The sounds we have just heard suggest that the eliminated harmonics have somehow been recreated.

Arthur H. Benade discusses this phenomenon in his book Fundamentals of Musical Acoustics. He speaks of mysterious components which he calls heterodyne, resultant, differential or summation sounds:

"Use of a probe microphone and wave analyzer to study sounds... show no sign of the mysterious components... Deeper probings... confirm that the hearing mechanism itself is creating new components. Furthermore, we learn that both the mechanical and neurological parts of our ears take part in this creative process." (p. 256)

Benade explains how we can calculate the frequencies of the heterodynes components produced by two sounds, and groups these components into three categories:

Original Components	Simplest Heterodynes Components	Next-Appearing Heterodynes Components
P	(2P)	(3P)
	(P + Q), (P- Q)	(2P + Q), (2P - Q) (2Q + P), (2Q - P)
Q	(2Q)	(3Q)

Curiously, if we apply Benade's method to a major triad, we can complete the harmonic series. In the following image you can see how to generate the first 10 harmonics:

Harmonic	Note	Frequency	Formula
1	C2	65	E4 - C4 = 325 - 260 = 65 Hz
2	C3	130	G4 - C4 = 390 - 260 = 130 Hz
3	G3	195	C4 x 2 - E4 = 325 x 2 - 325 = 195 Hz
7	Bb4	455	G4 x 2 - E4 = 390 x 2 - 325 = 455 Hz
9	D5	585	C4 + E4 = 260 + 325 = 585 Hz
10	E5	650	C4 + G4 = 260 + 390 = 650 Hz

We can now understand how the creative process of our auditory system is able to recreate the missing harmonics.

© 2017 José Rodríguez Alvira. Published by teoria.com