A natural sound, like the plucking of a string, produces a complex combination of the basic frequency and its overtones. The first overtone is an octave up, the second is a fifth above that, the third is an octave above the second, followed by a third, a fifth, and so on.
The crucial point is that the octave overtones are a major and repeating component of the overtones of a natural sound, so it is no surprise that we perceive them as the same.
I investigated this question a while ago while looking at a related question … why do we have 12 notes in an octave? It turns out that the division of an octave into 12 equally spaced parts is the closest fit to the natural overtones. To get a better fit, you have to go to 24 equally spaced parts, which is the “semi-tone” system used in some Eastern musical systems.
So not only perceiving octaves as the “same”, but also our 12-note musical system, are natural outgrowths of the physics of how sound is produced.
This is really the crux of it. I took a “physics of music” course in college and wrote a term paper on harmonics of stringed instruments.
If you play a tone, say A 400Hz, you will hear a harmonic sequence (even if it’s a sine wave, your eardrum will generate it) in whole number multiples of that frequency, 400, 800, 1200, 1600, 2000, etc., which form the intervals described by intention. If you play 800Hz, you will hear *exactly the same set of * frequencies but missing 400Hz. (Each of those frequencies will be of varying relative proportions, giving the sound a different tonal color, but they’re all still there.)
The interesting question is why the brain seems to be physically organized in a certain way regarding frequencies. I read the link posted in the OP but am not up for researching the references given therein.
Thanks for that. It looks to be a considerable (and appreciated) elaboration on what Groman posted. I think I’ve understood the gist, but if you see any errors in my reply to him, please tell me about them.
One point, though: I still disagree with things like “this doesn’t have to do with instinct, it has to do with physics.” Its instinct and physics. If there were some good reason for it, we could have evolved to hear fifths as “the same.” Its just that, as you and others have now explained very well, there is a good reason evolutionarily speaking for us to hear octaves as “the same.” This is exactly what I was asking, so again, thanks. You have no idea how relieved I am to have an answer to this question. It was driving me nuts last night.
I don’t see why there must be an evolutionary advantage to hearing pitches the way we do. It may just be a natural outcome of how the brain works, and evolutionarily neutral.
I also have a visualization, if it helps. You were saying earlier how, say, 200hz and 400hz sine waves which look completly different sound almost the same but 400hz and 425hz look almost the same but sound totally different.
I think the problem is that sound doesn’t look like anything. Graphs of waves are nice for some things, but a wave isn’t a line.
Imagine waves on a beach. If you get steady waves every 4 seconds and waves every two seconds what does it look like? wave, slightly bigger wave, wave, slightly bigger wave. So if you graphed these waves they’d look very different, with lines crossing, but looking at the ocean you can barley tell them apart.
This is why they sound “the same,” because when you’re not looking at a graph of lines they blend into each other.
I think the graphs are the root of the misunderstanding. But they are so cool. Hooray for oscilloscopes!
All intervals apart from the octave are discordant* to some extent, including fifths.
If you play a note with a tone that is not a sinewave then the note an octave up from the fundamental is already in there, so are the higher overtones that CookingWithGas wrote about. If you play a stringed instrument you can feel the beat ‘note’ when you play anything other than a perfect octave.
I doubt somehow that our hearing evolved to listen to music anyhow (speech to some extent maybe?). A more specific question could be “why we can discern pitch well at all?” After all the ability varies, many people are tone deaf to some degree or other.
*that is, overtones are produced that are not harmonically related at all. As a huge WAG it might be that the only reason that other intervals sound OK to us is that our hearing range is limited. If we could hear all the way from 0.1 to 500 kHz maybe everything would sound like it was being sung by Daleks.
Consider the mechanism by which we discern pitch. The outer ear transmits vibrations to the cochlea. This is an organ with different length beams with nerve cells at their bases. Each of these is mechanically resonant at a different pitch. They vibrate sympathetically with the incoming sound, and this vibration stimulates the nerves at the base. The thing is, however, that a simple structure that is resonant at 440 Hz will also be resonant at 880Hz, 1760Hz, etc. When we hear a 880Hz tone, the 440 Hz “sensor” in the cochlea is also being stimulated, but also the 880Hz one. Thus, while we can distinguish between the two, we also are aware of the harmonic relationship.
It might be possible to evolve weights or other structures on the beams to break up the harmonic resonances, but this would require that NOT hearing harmonically related tones as such have a advantage to drive natural selection. In modern society, those with musical talent often have a strong mating advantage, so this is not likely to happen unless dissonant music comes to dominate pop culture.
