Why does music become less harmonic if we transpose it down to the extreme low end of the piano?
The reason I'm asking this is, is so that I cross check an assumption in a philosophy post: https://philosophy.stackexchange.com/questions/100276/from-the-inverted-spectrum-to-the-music-transposed-by-12-problem/100281#100281
The linked answers are based on the assumptions that pitch hearing is the same over the whole audible range, and that pianos are in perfect tune. Both of these assumptions are not really true.
First, the pitch perception is most accurate in the human voice range and becomes more inaccurate towards low and high end. Additionally, the volume, duration, envelope curve and other sounds heard at the same time tend to affect the perceived pitch of low and high notes. These effects are smaller for trained musicians, but do not disappear entirely. I couldn't find a good web reference on simple search, so I refer to The Science Of Sound by Thomas D. Rossing here.
Second, pianos are not perfect tune. Because of the stiffness of the strings, the higher partials of low notes are sharp. To compensate for this, the low notes are tuned flat on purpose so that the partials would match higher notes. The low notes on piano also have relatively weak fundamental and a lot of overtones. Wikipedia has some material on this and searching for "piano inharmonicity" will find more. EDIT: The other reason for higher partials being out of tune is that the partials are slightly sharp from multiples of the fundamental frequency, but the notes are tuned to equal intervals to allow playing in all keys but produce intervals that are close enough, never exact.
Together these mean that a single note bass line can sound like it's in tune with itself and not clashing with higher notes on the piano but slightly flat compared to other bass instruments. Playing a chord on the low keys on the other hand produces a mess of harmonics that are very close to each other and slightly out of tune with each other. In some ways it is more like a tone cluster than a chord. I think that personal experience varies a bit, I would call the sound "growling" or "distorted" but mathematically it is interference from inharmonicity.
There are multiple aspects to this topic. For one thing if you take a justly intonated A major chord of sines down the octaves you will find that these keep sounding harmonic – but as we approach the lowest octaves we here very little of the actual notes:
http://petzel.at/chords/pure.mp3
This means that when you you play such chords you mostly hear overtones, not fundamentals. These overtones lie form many other intervals, leading to more harmonic clash. E.g. if you do the same with triangle waves the lowest octaves won’t sound as harmonic:
http://petzel.at/chords/pure2.mp3
But then we usually do not tune the piano justly. If we use equal temperament the chord will in itself have some amount of dissonance. Now, dissonance is not invariant under a pitch change: Intervals depend on the ratios of frequencies, while things like beat depend on the difference. This means that the beat of some dissonance is going to be a lot faster with high notes as compared to low notes. This makes this dissonance feel like "excitement" with higher notes, vibrato with middle notes and rumble with the lowest notes, as you can see here with sines:
http://petzel.at/chords/et.mp3
and with triangles:
http://petzel.at/chords/et.mp3
The next aspect is that a piano string hit with a hammer does not have anything that couples the overtones (as it would happen with e.g. a bow on a string instrument). This means that the overtones won’t be perfect multiples of a base frequency (so the overtones are not harmonic). This is in physics called the inharmonicity. This depends on multiple factors like the tension of the string, the length of the string, the stiffness of the string, the thickness of the string.
Now especially on smaller pianos you do not have the space to allow for very long strings, rather you get shorter but thicker strings. Such strings will have more inharmonicity.
So the low notes of the piano are less harmonic in the first place.
The final reason I’d like to give is a biological and psychological reason. Hearing happens by the ear essentially performing some sort of frequency analysis, with hairs resonating at different frequencies. Now, these hairs do not resonate at an exact frequency, but at a frequency range. The resonating frequency is just the frequency of highest response. This means that the brain needs to be trained to make sense of this data. Now human hearing is not trained on the whole frequency range equally well, which means that in the extreme parts like the lower end it is much harder for the brain and the hearing system to make sense of what it is getting.
There are three contributing factors.
Firstly, pianos are not in fact harmonic. The overtones are a bit off, a phenomenon known as inharmonicity which is why pianos use stretched tuning. Another effect of this inharmonicity is that the overtones of C2, E2, and G2 don't match up as closely as those of C5, E5, and G5, so you get a "muddier" sound.
You can hear this for example in piano music of the late 18th and early 19th centuries, which often calls for closely voiced chords in the lower registers. Listen to recordings of such a piece played on a modern piano and on a historical instrument. The strings in the old-style instrument are lighter and more flexible, so they are less inharmonic, and this is one reason for the greater clarity of such chords on these instruments than on modern pianos. Even within modern pianos, one reason longer instruments sound better is that their lower strings have less overwinding and are therefore more flexible and less inharmonic.
