending cadence of the first order D/. 

 So far as I can make out, the only discontinuity here is physi- 

 ological and due to the failure of the ear. 



The next train of sounds of marked prominence, though not 

 quite so clear and deliberate as the preceding, are the difference 

 tones of the second order (Z> 2 = P.—S, or = OS—P), due to 

 the combination of the first order of difference tones P 1 with 

 the notes of the ascending siren S ± as well as to the first over- 

 tone of the siren (OS) combining with the organ pipe (P). 

 On the same time-scale, these notes must appear in a descend- 

 ing cadence P 2 , vanish through beats, and reappear in an ascend- 

 ing cadence Z>/ and D 9 '" just twice as rapidly as the difference 

 tones of the first order, as is shown by the greater steepness of 

 the lines converging above t = 4 of the abscissa. Beyond this 

 what has been said in general about D 1 in passing through 

 unison applies to P r They are strong enough to mask some 

 of the first order of difference tones. They intersect the siren 

 notes (S) and the first order of difference tones (1)^ about in 



lr. The latter, coinciding with the b® of the lines D x and 

 OS and the f" of the organ pipe, is easily found. More 



marked however is the lr of D\ at its intersection with D^ 



coinciding with the V" of S and f" of the organ pipes. We 

 have here two ascending, one descending and one stationary 

 cadences, and the listening ear hears two orders of difference 



tones chromatically converge upon the common b v and there- 

 after diverge again. Other coincidences are given in the dia- 

 gram. Prominent intervals are apt to be in the key of B . 



In the second half of the diagram (t> 8) the second order of 

 difference tones separate into 2) a '" (=OS—P) and Z> 2 IV {=S — 

 I)/) coalescing with P vXf". 



In the left half of the diagram we have furthermore the 

 octave difference tones OD 1 (= OP—S) due to the first over- 

 tone of the pipe and the ascending siren, and the summation 

 tones, "Sum." (-=S+P). Neither of these high trains was I 

 able to hear. Finally the octave or first overtone of both pipe 



