PROCEEDINGS 
OK TIIK 
AMERICAN AC A DEMY 
OK 
ARTS AND SCIENCES. 
VOL. XXVII. 
PAPERS READ BEFORE THE ACADEMY. 
1. 
CONTRIBUTIONS FROM THE PHYSICAL LABORATORY OF THE 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY. 
XXXVII.— SOME CONSIDERATIONS REGARDING 
HELMHOLTZ’S THEORY OF CONSONANCE. 
By Chakles R. Cross and Harry M. Goodwin. 
Presented June 10, 1891. 
The present paper contains the results of a number of experiments 
relating to certain aspects of the theory of consonance and dissonance 
put forth by Helmholtz in the Tonempfindungen. In that work 
the author gives as the number of beats producing the maximum of 
harshness from thirty to forty per second, reaching this conclusion 
from a consideration of the amount of dissonance of various different 
chords in different octaves. 
Mayer* studied the relation of the number of beats causing the 
greatest harshness to the absolute pitch, and showed that there is 
a marked rise in this number as the pitch becomes higher. The 
beats studied by Mayer, however, as he himself pointed out, are 
in several respects different from those which occur in the case of 
mistimed unisons. They were caused by rotating a disk, which 
* American Journal of Science, Vol. CVIII. p. 241; Vol. CIX. p. 267. 
VOL. xxvii. (n. 8. XIX.) 
1 2 
PROCEEDINGS OP THE AMERICAN ACADEMY 
was provided with a circle of holes near the circumference, between 
a vibrating tuning-fork and a resonator, the number of beats per 
second being of course greater as the speed of the disk was greater. 
Hence the sound-wave reaching the ear was not at all a compound 
wave due to the superposition on one another of two simple sound- 
waves slightly differing in length, as in ordinary beats, but merely 
a series of simple waves with gaps, as it were; while the passage 
from maximum to minimum intensity in the sound reaching the 
ear was very different in its character from that actually occurring 
with ordinary beats. In referring to such heats as those studied 
by Mayer, we shall for the sake of brevity designate them as “ in- 
terruptional beats,” and the ordinary beats we shall call “inter- 
ference beats.” 
We have thought it desirable to ascertain what results would he 
reached in different registers with interference beats, such as are 
present in actual dissonances. It was of course necessary to em- 
ploy simple tones. Our method was a direct one, viz. to produce 
two such tones simultaneously, and to vary the pitch of one of these 
by known amounts, estimating by the ear when the sound became 
the harshest. 
As a source of sound we employed the cylindrical resonators of 
Koenig, which were sounded by gently blowing across the opening 
a stream of air from the brass, slit-shaped “universal embouchure ” 
made by the same maker. We found these tones to be very pure 
and loud, while their pitch could be varied with great readiness 
without altering the intensity of the sound emitted. The set of 
resonators employed ranged from Solx to Mi6> the ranges of succes- 
sive resonators overlapping slightly. The two embouchures were 
connected to a Y by rubber tubes, and placed on adjustable sup- 
ports in suitable relation to the apertures of the two resonators to 
he sounded. The strength of the blast was regulated by a stopcock, 
and the bore of each rubber tube could be closed more or less by a 
clamp, so that the force of the jet of air at each resonator could he 
adjusted with all needed delicacy. Thus the intensity of the notes 
given out by the two resonators could be adjusted so that they were 
exactly equal in loudness. This was done by causing the tones to 
heat slowly, and so adjusting the apparatus that there was ap- 
parently an absolute silence when the sound fell to its minimum 
intensity. No difficulty was found from any whirring of the sheet 
of air as it struck the edges of the aperture, since by proper adjust- 
ment this could be made quite inappreciable. Having once made OF ARTS AND SCIENCES. 
3 
the necessary adjustments, the pitch of either note could be varied 
by drawing out or pushing in the lower half of the resonator. The 
actual pitch of the note given by the resonator was in all cases 
determined by comparison with a tuning-fork of known rate. As 
the pitch of the resonator when blown is to a certain extent de- 
pendent on the position of the embouchure and the strength of the 
blast, it would not be safe to assume that the actual note given 
by the resonator was that marked upon it. Neither resonator 
when sounded sensibly affected the pitch of the note given by 
the other. 
