292 H. W. Poole on the Musical Ratios. 
This difference is barely under the number 1382, at which the beats 
vanish ; consequently the ci anya must be very slight indeed. 
Ta aking the fourth, the numbers 
384—312; nye ae 
Here we are clearly within the limit where the beats vanish, the 
consequent roughness being quite sensible. 
Taking the major third, “the numbers are 
320—256; difference=64. 
Here we are still further within the limit, and, accordingly, the 
nets | is more percept 
us we see that the seperti of our tuning forks is entirely 
in accordance with oP seek aon which assigns the dissonances 
to beats. pp. 299-30 
There is an RAs error in the numbers assigned to the 
Yourth. As four tuning forks are mentioned, they are probably 
those of the key-note, its octave, fifth, and major thirds. The 
interval of the fourth, then, would be obtained by sounding 
the fifth with the octave; giving 
512384: difference =128 
sate would give to the fourth the same “ roughness ” as to the 
If coincidences in the vibrations of simple chords be what 
is meant by “beats,” it would be easy to show that 33 such 
beats per second are not necessarily disagreeable. It is stated 
that 33 vibrations give a “perfect musical tone.” If we sound 
with this its octave with 66 vibrations, we shall have a still more 
agreeable musical effect, as always results from such addition 
toa very low note—and there will result just 33 coincidences, 
or so called beats. But if we should add a fifth, with 49°5 vi- 
brations, the effect would not be so good. A major third, with 
41-25 vibrations would be strug yet the ‘ beats” would 
be just 33 in each case. Why is this differen mce? Theanswer 
is found re examining the harmonic series: 
1:2:3:4:5:6:7:8:9: 10. 
to the double octave, with five times 
a with seven times 33 or 231. The rule 
