Not sure I see what you mean. Here's the infamous I – V – VI – III – IV – I – IV – V progression from Canon in D in just intonation, with square brackets marking tones that make up a chord (invert the chords to your taste):
[1/1 5/4 3/2] – [3/2 15/8 9/8] – [5/3 1/1 5/4] – [5/4 3/2 15/8] – [4/3 5/3 1/1] – [1/1 5/4 3/2] – [4/3 5/3 1/1] – [3/2 15/8 9/8]
Of course, you can't do this for every progression you might enjoy playing, but as an example this one works out quite nicely, with no serious harmonic ambiguity or conflict.
At any one time, yes, there's always a root tone and the tuning system is generated with respect to that root. Such is the nature of pure harmony. This is one of the reasons (I would say the main reason) why equal temperament tuning rather than just intonation is dominant in our culture.
You can change the frequency of the root tone in the settings though (gear icon, top right). You have to specify it in Hz which is a bit inconvenient, sorry about that. To compute the frequency in Hz for any tone on the usual piano keys, compute 2^(n/12) * 440, where n is the number of semitones above (n>0) or below (n<0) the mid A that you want the root tone to be. If that's not clear, let me know and I can give you list of frequencies for various tones.
Weellll, yes and no. Yes, modern music is (mostly) in tune with respect to the chosen tuning system. However, there really is a sense in which just intonation is the harmonically most pure tuning system. The reason musical harmony is a thing at all for human ears is that frequencies related by simple ratios have a special sound to them when played together, which we call harmonious or consonant. geofft does an excellent job explaining why this is the case in one of the comments below, and I provide my own complementary explanation in that thread as well. And just intonation preserves this consonance, these simple ratios, exactly. Sure, it comes at the cost of things like modulation that composers and listeners enjoy, I am not contesting that. Nor am I saying that just intonation sounds prettier, that depends on your taste. But for having a chord sound as resonant and consonant as possible, something I think can be fairly described as being "in tune" without reference to a particular tuning system but only to the physics of sound waves, just intonation is the real deal, and everything else is approximating it.
Also, while I agree that things like aesthetics of modulation are one reason why just intonation is such a marginal thing, another big reason is the difficulty of making and tuning instruments for just intonation, when there's in principle an infinity of tones within an interval, and the whole system changes when you change key. I thought this latter reason is an unfortunate one, and something we could try to overcome with digital technology, hence Jintone.
Sorry, I could have been clearer in explaining how to actually use the thing. So the keyboard at the bottom is in the regular 12-EDO tuning that we are all used to and cubase does as well. It's the red dots that are the just intonation tones.
To hear the difference, try playing a just intonation C major chord, by clicking the red dots labeled 1/1, 5/4, and 3/2 (hold Shift for sustain to have them ring at the same time), and then the same chord in 12-EDO by clicking the piano keys directly below the tones you just played, so C, E, and G. The difference isn't massive, but you can probably hear it. If you want to make it more obvious, click the gear icon on top right and choose the sawtooth waveform, and try the same thing again.
You can also try playing a 12-EDO E from the piano keys and a 5/4 from the just intonation dots. These are the same tone in the two different tuning systems, and you can hear the "beat" between them, the slow oscillation that comes from them not being quite at the same frequency.
Ah, I see. Indeed, I did not mean to imply that the average listener is aware of the short-comings of equal temperament without somebody explicitly providing the contrasting example of just intonation. I merely meant to say that the difference between just intonation and 12-EDO is large enough that even an untrained ear can spot it, when e.g. the same chord is played in both tunings back-to-back. Something that probably wouldn't be the case if for instance all of the 12-EDO approximations would be as close to the just intonation interval as perfect fifths and fourths are.
I would hypothesise that even the average listener might enjoy some pieces of music somewhat more if they were played in just intonation (depends of course massively on the piece whether this is even realistic or a good idea, but given a suitable piece), but I doubt they would be concious of what's making the difference. Not saying I have any evidence of this, but that's my guesstimate of what the answer to the question "how much of a difference would just intonation vs 12-EDO make for the average listener" would look like.
geofft gives an excellent answer to this, but I'll try to complement it a bit.
