458 
NAD ORE 
| March 3, 1870 
Professor Tyndall * undertakes to determine the consonances 
of the octave, fifth, fourth, and major third for two simple tones, 
without employing combination-tones. He writes as follows :— 
“Bearing in mind that the beats and the dissonance vanish 
when the difference of the two rates of vibration is 0; that the 
dissonance is at its maximum when the beats number thirty-three 
per second; that it lessens gradually afterwards and entirely dis- 
appears when the beats amount to 132 per second—we will 
analyse the sounds of our forks,+ beginning with the ocfave. 
Here our rates of vibration are— 
512—256 ; difference =256. 
Tt is plain that in this case we can have no beats, the diffe- 
rence being too high to admit of them. 
“ Let us now take the 7/¢. Here the rates of vibration are— 
384-256; difference =128. 
This difference is barely under the number 132, at which the 
beats vanish; consequently the roughness must be very slight 
indeed. 
* Taking the fourths, the numbers are— 
384-288 ; difference=96. 
Here we are clearly within the limit where the beats vanish, the 
consequent roughness being quite sensible. 
“Taking the mayor-third, the numbers are— 
320-256: difference=64. 
Here we are still further within the limit, and accordingly the 
roughness is more perceptible. Thus we see that the deport- 
ment of our four tuning-forks is entirely in accordance with the 
explanation which assigns the dissonance to beats.” + 
Tt will not be difficult to test the value of the above reasoning. 
Starting from the rate of 256 vibrations per second selected by 
Prof. Tyndall, all that can be deduced from his definition of 
beats and dissonance at the head of the extract is that the maxi- 
mum of dissonance will fall on the interval 256 :256 + 33, z.¢. 
256 : 289 ; and that all intervals larger than 256 : 256 + 132, Ze. 
256: 388, will be free from dissonance. These numbers indicate 
almost exactly a whole tone and a //t/ respectively. Each of 
these results is contrary to experience: the dissonance of a whole 
tone is less harsh than that of a half tone ; and intervals greater 
than a fifth are by no means equally free from dissonance. 
Moreover, it follows that the determination of the octave by this 
reasoning is delusive, for the process would bring out, as perfect 
consonances, a seventh or a flat ninth, which are extreme dis- 
cords, just as readily as an octave. If we apply the same method 
to other parts of the scale than that to which Prof. Tyndall has 
restricted himself, the results are even more remarkable. Thus 
starting from the higher octave of 256, viz. 512, the maximum 
roughness falls on 545, a half-tone, and dissonance ceases after 
644, which lies between a major third and a fourth. For the 
next octave, z.e. starting from 1,024, dissonance ceases before we 
reach the interval of a whole tone. If we take lower positions 
on the scale we obtain opposite results. With 128 as our funda- 
mental note, the maximum dissonance falls on 161, slightly above 
a major third, while roughness extends ‘to 260, just beyond an 
octave. 
With 64 the worst discord is at 97, just above the f/##, and 
roughness reaches 196, another ocfave higher. Starting from 32 
the worst dissonance 65 is just above the ocfave, and roughness is 
not got rid of until 164, two octaves and a major third trom the 
fundamental note. 
The following octave exhibits at one view the results we have 
reached. The lowest note of each triad represents the fundamental 
note, the middle one the position of maximum dissonance, and 
the highest the limit of roughness. The middle note necessarily 
falls out of the last triad, as it lies too near the fundamental note 
to be represented. 
| Ps | 
= a 
ee 
I sete = 
= a 
Sale 
Prof. Tyndall’s method leads to the following conclusions :— 
The interval of an octave from the 16-foot C is the harshest 
possible dissonance ; so is that of a f/th from the 8-foot C ; so 
is that of a sazor-third from the lower C of a baritone voice. 
On the octave above the high C of a soprano voice, all the 
* T. p. 296. 
+ Tuning-forks produce, of course, sz7zfe tones. 
¢ T. pp. 296, 297. 
intervals beyond D are perfect consonances, while in the 16-foot 
octave there are no perfect consonances at all. 
