[mhauru]

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.

[palimpsests]

thanks and this isn't what I'm asking about, I'm aware of all of this (background in physics and in music performance).

of course people can hear and distinguish mathematically pure intervals especially if demonstrated in these ways.

you said,

> This brings some practical musical advantages, but even the average listener can hear the difference, the loss of purity of harmony

which I interpreted as meaning that average listeners are both hearing and actively aware of this difference in their day-to-day exposure to equal-tempered music.

I am very skeptical of this claim, but perhaps this is not what you intended to imply? I can understand why my question would lead to yours and geofft's answers.

[mhauru]

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.