The fact that integer relations are mathematical relations that bear so heavily on physical phenomena is precisely the point. That's what the ancients got so damned excited about. You may not find meaning in it, but it certainly drove the development of science and philosophy from Plato to Galileo to Kepler to Descartes to Newton, etc.
Now, in recognizing that integer relationships are, in fact, an imperfect description of a pervasive phenomena -- that's why the ongoing investigation is so interesting and challenging. Don't we expect what is pervasive in physics to apply to psychology, culture and economics? I'd argue that the mathematical "imperfections" of modern music point to better models of what universal harmony really is.
Overall, the point is that we haven't yet solved harmony. Not even remotely. And i'd love a reference on why you think Byzantine music doesn't have octaves when the Greek music it grew out of most certainly did.
1) "Do not mistake your models for reality" (Lord Kelvin, sometime in the late 1800s) ... many of things that we describe with "small integers and ratios" are not in fact small integers and the ratios are rooted in our observations and thinking. The natural world is more fractal than integral, but the notion of "oh! that thing is a lot like two times that other thing" was a notion more accessible to natural philosophers and early philosopers.
2) No, I see absolutely no reason to "expect what is pervasive in physics to apply to psychology, culture and economics".
3) I don't have a citation for you on the Byzantine stuff, but have been discussing it a lot recently with a musician who grew up on it and continues to perform it, and he was explaining to me how they have no notion of octave equivalency and that because of their tuning and scale systems, when you go up or down the number of steps that "should" correspond to an octave, you end up somewhere other than 2*freq or freq/2. Remember, this is in part why equal tempered tuning was developed: if you stick to "pure" just intonation (precise integer ratios), you can't construct intervals that fit nicely into the octave (e.g. pick a note, go up some number of intervals. Pick the ending note, go down the same intervals, you don't end up back where you started). This stuff is all covered in basic music theory. My understanding from this Byzantine musician is that their musical tradition basically just said "we don't care", and went with a tuning/scale system where you don't just (as in our western system) go up or down N (12 in our case) and end up an octave from where you started.
* Your friend seems to be discussing the Pythagorean pure interval tuning system where, indeed, going up is different from going down --
* In my own empirical work, I've found that pure intervals are not preferred compared to slightly dissonant intervals (in 3 tone sawtooth chords). You can try it yourself -- perfect consonance sounds much worse! Understanding this conflict with mathematical intervals is part of trying to "solve" harmony
* I agree we should remain disinterested in our models, such that we are driven to improve them and not espouse them as reality.
* Yet, I think it is a mistake if we aren't inspired by simple models -- or at least take their hypotheses seriously enough to test empirically
* For instance, is cognitive dissonance involving actual dissonance of some kind? The brain is incredibly rhythmic and even has octaves in the coupling between brainwave bands. It would seem to be natural to test these theories that Plato laid out thousands of years ago -- for instance, that musical rhythm entrains neural rhythms through resonance effects. Maybe that sounds silly, but I'd argue that we are foolish to avoid gathering empirical evidence for these ideas!
* Similarly, harmony had a major effect on astronomy. It actually still does, at the level of the cosmic microwave background radiation, where the presence of perfect harmonic peaks in the signal were the conclusive proof that the universe is "flat"
* Using the concept of sympathetic resonance has been extremely generative in psychology (see Adam Smith's first book) as has harmony in economics. Expecting what is pervasive in physics to apply to these domains doesn't mean it should apply in exactly the same manner -- just that one should expect analogous effects -- at least to the extent that one is looking for explanatory theories to test! We are so far from understanding these domains that, if we don't at least consider these natural and ancient theories (because they seem silly), we are doing ourselves a disservice. Let us be inspired by the past and test, test, test!
The fact that integer relations are mathematical relations that bear so heavily on physical phenomena is precisely the point. That's what the ancients got so damned excited about. You may not find meaning in it, but it certainly drove the development of science and philosophy from Plato to Galileo to Kepler to Descartes to Newton, etc.
Now, in recognizing that integer relationships are, in fact, an imperfect description of a pervasive phenomena -- that's why the ongoing investigation is so interesting and challenging. Don't we expect what is pervasive in physics to apply to psychology, culture and economics? I'd argue that the mathematical "imperfections" of modern music point to better models of what universal harmony really is.
Overall, the point is that we haven't yet solved harmony. Not even remotely. And i'd love a reference on why you think Byzantine music doesn't have octaves when the Greek music it grew out of most certainly did.