82 Prof. Helmholtz on Vowel Sounds. 
the resonant tube, and thus, not being reinforced by the latter, 
shall remain inaudible. 
But when the motion of the air during a period of oscilla- 
tion does not follow the simple law of the pendulum, but any 
other whatever, we may in general by due attention discover 
various notes, even when the motion of the air is produced by a 
single sounding body. Now, according to the well-known theorem 
of Fourier, every periodical motion of the air may be expressed 
mathematically by a sum of terms, each of which has the form 
A sin (2amt-+c), and therefore corresponds to a simple pendulum- 
like oscillation of the particles of air. In this expression, A and 
e are dependent upon the value of m, and m assumes successively 
the values n, 2n, 3n, 4n, &c., where n, as before, denotes the 
number of simple periods in a second. 
Now in all cases where the nature of the motion of the 
sounding body can be theoretically found and mathematically 
represented as a sum of such sines, the ear, when due attention 
is paid, can really distinguish notes of n, 2n, 3n, &c. oscillations, 
although, in all cases in which such a motion of the air is not 
actually produced by different sounding bodies, the coexistence of 
a number of simple pendulum-like oscillations of the particles of 
air is a pure mathematical fiction. 
The universality of this perception of distinct notes induced 
a celebrated member of this Academy, the late G. S. Ohm, to 
propose as a definition of a simple note that which is produced 
solely by a simple pendulum-like motion of the air of the form 
Asin (27mt+c). This definition of a note given by Ohm was 
vehemently attacked by Seebeck, who maintained that it was too 
narrow, and that the sensation of a single note might also be pro- 
duced by a motion of the air which differs considerably in form 
from that of a‘simple pendulum oscillation. _1 cannot here enter 
iuto a complete refutation of the objections raised by Seebeck, 
and must therefore return to the subject on another occasion. I 
will only remark that his objections are founded essentially on 
the difficulties which we often experience in distinguishing the 
higher notes. In fact, in all observations made by the senses 
two things must be kept distinet, viz. the immediate sensation 
or effect upon the auditory nerves, and the conception which 
arises therefrom by a psychical process, and which leads us to 
the conviction of the presence of a certain sounding body. In 
the immediate sensation the several simple notes may certainly 
be distinguished from one another when sufficient attention is 
given ; whilst in the mentalimage they become blended together 
into the impression produced upon the ear by the tone of the 
sounding body in question. The attention, mdeed, generally 
requires artificial assistance in order to distinguish the several 
