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My motivation for this question is related to biological phenomenon of our inner ear, so I initially posted in the biology community, but I suspect the answer has a DSP explanation, so I figured this might be a better place to get an answer.

I'm learning about pitch perception in an Intro to Music class right now, and learned about the case of the missing fundamental.

In the main image in that wikipedia page, it seems like in the bottom graph, with the fundamental frequency and its second harmonic removed, the wave is still very periodic at 100 Hz. Since the Organ of Corti has some area which should be excited when there's a sound wave of frequency 100 Hz, why does this soundwave, which seems periodic at 100 Hz, not directly excite that region of the organ?

(like, won't there be high air pressure every 0.01 seconds, which will sway that organ in some direction, and thus excite that region which is "resonant" for sounds at that frequency?)

Maybe another way to phrase the question: why does the top signal have a signal with frequency of 100, but the bottom signal does not have a part with frequency 100? It seems to me that the bottom signal also has a part with frequency of 100 Hz.

enter image description here

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    $\begingroup$ to get a dsp answer we need to have a model of the system inside the ear. Then it is realtively easy to answer. The difficulty is finding the model. The literature is actually pretty active in that area. For example if a loud low frequency and high frequency are on top of each other we tend to prefer the low frequency content which MP3 compression relies on such psychoacoustic rules. The problem is understanding the process of hearing perception system. It is really not that trivial about the frequency spectrum of a signal. $\endgroup$ – percusse Apr 17 '18 at 14:37
  • $\begingroup$ It might help your understanding to eliminate even more of the harmonics (not just the fundamental). Consider a wave that only consists of 1000 Hz and 1001 Hz, for instance. These are both harmonics of a 1 Hz tone, and the combined wave will repeat at 1 Hz, but there will be no lower frequencies present. $\endgroup$ – endolith Apr 17 '18 at 16:33
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There are two distinct areas or domains to keep in mind when thinking about the idea of pitch, the acoustic signal (in your case a complex tone composed of a set of harmonics based on a fundamental frequency) and our perception of that acoustic signal.

Let's say you compose two complex tone signals, one signal includes the fundamental and a set of harmonics and the other signal has only a subset of the harmonics with no fundamental frequency. Using a 100 Hz fundamental frequency, an example set of frequencies for the first signal might be (100, 200, 300, 400, 500, 600; all values in Hz), while an example set of frequencies for the second signal might be (400, 500, 600; all values in Hz). I've generated spectrograms and links to WAV-files for each of these signals.

The spectrogram of the first signal is shown below. This signal has energy at six frequencies - the fundamental and five harmonics. A 3 second WAV-file of this signal.

enter image description here

The spectrogram of the second signal is shown below. This signal has energy at only four of the harmonic frequencies of the first signal. A 3 second WAV-file of this signal.

enter image description here

When each of these signals is individually presented to a listener, the only regions of the basilar membrane (BM) that are excited are the ones where there is acoustic energy present. Since the first signal includes the 100 Hz fundamental frequency and all of the harmonics, it will produce mechanical activity on the BM at these frequencies. The second signal will only produce activity at the frequencies that make up its composition.

If you listen carefully to these two signals they'll sound similar but they will have slightly different acoustic coloring, which is called timbre. Importantly, the 100 Hz fundamental is noticeable in both signals. Although I can detect the fundamental frequency in the second signal, the fundamental frequency is more pronounced in the first signal because the fundamental frequency is present in that signal. Pitch is influenced by the composition of a complex tone as well as other factors such as the intensity at which it listened to. These elements may play a part in how these two signals are perceived. For reference, here is a 3 second WAV-file of just the 100 Hz fundamental frequency.

Our perception of these two signals and our detection of the 100 Hz fundamental in the second signal, often called the residue pitch, is a function of our auditory system that extends beyond the BM of the cochlea.

There are two classes of models that attempt to explain the residue pitch, pattern recognition models and temporal models. For information about these models I suggest the textbook "An Introduction to the Psychology of Hearing", Fifth Edition, B. C. J. Moore, Emerald, 2008, specifically. Another very good textbook is "Psychoacoustics, Facts and Models", Third Edition, H. Fastl and E. Zwicker, Springer, 2007.

I hope this helps you.

Michael.

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Here is roughly how it works.

  1. A typically musical note is made out of harmonics and the fundamental. The fundamental is also the spacing between the harmonics.
  2. For 100 Hz fundamental you would get 100Hz, 200Hz, 300Hz, 400Hz, 500 Hz, etc.
  3. For 200 Hz you would get 200Hz, 400Hz, 600 Hz, which is simlar to 100Hz but not the same. Note that specifically 300Hz and 500Hz are missing. The spacing here is 200 Hz
  4. In practice you rarely get all the harmonics, some are louder than others, some are missing all together, however, you still can easily hear the pitch
  5. That's because you determine pitch by the spacing of the harmonics, not by the lowest.
  6. For example, if you generate a sound by adding 500, 600, 700 & 800Hz, you will hear the pitch corresponding to 100Hz, although no frequencies below 500 Hz are in the signal at all. 100 Hz is the periodicity, not necessarily the frequency content.
  7. The organ of corti does actually a mechanical Fourier transform. The excitation through the oval window creates a bending wave on the basilar membrane. Due to mechanical resonance of the membrane and some active feedback mechanism controlled through efferent nerves and hair cells, there is a strong localized peak in the bending wave, the position of which corresponds with the frequency
  8. In our example above you would see peaks at 500, 600, 700 & 800 Hz positions, but NOT at 100 Hz. There is no 100 Hz in the signal, so there is no excitation at the 100 Hz location.
  9. The actual pitch detection is done by identifying correlated spectral parts parts of the signal and determining if there is a constant spacing. The spacing is the pitch. I don't know of the top of my head where in the periphery that happens but I'm guessing it's most like down through some correlation mechanism
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  • $\begingroup$ i guess it would say what is more fundamental than your point #5 is that you determine the fundamental frequency by the reciprocal of the smallest period. actually, the pitch is the logarithm of the fundamental frequency, relative to a reference frequency: $$ p = \log_2\left(\frac{f_0}{f_\mathrm{ref}}\right) = \log_2\left(\frac{1}{T \, f_\mathrm{ref}}\right)$$ but you're correct that the spacing of the harmonics is the fundamental frequency, $f_0$, which is equal to the reciprocal of the shortest period. $\endgroup$ – robert bristow-johnson Apr 18 '18 at 11:15
  • $\begingroup$ Thanks for your answer! I'm afraid I'll be revealing my complete lack of knowledge of DSP here, but my follow-up question is about '8. In our example above you would see peaks at 500, 600, 700 & 800 Hz positions, but NOT at 100 Hz. There is no 100 Hz in the signal, so there is no excitation at the 100 Hz location.' Why is that? Why does the 100 Hz section on the basilar membrane react to the top signal but not the bottom? Isn't there still spikes of air pressure with periodicity 100 Hz that would evoke for something that resonates at 100 Hz? $\endgroup$ – Kevin Wang Apr 18 '18 at 11:49
  • $\begingroup$ Is the answer just that the fourier transform of that signal wouldn't have any 100 Hz components, and that's that? $\endgroup$ – Kevin Wang Apr 18 '18 at 11:49
  • $\begingroup$ @KevinWang: yes, that's pretty much it. $\endgroup$ – Hilmar Apr 19 '18 at 12:07

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