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.
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:
- First of all, transposition is not the same as "inversion," and our perception of pitch does not correlate to our perception of color. The original color hypothesis (which, as far as I know, is nothing more than a thought experiment) suggests that some people essentially "re-map" one color to another, not shift the whole spectrum up or down. And while the spectrum of visible light is dependent on wavelengths, just as the wavelengths of audible sound determine pitch, we don't perceptually draw sharp distinctions around certain segments, assigning them concepts like red, green, or blue.
- Maybe the biggest issue is the same one that's addressed in the philosophy post: You propose an objective measurement of a subjective thought-experiment. If some people "hear music an octave lower," you won't "match their hearing" by playing an octave lower, or "correct" their hearing by playing an octave higher. Rather, whatever they perceive will still be shifted to whatever their experience of "lower" or "higher" is—just as you can't prove the inverted spectrum hypothesis by showing an image with the colors inverted.
- And finally, perhaps it's a bit tongue-in-cheek to suggest that one way to measure experiential perception is by the listener's failure to headbang to a "groovy bass line solo," but there's a lot more that goes into affective response to music than neurology—individual inclination, enculturated patterns of response, etc.