The determination of the point at which the harshness of the 
dissonance produced by the tones of the two resonators reaches a 
maximum is of course a very difficult problem to be solved exactly. 
It is, moreover, a question still to be investigated, whether pre- 
cisely the same results would be reached by different observers 
possessed of equally good perception. The following figures given 
in Table I. were ascertained from a long series of observations by 
one of the writers (Mr. Goodwin). They are liable to an error of 
a single beat in either direction, possibly somewhat more in one or 
two cases. They furnish, however, the first series of direct meas- 
urements of this kind which we know. 
TABLE I. 
Ut* 
Ut% 
Sols 
Ut\ 
Solt 
Uu 
Mis 
19 
26 
29 
32 
36 
41 
45 
It will be noticed that the result for 32 heats per second, 
agrees very closely with the number 33, which Helmholtz gives 
for maximum dissonance in the middle registers. He found that 
of the ordinary musical intervals, b' c", as he designates it, was the 
most dissonant, and as this gave 33 beats per second he considered 
this number as giving the harshest effect for tones of corresponding 
pitch. 
The notes furnished by the resonators are also well adapted for 
the direct study of the limit at which the beats cease to be audible 
when their number is increased. As the interval between the beat- 
ing notes is increased, the beats heard become fainter and fainter 
until they disappear. It is all but impossible to fix upon the pre- 
cise point of disappearance, since the change in loudness of the 
beats is so gradual. But, as will presently appear, our results 
clearly show that the maximum number of beats perceptible, like 4 
PROCEEDINGS OF THE AMERICAN ACADEMY 
the number giving the maximum dissonance, varies greatly with 
the pitch of the notes sounded. 
The pitch of one of the resonators was varied so as to run upward 
and downward through the vanishing point of the beats. This pro- 
cedure was extremely tiresome, and the values obtained were found 
to be considerably influenced by fatigue of the ear. The following 
table (Table II.) gives the results obtained when the ear was in 
good condition. The first column gives the rate of vibration of 
the lower note, the next four the rate of the upper note as found 
in corresponding series of observations; the seventh column gives 
the number of heats, and the last the interval between the two 
notes. 
TABLE II. 
Lower 
Note. 
Upper Note. 
Mean. 
Beats. 
Interval. 
1. 
2. 
3. 
4. 
s 
to 
GO 
175 
170 
177 
173 
174 
46 
Fourtli +. 
Utz 256 
328 
321 
326 
329 
326 
70 
Major third +. 
Sol3 384 
467 
458 
469 
460 
464 
80 
Major third — 16 vibrations. 
UtA 512 
620 
616 
618 
611 
616 
104 
Minor third (nearly). 
a? 
a 
00 
882 
890 
886 
890 
887 
119 
Tone + £ tone (nearly). 
L\ 1024 
1160 
1167 
1160 
1163 
1163 
139 
Tone + 11 vibrations. 
J/i, 1280 
1420 
1424 
1421 
1418 
1420 
140 
Tone — | tone (nearly). 
These figures abundantly confirm for interference beats the results 
already reached by Mayer for interruptional beats, viz. that the ear 
is capable of appreciating a greater number of beats as the pitch of 
the beating notes rises. They also show very beautifully the in- 
creasing consonance of small intervals as the pitch rises, noted by 
Helmholtz. 
The numerical results reached by us differ, as a whole, consider- 
ably from those given by Mayer for interruptional beats in either of 
his papers, though the divergence is less than that between the two 
series representing his own observations and those of Mrs. Seiler. 
In some cases, however, the agreement is very close. Thus, for Ut2 
we find the last trace of beats to occur when these are 46 in number. 
The number as observed by Mayer is 26, and as observed by Mrs. 
Seiler 45. Tor Ut3 our figure is 70, and Mrs. Seiler’s is 70 also. 5 
OF ARTS AND SCIENCES. 
For the other notes the divergence is wide, except for Ut&, for which 
our results agree quite well with those obtained by Mayer himself. 
These differences may be due to the different way in which the vari- 
ations of intensity in the beats progress, to the presence of pitch 
variations in interference beats, to the fact that in our method the 
musical interval of the beating notes increases as the beats increase 
in number, to mere differences in the estimation of the observers, 
or to some other less evident causes. 
We find that the ratios between the number of beats producing 
the greatest harshness at different pitches and the maximum num- 
ber of beats discernible at those pitches are as follows (Table III.). 
The numbers given are obtained by dividing the former number by 
the latter. 
TABLE III. 
Ut% 
fj f8 
Sola 
Ut 4 
SoU 
Uh 
Jfi, 
0.41 
0.37 
0.36 
0.31 
0.30 
0.30 
0.32 
These figures, as a whole, agree very well with the ratio (four 
tenths) given by Mayer, although the absolute numbers whose ratios 
are taken are quite different from his. Our results also seem to 
indicate a clearly marked fall in this ratio as the pitch rises. 
A few experiments were made using Professor Mayer’s method, 
with some slight modifications, in order to observe the judgment of 
the same ear upon beats of different kinds. The tuning-fork em- 
ployed was kept in vibration continuously by electricity, and the 
rotating disk was driven by an electro-motor, which secured great 
constancy of speed. We were seriously troubled by the sound made 
by the rotating disk, which produced a sort of siren effect as its 
openings passed before the aperture of the resonator. For Ut3 the 
maximum number of intermittences perceived was 50 per second. 
For Utt the number was about 10G. These figures are liable to an 
error of perhaps two beats, or even a little more for the higher pitch. 
Mayer’s figures are 47, 78, respectively, for his own observations 
on these notes, and 70, 130, for those of Mrs. Seiler. It is clear 
that there are very great differences in the sensitiveness of different 
ears for the perception of rapid beats. 
In another series of experiments, interruptions were produced by 
a break-wheel placed in a telephonic circuit with a magneto-telephone 
at each end. The sound of a tuning-fork was transmitted through 
the line, and any desired number of interruptions could be produced 
by varying the rate of rotation of the wheel. Considerable annoy- 6 
PROCEEDINGS OF THE AMERICAN ACADEMY 
ance was experienced at times from sounds which apparently arose 
from microphonic action at the break-wheel. These can probably 
be avoided in future experiments by causing the wheel to divert the 
current from the receiver periodically, instead of interrupting it. 
The limits at which the beats disappeared, with the notes experi- 
mented upon, were as follows: Ut± 48, MiA 5G, SolA 69. These fig- 
ures are the means of a number of observations, and are apparently 
true within less than one beat per second. Comparing this kind of 
interruptional beat with those given by Mayer’s method, the maxi- 
mum number perceptible seems to be smaller for the former than 
for the latter. This is probably due in part to the greater loudness 
of the sounds observed in the latter case, but there is another marked 
difference which must exercise an important influence upon the phe- 
nomenon under consideration. With the beats produced by the 
break-wheel the sound begins and ends almost instantaneously, so 
that sharply marked intervals of sound and silence of equal du- 
ration succeed each other. One would naturally expect the tele- 
phonic beats to be the more distinct of the two, which is not the 
case. The subject is one which requires further study as to the 
influence of the relative duration of sound and silence in the tele- 
phonic method, and of the relative size of the apertures in Mayer’s 
method. 
In the matters which we have thus far discussed in the present 
paper, our results are fully in accordance with the theory of conso- 
nance proposed by Helmholtz. In some other particulars, however, 
we are led to conclude that this theory is incomplete, at least. 
Some years ago, in a paper read at the Philadelphia meeting of 
the American Association for the Advancement of Science, and of 
which an abstract was published in the Proceedings of that Society 
for 1884 (page 113), one of us called attention to the bearing of 
certain phenomena of binaural audition upon Helmholtz’s theory 
as follows: — 
“ In connection with his.study on the effects of beats in causing 
dissonance, Helmholtz considers the condition of the vibrating por- 
tions of the inner ear as to resonance and damping. In his remarks 
on the subject, he assumes that no notes are capable of beating with 
each other unless they both affect the same vibrating element of 
the inner ear. This view of the matter gives a purely mechanical 
action in the ear as the explanation of the physiological phenome- 
non of beats. 
“In addition to the experiments of Koenig, in which beats were OF ARTS AND SCIENCES. 
