I took some classes on experimental electronic music and such when I was in college and did some reading on my own since then... and so I've been exposed to some ideas in music that are way way way distant from the popular music norm. And I'm still not entirely sure what I really think about them so I'll write some thoughts up instead.
The musical tuning behind the notes you hear is inherently flawed. We only hear it as normal because we are used to it... which is kind of a controversial thing.
Let's start with the A frequency. The modern standard is that the A above middle C is 440 Hz. This has not always been the standard... if you examine tuning forks and relic instruments, you'll rapidly discover that this has changed over time, was not really standardized until the mid twentieth century, and even today, not all orchestras tune their concert pitch to 440 Hz. This is largely not a problem as long as all instruments actually tune to the same agreed standard (unless you have perfect absolute pitch) and that your instruments are able to be tuned properly in the first place.
You do hear a doubling of frequency as an octave... and maintaining this is important, because you can hear 'beats' if you get the octave ever so slightly off. And you also see a tripling of the frequency as a perfect twelfth -- a fifth interval, one octave up.
After this, everything goes to hell.
It turns out that, as you add notes to the scale so you are not just playing octaves and twelfths, certain intervals can sound extremely dissonant, depending on how you lay out the tones. Now, when you are talking about singers who are singing without any external references, you can have them sing in exact relative intervals off of each other. But pianos have keys and you can't easily and quickly re-tune them. Stuff like that.
You can either make one or maybe a small number of keys such that they sound extremely consonant and such that certain intervals will produce 'wolf' tones. Or you can make everything not quite properly tuned such that no key is especially out of tune but no key will ever be perfectly in tune. Keeping everything out of tune turned out to be the best option because tuning a piano is a lot of work and even splitting the black keys to let the player work around wolf intervals made the instruments too complicated.
There's a lot of argument about this in terms of historically influenced performance because there were no frequency measurement gauges, just instructions and preserved pitch-pipes and tuning forks, so it's hard to pinpoint exactly what a given composer meant the tuning to be.
Things have been mostly settled for a while now. Pitches haven't been changing. Like I said, barbershop quartets tend to use purely relative pitches instead of equally tempered pitches, but that's also because the voice is flexible. It's also part of their sound. Classical musicians have to be extremely careful about tuning and intonation because you are dealing with ten musicians playing the same unison note instead of maybe one or two.
Jazz, on the other hand, has represented a 'freer' tone, having been created as an intersection between multiple cultural traditions that was jammed up against classical music. Thus, jazz and blues and rock have the notion of a 'blue' note, which you can't play on a piano but can sing or play on certain horns or stringed instruments. And if you take a jazz-trained trombone player and a classically-trained trombone player, you'll discover that there are subtle intonational differences between them.
Now, these days, you can create microtonal music with a synth in ways that you couldn't in the past. So we don't need to be limited quite as much by a lot of those restrictions from the past. Retuning a synth is a config setting, not an annoyance.
I've found that part of the problem is that, if you propose to change the world of music, classical music isn't where you'd really want to start. Jazz changed music in all sorts of crazy and often dramatic ways with twelve-tone explorations that sounded pretty damn good. Whereas classical music's twelve tone experiments are, to me, wretched and unlistenable. I've always felt that Ianis Xenakis did great work but when I hear a tone row, I hear a kid banging on a piano.
If you completely dispense with the need for things to sound consonant and exist on a simple scale, you can write xenharmonic music. But I feel like there's certain inevitable restrictions that you can't get past. The modern music construction, albeit frequently a bit silly and pointless, exists for a reason. You really need about 12 tones, maybe a few more or a few less, so you can make keyboards and fretted instruments. And you really want to be able to play real instruments, not just screw with synthesysers. And constraints build art. Architecture would be really easy to get artsy with if you are making ornamental bubbles in zero-g space instead of real places that people live in that exist on earth from real materials.
I'm trying to keep an open mind. As such, I did some listening and researching and stuff. I found some 19-TET music and it sounds pretty nice, actually. Different. But good. It still generates consonant sounds. 19-TET is approaching the high-end of what you can do with physical instruments. Pretty much, it's the same distribution of notes that gives you the standard 12 tone music scale, but with 19 notes per octave instead of 12, which makes some intervals more 'pure'. Pretty much, all that changes is that a sharp is not the same thing as a flat. Guitars just need a neck with a different fretting pattern. Pianos need to split the black keys in half.
You can also use a second keyboard to select a given key's intonation system, maybe one of the just or meantone systems. This works great on a synth and OK on some acoustic instruments... but not on the guitar, for example. For not especially harmonically adventurous music.. say a I-IV-V pattern rock song, you'd just set the key and leave it and it would sound a smidgen sweeter than before. Or you can modulate between keys by re-setting the base intonation.
But... like I said. Constraints build art. Part of what distinguishes a guitar from a keyboard is that a guitar can fluidly bend notes to create blue notes and slides and a keyboard cannot. I think my time-wasting-research on the subject showed me that it's not just the xenharmonic stuff that careers as professors of modern music are built upon... but I'm still not too interested in hunting down a 19-TET neck for my guitar.