Perhaps it would help to use the analogy of sight.
The reason we see in a certain frequency of electromagnetic radiation is that this is the dominant radiation in the spectrum given by the sun. The physics of light waves are there first. It might be helpful to see in the infrared or ultraviolet but evolution is about being good enough not about being perfect. However, the ability to see colors is a mathematical consequence of the ability to see light in the first place. There’s no meaning is asking why we evolved to see blue.
This is another way of saying that it’s all about physics, not biology. We are evolved to hear sound waves, not specific frequencies. The mathematical relationship between octaves ensures that we will perceive them as such within the range of our hearing. But this is a second-level consequence of hearing. It has nothing to do with evolution or biology. If it did, then there wouldn’t be the dozens of tonal systems in use around the world. Instinct plays no part in it at all.
Your question, therefore, is presented backward. The ability to perceive sound waves is an evolutionary advantage that is biologically beneficial. The ability to react to sound waves that are related mathematically is not.
Right, you’re saying it might just be an evolutionary accident. But I was expressing amazement at the “accident” that evolution hit on a way of doing things which has this co-incidence with some mathematical and physical facts that don’t, themselves, seem to have any implications for fitness for reproduction and so on.
But others here have shown me what the reason is. Octaves are in fact naturally “similar” in a real sense, and so, there would have to be a special reason for us to not hear them as similar, just as Groman has said. We are evolutionarily fit to hear octaves as the “same” for reasons similar to those we are evolutionarily fit to find simple arithmetic intuitive. In both cases, what we find intuitive fits a pervasive fact about the physics of the situation around us. That is what I did not know about in posting my OP. I did not know there is a fact about sound which makes it the case that frequencies that are multiples of each other produce waveforms that are more similar than do frequencies which are not multiples of each other.
Seconded, as learned in “Wave Propagation” last semester. To elaborate, in a one-dimensional string vibration (such as a guitar). The first harmonic of a plucked string will have two nodes, one at each end. The second will have three nodes, with the third being in the middle. This will cause the effective wavelength to be halved, therefore doubling the frequency. The third harmonic will have four nodes, the inside ones being spaced 1/3 of the length from each end. Again, this will third the length, thirding the wavelength and trippling the frequency.
The amplitude of successive harmonics decays logarithmically, so higher harmonics become negligible in percieved sound.
Right, I’ve seen this explained, and I think it is a summary of a good answer to my question.
This still doesn’t mean “its physics, not biology” or “its math, not biology” as some on this thread have said. Its physics/math and biology. There are good reasons, from the evolutionary perspective, for our biology to be such as to cause us to represent and respond to our environment in a way that meshes with the physical/mathematical facts that have been described in this thread. Namely, since octave notes’ waveforms really are similar, the simplest engineering solution is one which treats them as similar. And since there’s no advantage to be had in distinguishing them more strongly instead (thereby, incidentally, losing instinctive access to the fact that they are similar, for whatever that access might actually be worth,) the simplest engineering solution is the one to go with on this one.
The physics/math, therefore, determine the best way for organisms to evolve. I was asking how in this particular case it was determined physically/mathematically that our organisms should have evolved as they did.
As I’ve said several times, my question has been answered to my complete satisfaction.
I’m displeased that many people are still approaching the conversation as though I was asking something like a “dumb question” in the first place. It was not a dumb question. It was a well founded and well motivated question. And it is a well answered question.
There was a sort-of answer to your question. It was somewhat tongue-in-cheek, as are all entries in the column Daedalus, which appeared (still appears?) in the journal Nature. All the entries are speculative, off-the-wall, but at least su[perficially plausible scientific and engineering ideas.
The suggestion in this case (made, IIRC, sometime in the late 1980s or early 1990s) was that our simian ancestors developed the ability to tell octaves (and to dislike the lack of them) as an aid to brachiation. If a tree limb made a sound that consisted of creaks and squeaks separated by neat octaves, then you were hearing the sound of an intact branch, which consisted of harmonics of the fundamental. A cracked branch, however, would have a more complex tonal quality, inculding terms that weren’t harmonics, and they could recognize a potentially broken branch in this way. They’d know not to place their full weight on it, and would come to is like it. Or natural selection would have those who liked the non-harmonic chord made by a broken branch dying, and only those who hated it (and wouldn’t swing on such a broken branch) surviving.
Certainly a simply-supported board (one just resting at each end, unconstrained) and a single-side supported bar (like a branch) have harmonics that are integral multiples of a fundamental.