The second factor is that our ears are much less sensitive to lower frequencies, so the overtones become more and more overrepresented as we move to lower notes on the same instrument. Which actually doesn't have as big of an effect on how we perceive the sound as one would expect, because our brain will even create the illusion of hearing a completely imaginative fundamental frequency based on frequencies it interprets as the overtone frequencies. That's why we can hear the low notes of a double bass on a speaker way too small to perceivably reproduce the fundamental note, and church bells make use of this phenomenon too, to drastically save on material, weight and size. However, for inharmonic overtones this works much less well, and our brain will stop perceiving them as overtones when the fundamental frequency and lower overtones become too quiet.
And when multiple low notes come together, this overrepresentation of overtones becomes problematic even for instruments with very harmonic overtones (bowed strings, woodwinds, brass, organ pipes, etc.) which tend to sound rather "muddy" for low chords. If they are tuned very precisely to just ratios, they will sound less muddy because the audible overtones of the different pitches will match, but if they are tuned to equal temperament (or anything else for that matter) the non-alignment of the overtones will be more significant than it is in higher octaves.
The third factor is that pianos are generally tuned in equal temperament, or something close to it. For example, C2 and E2 have common overtones at E4, E5, B5, E6, etc., easily heard by most people, whereas the analogous common overtones of C5 and E5 are E7, E8, B8, and E9, of which only the first is within the range of the piano. Clashes caused by deviations from just ratios are less noticeable in the higher octaves.
You've got a lot of answers that get rather detailed, talking about tuning temperaments and such, but looking at the crosslinked question on Philosophy SE—in which you are the original asker as well—I think the most appropriate answer is "there's some confusion underlying your premise; music doesn't work the way light does."
To summarize that post in case the link dies: The OP quotes some content discussing the Inverted spectrum concept, i.e. "How do I know that when I call a color 'red', and you do too, that we're both experiencing the same experience?" The content seems to be refuting the theory by talking about rods and cones, though it's not terribly clear. The OP then "transposes" the theory to a musical setting, asking whether there could be someone whose hearing could be "transposed by 12":
So basically if I play a groovy baseline solo. This individual would not headbang to it and I would know aha! This person is hearing transposed by 12.
Or
I can play chords which when transposed to the extreme left end of the piano will no longer sound like music. I think this fact of music as a kind of universality.
So for this present post you've doubled (tripled?) down; the original version of this present post suggested "transposing by -36". We must assume the unit in all of these is semitones.
Here's the sound part of your argument: A given bit of musical content will "feel different" in different ranges. That is, it will produce a different emotional affect in the hearer. If we play a simple tune in the middle range of the piano and ask a hearer what they think of it, they might have a neutral response. Transpose it to the very low register, and they might use words like dark and foreboding; in the extreme high register they might use words like light and airy. These are not scientific or measurable responses, but it is true that we hear the difference.
Transpose it high enough and the hearer will stop perceiving it at all as it moves out of the range of human hearing; this is measurable. Transpose it low enough and you may find a point, not where it becomes inaudible, but at which the individual waves that make up the pitch stop being perceived as a musical pitch and are instead perceived as a series of regularly spaced clicks or beats.
The original version of this question talked about "sounding like noise," and the "noise" tag remains. This enters into a different kind of philosophical discussion, not about "the subjectivity of objectivity," but about semantics and aesthetic constructs: what operative definition people use for the word "music", and for the word "noise." This is a largely fruitless exercise that we shouldn't entertain here, because in short there can be no universal agreement. Some people use "noise" to mean "genres they don't like." Others might draw the line at the absence of organization: sit in a forest, listen to the wind in the trees, and they call it noise rather than music. But John Cage would disagree, and indeed if you played a snippet of tree-wind in a concert hall as "found art" it would be hard to argue with the notion that you have turned it into an aesthetic object. But between these two extremes would be many people who would hear the series of clicks as unpitched but as "music" since it's rhythmic. Even if you slowed the clicks to the point that the listener might not notice the periodicity, one might be moved to refer to it as music (in an article about gravitational waves a physicist waxes poetic, “I like to think of it as a choir, or an orchestra,” even though this literal "music of the spheres" has to be sped up 40 million times to be audible (as I understand it)).
So to your original question on the Philosophy SE: Is it possible that what one person experiences as "A 440 Hz" is a different perceptive experience than another person's, but they have no shared objective framework to describe subjective perception? Uh... sure, I guess. But those are questions of neurology, verging into existentialism and semiotics an' stuff, not actually about music. And "transposing by 12 [semitones, i.e. an octave]" is not really a way to "diagnose" such perceptual differences, should they exist. To elaborate:
