As a temperament junkie you should understand that it is not possible play harmonic music in even one diatonic key on a keyboard tuned in just intonation unless you are willing to accept either a very sour fifth from D to A or a Pythagorean major third from F to A. If you compromise between those two, then you have a temperament rather than just intonation. So just intonation does not "increase quality" because there isn't a single nice key to use.

@phoog - Yes, I know exactly what just intonation entails. Among other things, it means not expecting to be able to have a perfect fifth above every scale step. So what? It's a trade off, just as I said.

Scott Wallace: Your answer implies that just intonation has a smaller number of keys that sound better compared to, for example, meantone temperament. But that's not true. If you tune a keyboard in C major just intonation, C major will sound worse than it does in 1/4-comma meantone temperament. @Dekkadeci yes, my example concerns C major. Just intonation typically puts D at 9:8, F at 4:3, and A at 5:3 for a just 5:4 major third between F and A. This puts the D to F minor third at 32:27 instead of 6:5 and, far worse, the D to A fifth at 40:27 instead of 3:2.

@phoog, this depends somewhat on C major is being used, of course. The nice thing about the standard 5-limit tuning of C major is that every note in the C major scale is consonant with respect to every note in the C major triad. So, if your chief concern is the consonance of the melody with respect to an underlying chord progression, the JI C major scale reigns supreme. I concede that in some contexts (e.g. contexts that are less chord-driven and more melodic in nature) other temperaments can actually sound better, especially if the problematic intervals you've identified feature prominently..

...in the piece. I'll add that, in my opinion, this was the kind of complexity musicians wanted to get away from my adopting 12TET. "Theory is so boring... just let me play music!" is definitely an attitude many contemporary performers adopt, and we can expect that the same attitude existed in times gone by. (Though it's not my attitude, of course, nor yours, from what I gather.)

@phoog, also, as yet another temperament junkie on this site, I'll make the following comment. There's two ways to think about scales. One way is: "These are the notes I'm going to use for this entire phrase of music." Another way is: "These are the notes I'm going to use while this particular chord is playing. However in a few moments, I might go to another chord, and in that case I'll need to retune some of my notes." My opinion is that JI scales are only practical if our ideas about what a scale is fit more with the latter point-of-view. If you're stuck with one scale for the duration of...

...an entire phrase, then JI becomes much less practical.

@phoog - I agree with goblin, it depends entirely on what you deem necessary for a key (or mode) to sound good. For some music, I can either do without the D-A fifth altogether, or use it as a dissonance. I recognize that JI with a fixed scale is never going to be the best choice for, say, Chopin, but as I said, it has its place. And a 4/5/6 triad is simply heavenly, imho. Suum cuique.

@goblin and Scott Wallace you are mostly right. But "in a few moments, I might...need to retune some of my notes" means that you're not talking about a keyboard tuning system, and a temperament is first and foremost a keyboard tuning system. "Just intonation further increases quality at the expense of quantity" perpetuates the myth that tuning a keyboard in just intonation (whatever that means) will make it sound optimally good in one key at the expense of other keys, while the truth is that even the best possible tuning system for the diatonic scale of a single key is a temperament.

@goblin "every note in the C major scale is consonant with respect to every note in the C major triad. So, if your chief concern is the consonance of the melody with respect to an underlying chord progression, the JI C major scale reigns supreme": this is simply not true, because consonance is not transitive. For it to be true, you either must accept a C-major scale in which there's a 40:27 fifth or an 81:64 major third, or must avoid those intervals. For playing diatonic harmonic music, a system that requires doing either of those things cannot be said to "reign supreme."

@phoog, I think we're mostly in agreement, but I don't understand your point about "this is simply not true, because consonance is not transitive". I agree with the premise that consonance is not transitive, for exactly the same reason the "being close to" relation isn't transitive in geometry; namely, for any cutoff $k$ that defines a distance that's considered short, we can find two lines segments with length less than $k$ whose lengths sum to a value greater than $k$. However, I don't see how this premise warrants your conclusion.