There are reasons grounded in the physics of soundwaves for why frequencies related by simple ratios sound "special" when played together. As geofft explained, they naturally arise in any physical instrument due to how things like strings (guitar, piano, etc.) or air in a tube (wind instruments, etc.) vibrate. Another way to look at this is to think of two waves with wavelenghts that are related in say a 3/2 ratio (wavelength of a wave is essentially just 1 over the frequency of the wave, so you can think in terms of either, but wavelengths are easier to visualise). If you combine two sine waves with such wavelenghts by summing them up, they form a very clear repeating structure, where the crests and valleys of the sum-wave have a regular pattern to them that repeats: https://www.wolframalpha.com/input/?i=plot+sin%28x%29+%2B+si...
Whereas if the wavelenghts of the two waves being combined are not related to each other by a simple ratio, the crests and valleys of the sum-wave keep moving and changing instead of repeating: https://www.wolframalpha.com/input/?i=plot+sin%28x%29+%2B+si...
When heard as sound, the fixed repeating structure of the combined wave when the two component waves "line up" with each other has a special quality to it, that we would call harmonious or consonant.
So in the above sense, and because of overtones as explained by geofft, just intonation really is special among various tuning systems. Of course there's still the separate question of if the difference between just intonation and 12-EDO (what most music uses) is big enough for the average listener to notice. This you can test for yourself by playing on Jintone e.g. a just intonation major chord, for instance 1/1, 5/4, and 3/2 together, and then the same chord in 12-EDO by clicking the piano keys directly below the tones you just played, so C, E, and G. The difference isn't like night and day, but the latter pretty clearly has an unstable wobbliness in it that's not there in the just intonation version, a difference that I think most people would detect. If you want to make the difference super obvious, click the gear icon on top right and choose the sawtooth waveform, and repeat the exercise. The timbre of the sawtooth waveform makes the difference more obvious for exactly the reason geofft was talking about: It's a waveform that has a lot of very strong overtones in it (kinda the opposite of a pure sine wave), and in the just intonation chord the overtones of the three tones in the chord match exactly.
As far as I know, no one has so far made a piece using the Jintone instrument (it's admittedly quite clunky for actually playing rather than toying around), but there's plenty of music in the just intonation tuning system. To completement ivanmaeder's answer, and to counter balance the fact that most just intonation music tends towards experimental classical stuff, here's a chiptune game soundtrack mostly in just intonation: https://mayazimmerman.bandcamp.com/album/galactic-refugees-o...
Agreed. Although as you point out, the singers being unaccompanied is important. Mixing keyed (or fretted, valved etc.) instruments into the ensemble quickly starts to push everyone towards 12-EDO.
I also wonder how much continuous pitch instruments do this, when playing pieces of music and with ensembles that would allow adjusting to just intonation. If I walk up to a good violinist and ask them to play a simple melody on a single string, do they gravitate towards just intonation, or does muscle memory make them place their fingers for 12-EDO? What about a superbly good violinist?
Musical harmony is all about sound frequencies related to each other by simple ratios, like 3/2 and 5/4. For the past few hundred years Western music has been using a tuning system in which these simple fractions are approximated by powers of the twelfth root of 2. This brings some practical musical advantages, but even the average listener can hear the difference, the loss of purity of harmony. I made an instrument that instead uses a tuning system called just intonation, that uses pure, exact fractions only.
The link opens with the help overlay of the instrument. It reads like a blog text explaining tuning systems and just intonation. If you would rather poke at things than read, then click the X on top left to close the overlay and just play with the thing.
Source code is at github.com/mhauru/Jintone. Comments are very welcome.
Of course, you can't do this for every progression you might enjoy playing, but as an example this one works out quite nicely, with no serious harmonic ambiguity or conflict.