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We know that frequencies that follow a base two logarithmic relationship sound as the same tone. This seems to be one of the fundamental principles that underlies music theory. For example, frequencies of 220, 440, 880, 1760, ... all sound as the same tone: 'A'.

f_n = f_0 • 2^n

After searching through several different music theory sources, I have not found a convincing physical or mathematical explanation for this phenomenon. Many sources say that human's perceive pitch on a logarithmic/exponential scale, or that the waves 'line up' in a way that creates a sense of 'sameness' between the two frequencies.

It is certainly clear that humans do perceive pitch in a logarithmic manner. My question: why? There must be a physical reason for why humans perceive frequencies that follow such a well defined mathematical relationship as being of the same tone.

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  • Um, they DO NOT SOUND THE SAME! Who ever told you that? C5 DOES NOT sound the same as C4, C3, C8, C12, etc. The reason why they sound so similar is because they are so similar, because the vibrations are exactly some power of 2. It is not physical, it is mathematical. There is no way to tell the difference between them. Could you tell the difference between to perfect sinusoidals at the same frequency? Of course not. Well, when you compare a note and it's octave they differ the least.
    – Stretto
    Nov 30, 2020 at 23:51
  • So if you add a metric to R to get "pitch space" then octaves would be the closest notes. It is sorta like p-adics. Ck is closest to Cn. Why? Because Any other note Xn when compared with Ck will produce "intermodulation distortion", e.g, 320hz and 353hz creates all kinds "extra" frequencies". Why? Because our brains can deduce all the combinations when it filters things. Take a metronome at 100bpm, we also can deduces 200, 400, 300, 50, 75, etc but the further the ratio the less likely to be(and it's imaginary anyways). Take another one though, 125bpm, now what?
    – Stretto
    Nov 30, 2020 at 23:55
  • well, we have two quick pulses then a space. Our brain will fill in the space with 25bpm to make a periodic sound. Hence 100 + 125 will sound much faster, but do it with 130 and it will sound even faster when you fill in the gaps because the ratio is not as simple(or it might sound slower in triplets). Our brains, for some reason, tries to simplify and align things probably to reduce complexity so it can understand things... so it tries to minimize certain things(I don't think anyone has figured out how it actually works so but we know what it does).
    – Stretto
    Nov 30, 2020 at 23:58
  • Our brains work a certain way, it is mathematical, but octaves do not sound the same, they are clearly different, it is just that the mathematical nature of the brain represents sound on a manifold where they are the closest notes. I believe this is probably due to the overtone series, since that is what we are conditioned to by nature and mathematics. So octaves are the closest together due to the OT series which we hear in almost everything and our brains pick up on that connection and treats octaves as the most similar pitches.
    – Stretto
    Dec 1, 2020 at 0:01

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There are lots of papers about octave equivalence. Some lean toward a cultural explanation, others (citing birds songs) prefer a physiological explanation.

https://www.quantamagazine.org/perceptions-of-musical-octaves-are-learned-not-wired-in-the-brain-20191030/

https://www.researchgate.net/publication/330865079_Octave_equivalence_perception_is_not_linked_to_vocal_mimicry_Budgerigars_fail_standardized_operant_tests_for_octave_equivalence

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5479468/

These papers prefer a cultural explanation.

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  • I swear I've read that at least some non-human primates also respect octave equivalence.
    – Dekkadeci
    Dec 1, 2020 at 14:30
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I do not believe that there is a convincing argument for this. In fact recent research in cultural anthropology suggests that our tendency to hear octaves as "the same note" is due to cultural brain washing. I cannot recall or locate the article at this moment (I will edit when I do) but there was a recent article that introduced data taken by interviewing people from some indigenous tribe that is not influenced by Western culture and they simply do not hear octave as the same note. And they're not in reality.

I am not sure why we do this except to say that it must have had some evolutionary advantage like not misinterpreting calls from people when they raise their voice, or something.

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  • Although as I recall that article, the general tone was "Wow, there are actually humans who don't have octave equivalence, this is very surprising"
    – AakashM
    Dec 2, 2020 at 15:50
  • Could you point us to some of this research? From what I know, octave equivalence is almost the rule, especially in cultures where men and women sing together. It's unlikely to be a coincidence that the mathematically simplest ratio after the unison, 1/1, is the octave, 1/2.
    – Scott Wallace
    Dec 4, 2020 at 12:33
  • @ScottWallace, here is an article. quantamagazine.org/… I do not understand your comment though. Are you suggesting that light of double frequency would be seen as the same color? Maybe, after all we cannot see an octave worth of the EM spectrum.
    – user50691
    Dec 4, 2020 at 17:07
  • I'm suggesting, and this is far from original, that the simpler the ratio, the more consonant/seen as identical intervals are. And I don't think the comparison with the EM spectrum applies to sound usefully, since our color receptors work completely differently than our ears do.
    – Scott Wallace
    Dec 18, 2020 at 16:24

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