7 
obtained between notes of very great intervals, there are certain 
phenomena of binaural audition which appear to prove that this 
view is incorrect, and that rather than, or at least in addition to, 
this mechanical interference of vibrations in the ear, there is a more 
obscure operation within the sensorium itself. I refer to the fact 
that beats may occur when the exciting sounds operate upon different 
ears, and also apparently between the after-sensation and the suc- 
ceeding sound of mistimed unisons. Whether this last phenomenon 
can be observed when the after-sensation results from an impression 
made upon the opposite ear to that which is impressed by the suc- 
ceeding sound, I do not know; but in view of the fact that two 
simultaneous sounds acting upon different ears may beat, I do 
not see why this result should not be possible. 
“Now if beats arise between the after-sensation and a following 
sound, Helmholtz’s view can be true only upon the supposition that 
the vibrating parts of the inner ear continue in motion as long as 
the after-sensation persists; and that there is no residual sensation 
capable of giving beats other than that which persists only as long 
as the actual vibration in the ear itself continues. Such a. sup- 
position seems quite improbable. Moreover, the phenomenon of 
beats in the case of sounds acting simultaneously upon different ears 
cannot be explained even upon this supposition. There must "be 
some kind of vibration, using this term in the most general sense, 
or some kind of alternation of phase or state within the sensorium 
itself. 
“There is no question as to the fact of the production of beats 
under the circumstances last mentioned. I have not only verified 
the fact by experiments conducted in the usual manner, but have 
also studied the beats produced when a tuning-fork is held close to 
one ear and the sound of a second fork, not quite in unison with it, 
is transmitted by telephone to the other, taking suitable precautions 
that the ear against which the receiving telephone was held was not 
affected by the sound of the neighboring fork. Beats were readily 
obtained, which grew weaker as the fork was moved away from the 
open ear and approached to the ear which was closed by the receiver. 
The same result occurred when, instead of mistimed unisons, har- 
monic forks were used, which gave beats with the fundamental.” 
In the experiments just cited, the fact that the fork gave feebler 
beats as it was moved towards the ear to which the telephone was 
applied seemed clearly to indicate that the beats were not likely to 
be due to sounds transmitted through the head from one ear to the 8 
PROCEEDINGS OF THE AMERICAN ACADEMY 
other, or to the passage of the vibrations through the skull from 
the outer air. These experiments did not differ in other respects 
from those of previous observers. The beating of the sound per- 
sisting in one ear after excitation had ceased with a second sound 
falling upon the other ear was, in fact, observed by S. P. Thompson, 
in 1881. 
The existence of such binaural beats as are under consideration 
was noted as early as 1874 by Mach, who assumed that the sound 
was conducted from one ear to the other through the bones of the 
head. S. P. Thompson has studied the subject very carefully;* 
and while he is evidently strongly inclined to believe that the phe- 
nomena are due to an interference of sensations not produced by the 
same ear, nevertheless he does not appear to consider this as a fact 
finally settled beyond all question, although to us his results seem 
to leave small room for doubt on the matter. 
The important bearing of the existence of binaural beats upon 
some parts of Helmholtz’s theory of audition does not seem to have 
been very fully recognized. This may be because of some question 
as to whether each sound did not after all really act upon both ears, 
even when apparently applied to one alone. From the nature of the 
case, it is very difficult to prove finally that there is no possibility 
of such conduction of sound from one ear to the other, but we have 
sought to study the phenomena of binaural beats under circum- 
stances which render such conduction improbable in the highest 
degree. With this end in view, we have repeated most of the ex- 
periments of Thompson, and made such additional tests as suggested 
themselves to us. 
The sounds of two tuning-forks placed in a distant room were 
conveyed to the ears telephonically over two separate circuits. 
These forks were struck by hammers actuated by electro-magnets 
placed in circuits separate from those containing the telephones, 
and governed by keys operated by the foot of the observer at the 
receiving end of the line. This method was found to be preferable 
to running the forks continuously by electro-magnetic means, as 
the currents used in the latter method were liable to act directly 
tipon the transmitting telephone and cause disturbing noises. Very 
powerful magneto-telephones were used as transmitters and receivers, 
forks were used throughout the experiments, unless otherwise 
* Philosophical Magazine, Vol. IV. p. 274 ; Vol. VI. p. 383, Vol. XII. p. 351; 
Vol. XIII. p. 406. OF ARTS AND SCIENCES. 