@phoog, perhaps I didn't make my point very clearly. What I meant was that if the C major triad is tuned as T={1/1, 5/4, 3/2}, and if the C major scale is tuned as S={1/1,9/8,5/4,4/3,3/2,5/3,15/8}, then for all notes t in T and all notes s in S, the interval between t and s is consonant. My claim is definitely true, so long as we define (admittedly very simplistically) firstly that the dissonance of a JI ratio x/y is max(x, y), so long as x and y share no common factors (which is a reasonable assumption from a mathematical standpoint) and so long as x and y contain no factors of 2 (which is...

... a reasonable assumption from a musical standpoint, because up to octave equivalence we can multiply and divide by 2 will nilly). And secondly as long as we define that an interval is consonant so long as its dissonance is at most 15. For example, under the aforementioned definitions, the intervals 15/8 and 16/15 both have a dissonance of 15, and are consequently considered consonant.

@goblin I misread the statement about the C major triad and scale. But that property is not sufficient for the tuning to be useful, let alone optimal, because chord progressions in C major use other pitches than C, E, and G. The point is that not all nominally consonant intervals are consonant. Namely, the perfect fifth between D and A is terrible. A tuning system that has a terribly out of tune fifth between D and A cannot be said to "reign supreme" in "the consonance of the melody with respect to an underlying chord progression" in C major.

@phoog, but don't you agree that it depends on your notion of what a scale is, as described earlier? If by "scale" we mean "something tells you how to tune your notes with respect to a particular chord", then the JI major scale reigns supreme for the major chord. If by "scale" we mean "something that tells you how to tune your notes for entire phrase, or an entire piece", then you're correct that the JI major scale has major problems (pun intended). So, are we in agreement, or is there still a point of contention on the table?

@phoog - well, it's obvious that you're no fan of JI. That's fine. But that doesn't render JI useless for others who don't share your preference, does it? I could say that the sour JI fifth of D-A is more than countered by the sour thirds (all of them) of 12TET. In the JI discussed here, we have three perfectly in tune major triads 4/5/6, and two perfectly in tune minor triads, 10/12/15. Can't do that in any equal temperament. But as I said, there's a place for all these temperaments.

@ScottWallace I love JI. My point is only to debunk the myths that it is useful for tuning keyboards, that it ever would have been used to tune keyboards, and that using it makes the keyboard sound good in any key. JI only works as a tuning system for instruments that can adjust pitch dynamically based on the harmonic context, or for music that avoids certain intervals. For keyboards, for music that doesn't avoid certain intervals, meantone temperaments give pure or nearly pure major thirds and have no sour fifths if you stick to a small number of keys.

@goblin but the first definition of scale is useless for music with harmonic changes. Further, one doesn't tune a keyboard for a phrase. One usually tunes it for at least several pieces. If the keyboard is an organ, one normally expects each tuning to last for a period of time ranging from one performance to months. A temperament has to take this into account. Are you aware of any keyboard music that can be played on a justly-tuned keyboard without sounding a 40:27 fifth?

@phoog, look, I think we're basically in agreement, but just emphasising different things, namely static versus dynamic tuning.

@goblin that does appear to be correct. But since the context here is a question about keyboard tuning, it doesn't make sense to discuss dynamic tuning, let alone emphasize it. I can see that you understand everything I'm saying; I'm mostly concerned about other readers who might be misled by this idea that just intonation is somehow optimal for tuning a keyboard if you only play in one key.

@phoog - what is "optimal" tuning for a keyboard (or any other fixed pitch instrument)? You are obviously of the opinion that the sour D-A fifth disqualifies the JI scale under discussion from being "optimal". As I've pointed out, with all due respect, that's simply your opinion.

@ScottWallace every opinion about the suitability of any tuning system for any instrument and any piece of music is subjective. The thing that troubles me is the idea that it's possible to tune a keyboard so that all intervals in a single key are acoustically pure, and this idea is objectively false. Yet the idea persists nonetheless.