To prove the incorrectness of Helmholtz's statement that beats do not
colesce into musical sounds, but that the ear will distinguish them as a
rumbling noise, even when their number rises as high as 132 vibrations
per second, Rudolph Koenig has constructed a series of tuning forks,
recently presented by President Morton to the Stevens Institute of
Technology. The following table exhibits the number of vibrations per
second of these forks, the ratios of their vibrations when two are
sounded together, the number of beats produced, and the resultant sound:
Vibrations per second. Ratio. Beats. Sounds.
3840 :4096 15:16 128 Ut_{2}
3904 : " 61:64 96 Sol_{1}
3936 : " 123:128 80 Mi_{1}
3968 : " 31:32 64 Ut_{1}
3976 : " 497:512 60 Si_{-1}
3989.3 : " 187:192 53.3 La_{-1}
4000 : " 125:128 48 Sol_{1}
4010.7 : " 47:48 42.7 Fa_{-1}
4016 : " 251:256 40 Mi_{-1}
4024 : " 503:512 36 Re_{-1}
7936 : 8192 31:32 128 Ut_{2}
8064 : " 63:64 64 Ut_{1}
8096 : " 253:256 48 Sol_{-1}
8106.7 : " 95:96 42.7 Fa_{-1}
8112 : " 507:512 40 Mi_{-1}
8120 : " 1015:1024 36 Re_{-4}
8128 : " 127:128 32 Ut_{-4}
On sounding two forks nearly in unison, the sound heard corresponds to
a number of vibrations equal to the difference of the numbers of
vibrations of the forks.
On sounding two forks, one of which is nearly the octave of the other,
the ear perceives a sound, which is that given by vibrations whose
number equals the difference in the number of vibrations of the higher
fork and the upper octave of the lower fork.
Koenig has also found out the laws of the resultant sounds produced
by other intervals than the octave, and has extended his researces to
intervals differing by any number of vibrations, as may be seen from the
above table.
His conclusion is that beats and resultant sounds are one and the same
phenomenon.
Thus, for example, the lowest number of vibrations capable of producing
a musical sound is 32 per second; in like manner, a clear musical sound
is produced by two simple notes of sufficient intensity which produce 32
beats per second.
Koenig also made a very ingenious modification of the siren for the
purpose of enabling Seebeck to sound simultaneously notes whose
vibrations had any given ratio. It is furnished for this purpose with
eight disks, each of which contains a given number of circles of
holes arranged at different angular distances. A description of this
instrument, which is also the property of the Stevens Institute, and of
Seebeck's experiments is thus given in a letter by Koenig himself.