These conclusions are so utterly at variance with facts, that 
the method by which they have been obtained must be pro- 
nounced erroneous. In fact, Prof, Tyndall is himself a witness 
that this is so; for, in speaking of the oc/ave, he remarks 
that if this interval be slightly impure, deats of the fundamental 
tone are heard.* Now this does not square with his 
own theory ; for suppose two simple tones with 513 and 256 
vibrations per second, which would form an impure octave : the 
difference is 257, which is as much ‘‘too high” as 256 was in 
the case of the pure octave. This interval should, therefore, 
give no beats, and an impure octave be as harmonious as a pure 
one. But according to Helmholtz’s view, the first combination- 
tone of 513 and 256, viz. 257, will produce one beat per second 
with the fundamental tone, as stated, but not satisfactorily ex- 
plained + by Professor Tyndall. This amounts to a practical 
admission by him that the beats of two simple primaries are not 
adequate for the determination of their consonances, and that - 
recourse must be had for this purpose to combination-tones. 
I claim to have shown that the method by which Prof. 
Tyndall appears to determine the consonances of simple tones is 
erroneous, and the determinations themselves fallacious. I pro- 
ceed to point out the defect which vitiates his reasoning. He 
enunciates but one condition for the production of audible beats, 
that their number should not exceed 132 per second. Helmholtz 
lays down a second, quite as important—that the tones produc- 
ing them should not differ too much in pitch. ‘‘ These beats,” he 
writes, ‘‘are powerful when their interval amounts to a half-tone or 
a whole tone, but weak and audible only in the lower portions of the 
scale, when it ts equal to a third, and they diminish in distinct- 
ness as the interval increases.” $ Were we see at once the reason 
why it is futile to attempt to determine consonances of a fifth or 
oclave by the beats of two simple primaries —viz. that for these 
wide intervals the beats are imperceptible. 
Let us now proceed to Prof. Tyndall’s theory of consonance 
for composite sounds. ‘Taking the octave C’, C” or 264 : 528, he 
writes :—‘* With regard to the octave C’, C’, its two fundamental 
tones and their over-tones answer respectively to the following 
rates of vibration :— é 
I J 2 
Fundamental tone 264 528 Fundamental tone 
Over-tones. . I. 528 1056 
2. 793 1584 
3. 1056 2112 
4. 1320 2640 
5. 1584 3168 
6. 1845 3696 
7; 2112 4224 
8. 2376 4752 
g. 2660 5280 
“Comparing these tones together in couples, it is impossible 
to find, within. the two series, a pair whose difference is 
less than 264. Hence, as the beats cease to be heard 
as dissonance when they reach 132, dissonance must be 
entirely absent from the combination. This octave, therefore, 
is an absolutely perfect consonance.”§ The same process 
is then applied to other intervals. For the th 264 : 396, 
the lowest difference between any two overtones being 132, 
the interval is ‘‘a]l but perfectly free from dissonance.” For the 
Jourth, 264 : 352, the least difference, 88, makes it ‘‘clearly in- 
ferior to the fifth.” Similarly the major third, 264 : 330, with least 
difference 66, is *‘less perfect as a consonance than the fourth, 
and the minor third 264 : 316°8, with least difference 53, ‘‘in- 
ferior as a consonance” to all the previous intervals.|| In each 
case the ‘least difference” is precisely equal to the difference 
between the vibration-rates of the fundamental tones ; so that, in 
spite of the array of figures, xothing is added by this process to that 
employed by Prof. Tyndall to fix the consonances for simple 
tones. Inasmuch, therefore, as that method has been proved 
erroneous, these determinations cease to have any validity. 
Here, again, the neglect of the second condition for the pro- 
duction of audible beats is at the root of the error. Helmholtz 
* T. p. 297. 
t piece Tyndall’s attempted explanation depending on difference of 
Phase between the two primaries is at once refuted by the general principle 
that azfference of phase in partial-tones has no effect on quality. See H. 
p- 193. A note and the octave above may obviously be treated as the ground- 
tone and first overtone of a composite sound. 
t H. p. 302. § T. p. 209. || T. p. 299—308. 
