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Proceedings of the Royal Society 
met with only one individual who could hear what are termed 
secondary and tertiary difference tones, and he could not hear these 
without apparently a strong effort of attention. They were as fol- 
lows : — On sounding 16, C 2 =512 vibrations per second, and 20, 
e 2 = 640, he heard C° = 128, that is, 20 - 16 = 4. By using resona- 
tor No. 12, he heard 12 = 384, that is 16-4 = 12; and on using- 
resonator No. 8, he heard very feebly 8, that is 12 - 4 = 8. When 
C 2 and e 2 , that is tones corresponding to 512 and 640 vibrations per 
second, were sounded in this person’s ears, he heard other three 
tones with the use of resonators, namely, those produced by 128, 
256, and 384 vibrations per second. 
But when two tones are sounded, in addition to a tone produced 
by a vibration number equal to the difference between the vibration 
number of the two, another tone is produced, the vibration number 
of which is equal to the sum of the vibration numbers of the two 
primaries. This tone is called a summation tone. For example, on 
sounding 4 = 128 and 6 = 192 vibrations per second, by using resona- 
tor No. 10 = a tone having 320 vibrations per second may be dis- 
tinctly heard. Thus, 2 : 3, and 3 : 4, and 5 : 9, will produce sounds 
heard by resonators Nos. 5, 7, and 14, respectively. I have found 
that non-musical people can hear these summation tones with great 
distinctness if increased by resonators. They can hear the lower 
order of summation tones much more easily than the higher order. 
For example, all could hear the summation tone 2 (64) : 3 (96) = 5 
(160)- , or 4 (128) : 6 (192)= 10 (320) ; but only four out of the 
ten could hear 7 (221) + 8 (256) = 15 (480), and 8 (256) + 9 (288) 
= 17 (544). Only one out of the ten could hear 30 (960) + 28 (896) 
= 58 (1856), and none could hear 32 (1024) + 30 (960) = 62 (1984). 
I observed also that they could hear the higher summation tones 
only when the intensity was increased to as great an extent as 
possible. The two musical persons examined were able to hear all 
these sounds with ease, with even diminished intensity. 
According to Helmholtz, there are secondary and tertiary sum- 
mation tones, which spring from combinations of the primary sum- 
mation tones with its elements. Thus, 3 (96) : 5 (160), with the 
overtone apparatus, give 8 (256); 3 (96) and 8 (256) give 11 (352); 
5 (160) and 8 (256) and give 13 (416). Therefore, when 3 (96) and 
5 (160) are sounded, according to this statement, the listener with 