An interesting, if far out, suggestion. But since reading it I haven’t been able to come up with an alternative one that gives some sort of advantage – evolutionary or otherwise – to recognizing and/or liking combinations of harmonics.
Interesting you should bring this up since, as you probably know, the way we see color does not directly reflect the physics of color in almost any way.
It turns out, as has been demonstrated to me on this thread, that the way we hear sound as sort of “looping” as you go higher through octaves actually does reflect the physics of sound.
But the way we see color as being such as to naturally be thought to form a “color wheel” does not reflect the physics of light. Light waves go up in frequency continuously, they do not loop around as do color wheels.
Furthermore, sounds naturally suggest an “ordering” to us, no matter what interval is played. But colors do not suggest an “ordering” to us instinctively except over relatively small intervals. (Of course we can piece together an ordering based on judgments of similarity, but thats not how it works with sound, even though the physical basis of the ordering in both cases is formally similar.) If our visual percepts matched the physics the way auditory percepts do, we would very likely see colors instinctively and immediately as “ordered” to the same degree that we hear sounds as so ordered.
I believe you have misspoken yourself.
It is not a mathematical consequence of the ability to see light that we see colors. We could just see brightness, without seeing hue.
And of course there is meaning asking why we evolved to see blue, just as there is meaning, as you’ve already said, in asking why we didn’t evolve to see infrared.
I’m sure you mean something other than what you’ve actually said.
See a couple of my previous posts, including the one just previous to this one. Its about physics and biology. Sometimes, it is advantageous to have a mind which represents things in a way accurate to the physics of the situation. Sometimes, its not.
That’s just not true. We could very well (though very improbably) have evolved to hear fifths as “the same” and octaves as “different.” This would not have reflected the physics of sound, but that’s beside the point. Or visual percepts in many ways do not reflect the physics of light. Our predictions of trajectory do not reflect the physics of motion. (There are good reasons for them not to do so–good evolutionary reasons. But the important fact is, they don’t reflect the physics of motion even though intuitively that’s what they’re “supposed” to do.) It is possible for us to have mental apparatus which do not reflect physics accurately. So there’s nothing about the physics of sound which determined that we must hear things one way and not another.
No, its not correct that if it had to do with evolution then different people would judge tones differently. Its precisely because of evolution and biology that we all hear them in the same way.
If similar reactions to similar waveforms is not evolutionarily advantageous, and yet we have ended up being such as to so react nonetheless, then the amazement I expressed earlier in this thread is called for. I hope this post has explained why.
These columns occur in, at the least, the visual system, auditory system and prefrontal cortex. The evolutionary advantage to them is that they are a single tool that serves diverse purposes. Nothing more, nothing less.
I think this sums up your post, and your more basic problem of not understanding evolution.
Nothing that did not reflect the physics of the situation could have evolved. I don’t understand any of the examples in which you claim that we do not use the proper physics. (E.g. “Our predictions of trajectory do not reflect the physics of motion.”)
My underlying point is that all evolution takes place to account for the physics of the world around us. To hold up a particular consequence of physics and ask why we were evolved to use it becomes a “Just-So” story, as in CalMeacham’s example.
Finding a true example of evolution which worked contrary to physics would be an amazing feat. As I said, I don’t believe you have any examples, but perhaps you could explain what you meant.
Guys, evolution does not always hard code everything! If you have a very general neural substrate, such as cortical columns, you can learn about the physics of the environment you are placed in, and furthermore, you will be more adaptive because of it!
On preview I see a lot of intervening discussion but I think this is still relevant.
I would disagree. Different animals have evolved the ability to see different parts of the EM spectrum; birds can see ultraviolet and can distinguish more colors than humans can. This is due to a different structure of light receptors in the retina that has four pigments, whereas humans have only three. Early mammals had lost a pigment, then re-evolved one. There is an excellent article from Scientific American July 2006 on this (need to pay to read full article).
But the OP didn’t ask, “How is it that we can we distinguish different pitches?” and not even, “How can we tell that the frequency of one note is twice that of another (i.e., octaves)?” It asked, “Why is our perception of notes that are an octave apart such that they sound to us like the same note?” And the question was spurred by a linked article that showed evidence that this is a consequence of the organization of the part of the brain that processes sounds–it may be a primary consequence of hearing.
An interesting question that turns this analogy around might be: The highest light frequency humans can see is less than twice the lowest frequency. If we could actually see a light frequency that was double another frequency, would we see them as “octaves” of light? Do birds see light in the ultraviolet range as an “octave” of red?
