Many just intonation chords (where ratios are actually integers, not approximations of them) will sound out of tune to most people.

Out of tune or just cold?

Typical person on the internet will be really used to 12 tone equal temperament, for just tuning most intervals are 10+ cents off even if you allow pretty large ratios: https://en.wikipedia.org/wiki/12_equal_temperament#/media/Fi...

But the whole point of 12-TET is it is a close approximation of those just intervals while still allowing multiple keys to be played without retuning the instrument. If the close mathematical approximation didn't exist, there would be no reason to tune your instruments with a logarithmic recurrence relation.

> allowing multiple keys to be played without retuning the instrument

That's another simplification, real pianos are not tuned to 12TET, but use stretch tuning (which can be over 30cents off in the lower range).

I'd rather argue that people like what they're used to and so "people like pure ratios" is seemingly only true if "pure ratios" are not really pure. And that's ignoring a lot of the music that doesn't have roots in Europe.

FWIW it doesn't get better than 12-TET until you get to (like) 53-TET.

Both statements are true. We have a strong tendency for integer ratios in harmony, and just intonation often sounds out of tune.

Integer ratios are the base upon which harmony is built. Temperament is a subtle modification that sounds very close to integer ratios, but allows more complex harmonic structures where dissonance is evenly spread out across all the relationships between tones.

Seventh and eleventh harmonics are way off. If ratios were the fundamental thing Bohlen–Pierce would be about as pleasant as 12TET to people.

Way off what? Complex ratios are likely to be heard as out-of-tune simple ratios, that's why they sound off. A concept sometimes called tolerance in music cognition. Note that by "complexity" and "simplicity" I'm referring to harmonic distance here.

7/4 ratio should be simple, but it'll sound out of tune (over 30 cents) in a normal context. Many BP intervals are just as simple and they'll sound very out of tune to people unused to them.

7/4 happens to be approximately 30 cents away from 16/9. It's hard to tell what's "simple" when looking at fractions, but 16/9 is indeed simple: divide by 3 twice and adjust the octave. If we assume octave equivalence, that means one step in the "7" direction is perceived as more complex than 2 steps in the "3" direction, so the second interpretation wins, but is perceived as out-of-tune.

That said, we're trying to isolate things that are typically not isolated. If you get to 7/4 by following the harmonic series, it will sound in-tune. If you get to 16/9 by playing and applying 4/3 twice, then that will sound in tune. Unsurprisingly the second option is more common in music.

Before 7th harmonic all you have is octaves, fifths and major thirds. If you want to stick to making other pitches out of stacked fifths and major thirds you'll end up with other compromises.

Integer frequency ratios in chords are probably favoured because of the way overtones line up and make the cilia of the inner ear vibrate