9 
stated, on account of the remarkable intensity with which their 
tones were transmitted, this being approximately the natural tone 
of the diaphragm. The singular phenomena of the localization in 
the back of the head of the sensation caused by two notes in oppo- 
site phases when heard by separate ears, and the apparent wander- 
ing of the sensation from one ear to the other in the case of binaural 
beats referred to by Thompson, were always clearly recognized. 
The method of experimentation precludes any fusion of the two 
sounds, except within the head; but to decide whether the beats 
arise from an interference of the sensations or from a mechanical 
interference in Corti’s organ, we must investigate further. We 
tried to throw some light on this point by the following experiments. 
It is well known that the vibrations of a fork, even when of too 
small amplitude to be heard when held opposite the ear itself, can 
be made audible by pressing the stem of the fork against the bone of 
the skull, or, still better, against the teeth, the latter transmitting 
vibrations to the ears better than any other part of the head. A 
still more effectual means of making audible very small vibrations is 
to close the ear with a bit of beeswax and press the stem of the fork 
lightly against the wax. In this case the vibrations are transmit- 
ted to the membrana tympani by the small amount of air enclosed 
within the meatus, as is clear from the fact that the sound of the fork 
is heard on touching the wax long after it ceases to be audible on 
touching its stem to the pinna of the ear. lienee in this case there 
is no conduction to the middle or inner ear through the bones of the 
head. Whatever sound reaches the ear follows the ordinary path 
through the meatus to the membrana tympani. Now we found 
that the vibrations of a fork could he heard longer when touched to 
the wax in the ear than when held against the teeth. We therefore 
took two small tonometer forks making four beats per second, struck 
them very gently, and held their stems against the teeth; loud beats 
were heard in the ears just as they are ordinarily heard when power- 
ful forks are sounded in the air. The forks were then held in this 
position until the beats had entirely ceased to be audible, when 
they were removed, and the stem of each was touched to the wax 
closing the two ears. Instantly the two notes were heard, faintly 
but distinctly, in the ears to which they were held, and accompany- 
ing them were faint heats seeming to wander in the head from ear 
to ear, as is always the case with binaural beats. In trying this 
experiment care was taken that no vibration should be imparted to 
the fork from a gentle blow given on touching its stem to the wax, 10 
PROCEEDINGS OP THE AMERICAN ACADEMY 
and it was found that the liability of such an occurrence could be 
greatly diminished by covering the end of the stem with a layer 
of wax. 
To make sure that these beats were not the result of imagina- 
tion, we took forks giving four, eight, twelve, and sixteen beats 
per second at random, struck them lightly as before, but in this case 
held only one of them against the teeth, so as not to know the 
number of beats which ought to be produced. When this fork had 
ceased to be heard, we touched both forks to the wax in the ears, as 
before, and in every case the correct number of beats was at once 
heard. The experiment was varied slightly as follows. One ear 
only was closed with wax; the other was immersed in a large basin 
of water. The experiment was then repeated as above, with the 
difference that one fork, instead of being touched to the ear, was 
touched to the marble basin, its vibrations being transmitted to the 
enclosed ear through the water. The same results were obtained 
as before. These experiments were all carried on at night, when 
there was not the slightest disturbing vibration in the air. To- 
gether with another experiment described below, they lead us to 
conclude that aerial vibrations acting upon the ear are not trans- 
mitted through the skull or bony parts of the head from one ear to 
the other. The experiments cited certainly seem to bear out this 
view, at least for tones of feeble intensity. Furthermore, the fact 
that differential tones have not been produced binaurally leads us 
to believe that it is true generally, even for very intense sounds, 
that binaural beats do not result from sound conduction within the 
head. 
