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Quora answer by Mr. Kennan Hye BS (Engineering), MEng. (Acoustics):

Pitch is sometimes defined as the fundamental frequency of a sound wave (i.e. generally, the lowest frequency in a given sound wave). For most practical purposes, this is fine, and pitch and frequency can be thought of as equivalent. On the other hand, for most practical purposes, amplitude can be thought of as volume.

However, technically, pitch (and volume) are human perceptions. Thus, our perception of pitch and volume are not solely based on frequency and amplitude respectively, but are based on a combination of both (and even other factors). Frequency overwhelming dictates perceived pitch, but amplitude also does have some small, small effect on our pitch perception, especially when it is very large. For example, a very loud sound can have a different perceived pitch than you would predict from its frequency alone.

That all being said, usually these effects are negligible, and pitch can be thought of as equivalent to fundamental frequency.

Please explain elementarily; I'm never studied natural sciences.

I still don't accept intuitively how Amplitude can affect pitch. Can the bold sentences be exemplified by anything simple? The example in my mind is opera singers: a competent one's loud singing, shouldn't affect the frequency, because then (s)he'll sound off-tune.

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The section you've highlighted is concerned with psychoacoustics, a branch of psychophysics. So, the problem doesn't involve amplitude causing the frequency of the wave to change. It involves amplitude causing our mind's perception of pitch to change (Typically we perceive lower pitches.)

Note that this change is quite small! In Brian C.J. Moore's 5th Edition of Introduction to the Psychology of Hearing he reviews existing literature to suggest that this change in pitch perception for tones between 1kHz and 2kHz is less than 1% (i.e., 10 - 20 Hz).

Consider this in terms of your Opera Singer: singing a C at 1047 Hz. Even with a decrease to 1037 Hz she'd still be quite far from the B below it at 987.8 Hz, and it'd probably be difficult for her to tell that much has changed.

Overall he concludes "at present there is no generally accepted explanation for the shifts in pitch with level". Both the rejected theories have to do with hair cells in the cochlea. The more promising has to do with auditory fatigue due to being exposed to a loud sound, which might cause a shift in excitation of the cochlea's tiny hairs towards lower-pitched (basal) side of the organ. These basal hairs vibrate with lower incoming frequencies. Thus a proposed mechanism for the perceived shift in pitch.

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    $\begingroup$ A 1% miss in a note can easily be heard above 1kHz, particularly in a chord. That's an interesting figure, because often deliberate detuning is used both by singers and by piano tuners to set up beats with other notes in chords or with the other strings in the same note (notes above 1kHz have three strings each on a piano). Presumably, however, amplitude induced pitch shift would sound quite different, because it wouldn't induce beats (which are physical, rather than psychophysical). $\endgroup$ Nov 10, 2017 at 4:04
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I wonder if this could be the result of a logical fallacy (affirming the antecedent) happening in your brain, at the level of instinct. Confusing cause with effect -

My conjecture: We know that "Loudness Contour" is a real thing on "Hi-Fi" stereo amplifiers, where they boost bass and treble when you want to listen to your music at a quiet listening level, so you don't just hear the mid-range. Probably because your ears (not brain) are less sensitive to highs and lows.

What if your brain automatically knows about this loss of perceived loudness of highs and lows, perhaps learned as you move closer and farther from the speakers as you're moving around a room or your house while listening to full-spectrum music over the years, and tries to compensate for it?

So now when you hear some music go quieter, your brain tries to "boost" your perception of the highs and lows, so you don't miss out on what you're listening to. Actually your brain is always looking for differences from what it expects to hear - so it adjusts expectations so that unexpected changes may stand out.

But what if your brain gets mixed up on cause and effect? Let's focus on the low frequencies below mid-range, say 20-500 Hz. (The higher frequencies above mid-range would have the opposite effect to what I'm about to say.) The truth is that at low overall volume, lower frequencies have lower perceived loudness than they have at normal listening levels (due to lower sensitivity in the physics of the inner ear). At these low listening levels, lower frequency causes lower loudness in a sense. But your brain may think that so some extent lower loudness also causes lower frequency. All it "knows" is that there is a correlation between lower frequency and lower loudness. So now when you adjust the amplitude of a low note, your brain may expect the frequency to change with it, and when it doesn't change as expected, your brain perceives an "unexpected" shift in frequency that isn't really there in the physics. So as a low note is brought up in amplitude (physics), your brain expects the frequency to go up too. When it doesn't, that's a note that is unexpectedly "flat", which is what you (I at least) hear - that the note shifts to the flat as the volume comes up.

Such an interesting topic. I have more questions than answers. But what I've written here could be tested simply: Have a subject listen to different notes and adjust the volume himself to make them all the same loudness. Then have him do the same as you vary one note in volume: have him adjust the pitch so that it is the same pitch to him. Then try to teach his brain the opposite loudness compensation: play full spectrum music softer and louder, while over-compensating the highs and lows, and then run the tests again to see if the results are affected. That would prove that it is a learned effect, and not physics happening at the vibrating hairs in the inner ear.

By the way, those hairs have exotic physics happening in them: hairs in water should be highly damped, yet they ring like little bells. That is because they have positive feedback with muscle molecules built-in that compensate for the damping of water and keratin so that they vibrate more like steel in air than hair in water. Freeman Dyson talks about research he did with Tommy Gold on this at one time. Search "Freeman Dyson Hearing" and look at the edge dot org article.

As far as re-training your musical perception goes - I have been able to re-learn harmonic relationships by listening to music that I have shifted (not scaled) in frequency, which initially sounds like banging on trash cans, but over time (several minutes) begins to sound musical again, harmonic. Then listening to the original music sounds initially like trash cans again, but gradually improves back to natural sound perception. So if our loudness-pitch anomaly is a learned behavior, we may be able to re-shape it and show that that may be at least one part of why some people hear a frequency shift with changing amplitude.

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