Our auditory system completes the harmonic series when some harmonics are missing. Even when the first 5 harmonics in sound 4 are removed, we continue to hear a C2 that is not present. The upper harmonics stimulate our auditory system to recreate the missing harmonics.

Arthur H. Benade in his book Fundamentals of Musical Acoustics talks about heterodynes components (also known as combination, resultant or subjective tones):

"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 generated by two sounds and groups these components in 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)

If we apply Benade formulas to the notes of a major triad (shown in red) we get the heterodynes components corresponding to the harmonic series (shown in black):

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 why the creative process of our auditory system is able to recreate the missing harmonics.

© 2017 José Rodríguez Alvira. Published by teoria.com


    
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