In his experiments Thompson was never able to obtain differen- 
tial tones \inless the sounds were allowed to mingle before falling 
upon the ears. Our experiments fully bear out this result, which 
we have verified in the following ways: (1) By listening to Uti 
and Mil forks through the telephones. These notes were trans- 
mitted with more than sufficient intensity to give rise to a differ- 
ential when heard in the ordinary manner; but with a telephone 
placed at each ear, not the faintest trace of such a tone could be 
heard. We also reached equally negative results on listening to 
Mii with a telephone, and holding an Ut4, SolA, or Sols fork close 
to the other ear, or on leading their vibrations to the ear through a 
long rubber tube. (2) By Thompson’s method of carrying the 
tones to the ears by means of rubber tubes (in our experiments 
about thirty feet long), the forks being struck, one in another OF ARTS AND SCIENCES. 
11 
room and the other outside of the window, to prevent their tones 
from mixing before reaching the ears. No differential tone could 
he heard. We also tried to use Koenig’s forks for illustrating 
beat-notes, but their very rapid vibrations were so deadened by 
the tubing that they could only be heard very faintly. (3) By 
the same process as above, with the exception that the ends of the 
rubber tube were connected with cylindrical resonators which were 
blown as already described. This gave almost deafening tones in 
each ear, but not the slightest trace of a differential. But in this 
case, as also in (2), if the two tubes were connected to two branches 
of a Y tube, and a third rubber tube leading to the ear was con- 
nected to the third branch, the differential tone came out at once. 
(4) The most striking proof was the following. The ears being 
closed with wax, a brass rod about five feet long was held lightly 
against the wax in each. When the stems of forks struck by two 
assistants were pressed against the farther end of the rods, very 
loud tones were heard in the ears, unaccompanied by any differential 
tone. If, however, one of the rods was removed from the ear and 
pressed tightly against the head, or, better, against the teeth, a loud 
differential tone was heard at once in the ear against which the rod 
was placed. If both rods were held against the teeth or head, the 
differential tone was heard in both ears. 
This apparent impossibility of producing differential tones when 
the pri inary tones are caused to actuate separate ears would seem 
conclusive of the point under discussion except for the fact that it 
might reasonably be argued that, if sound conduction through the 
skull occurred from one ear to the other, the vibrations thus con- 
ducted might act directly on the inner ear, and not indirectly 
through the membrana tympani and chain of bones, in which case 
no differential tones should arise, although beats could still take 
place in the manner assumed by Helmholtz. But our experiment 
with the brass rods, described above, shows that such tones are in 
fact readily produced when conduction to the ear takes place through 
the bones of the head. And if there were any material conduction 
of this kind, it certainly seems as if the very loud sounds used 
by us would have given evidence of it at least by the occasional 
production of differential tones when the sounds actuated separate 
ears. 
The production of beats without the simultaneous action of the 
beating sounds upon the same ear seems, therefore, to be clearly 
shown from these facts: (1) Beats are produced by sounds falling 12 
PROCEEDINGS OF THE AMERICAN ACADEMY 
upon the ears separately even when the greatest precautions are 
taken to prevent both from affecting the same ear. (2) Under no 
circumstances does it seem to be possible to produce differential 
tones by binaural combination, even when the separate tones are 
extremely loud. (3) In all cases in which the beating sounds have 
apparently acted separately upon the auditory apparatus of each ear, 
the singular phenomenon of encephalic localization of the beats has 
appeared, while this phenomenon has never been found present 
either when the two sounds acted simultaneously upon the same ear 
through the auditory canal, or when one of them was manifestly 
transmitted to the ear through the bones of the skull. 
We are therefore led to the conclusion that the mechanism in 
which the interference causing beats takes place, and through the 
operations of which we consequently recognize the character of con- 
sonant and dissonant intervals, lies, in part at least, beyond the 
auditory apparatus of the inner ear. It seems to us clear that notes 
may beat when they, do not affect the same vibrating elements of 
the inner ear, and that we must look to some more profound changes 
within the sensorium itself for a complete explanation of the phe- 
nomena under consideration. 
It may be possible that there are two distinct causes of beating; 
one, that suggested by Helmholtz, which wholly or principally 
obtains when the two notes act simultaneously upon the same 
ear, and the other due to some different and more obscure action 
which obtains in the case of binaural beats. In this case quite 
different laws as to consonance might hold. The exceeding harsh- 
ness of dissonances in binaural audition lends some support to this 
view. 
Rogers Laboratory of Physics, 
May, 1891.