What is Octave Equivalence?

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ISBN 978-90-816095-1-7 Applying physics makes auditory sense

       A New Paradigm in Hearing

       Willem Chr. Heerens

       and

       J. Alexander de Ru

       ©2010 Heerens and De Ru

“The incoming sound signal is transformed into the sound energy signal inside the cochlea. It is this signal that evokes both the mechanical vibrations in the basilar membrane and the corresponding electrical stimuli in the organ of Corti, stimuli that are subsequently sent to the brain in a frequency selective manner.”

Mathematically, this signifies that the mammalian cochlea differentiates and squares the incoming sound pressure signal.

In terms of physics, it means that a sound energy signal is offered to the organ of Corti. Functioning as a Fourier analyzer, the organ of Corti subsequently converts these incoming signals into the sound energy frequency spectrum that is transferred to the auditory cortex in a frequency selective way.

Salient experimental results so far • For residual tone complexes – harmonic series where the first harmonic or fundamental is missing – the differentiating and squaring process in the cochlea reconstructs perfectly the corresponding but missing fundamental. • Contrary to the conclusion that an early neural mechanism is responsible for the mystery of the inferential pitch, strong evidence exists that the cause for this reconstruction of the virtual or fundamental pitch is hydrodynamic in origin.

I think a good start would be Music: A mathematical offering by Dave Benson, particularly Chapter 4.

My recollection definitely agrees with yours that our perception of consonant octaves (and fifths etc.) is a phenomenon that occurs with complex sounds (pitches with overtone series included) rather than with pure sine waves.

I remember reading somewhere how investigating the specific waveforms of an oboe and a horn can show mathematically why a major third sounds good with the horn above the oboe, but a perfect fourth sounds good with the oboe above the horn. (Please consider every single specific detail in that sentence extremely suspect.)