Siren and an Organ Pipe. 89 



and vice versa, at the same time with the siren, supposing that 

 the same musical interval is retained. In order to exhibit the 

 continuity of these phenomena perspicuously (see chart), I have 

 modified this experiment by replacing one siren by an ordi- 

 nary open organ pipe of fixed and sufficiently high pitch. 

 This is mounted on the same bellows with a simple siren rising 

 continuously from low to high pitch. The advantage gained in 

 this way is best exhibited in the accompanying diagram, but con- 

 sists essentially in producing difference tones whose frequency- 

 changes vary at first in opposite direction to the pitch of the 

 continually rising siren. Thereafter with a passage through 

 zero, the difference tones change in the same direction as the 

 pitch of the siren. This contrast between ascending and 

 descending cadences, of which there are several successive 

 groups, facilitates recognition, while the apparent scattering of 

 the difference tones into beats beyond the lower limits of audi- 

 tion, and their subsequent emergence and coalescence from 

 beats, is of special acoustic interest. The method is thus pecu- 

 liarly adapted for demonstrating the whole scope of these com- 

 plex phenomena even to an indifferently musical ear. It is 

 not so useful in fixing the pitch of a single group of notes, 

 though this also succeeds after a little practice, for an ear 

 accustomed to recognize chords and musical intervals. 



2. To work out the diagram, I used an ordinary open 

 pipe, f\ (frequency n = 696) which happened to be available 

 and which was high enough in pitch to admit of difference 

 tones of a wide range below it. The siren was an ordinary 

 small single-voiced Konig instrument. By supposing the 

 angular velocity of the disc to increase at a small constant rate 

 in the lapse of time — a condition of things nearly enough 

 reached in the experiment — the diagram is constructed by 

 representing the frequencies (?i) of the different notes as ordi- 

 nates, and times, t, in arbitrary units (0 . . . 16) as abscissas. 

 In such a diagram the notes (chords) heard simultaneously will 

 lie on the same vertical. A number of these have been 

 accentuated by drawing the vertical lines in question from 

 note to note, because of their value in orientation. (See chart.) 



In the chart the continuous note of the organ pipe (P) is 

 given by the broad horizontal lines at the level n = 696, 

 extending quite across the diagram while its first overtone, the 

 octave, is shown by a similar line {OP) at n = 1392. The con- 

 tinuity of notes given out by the siren is shown by the broad 

 diagonal (S below f% S f above f") extending from the origin 

 at an angle of 45° as far as the height n =1392 orf /,f . Beyond 

 this the notes were difficult to produce and their succession too 

 slow. The first overtone of the siren is shown by the line 

 (OS) having double the slope of (#). The successive notes of 



