[palimpsests]

would love to hear some justification as to why you feel the average listener - who has been totally conditioned by listening to music in equal temperament for their entire lives - would have sensitivity to what you refer to as "loss of purity in harmony". especially non-musical average listeners.

or are you referring to the idea that people can be trained to hear these kinds of differences?

[geofft]

It's a pretty defensible claim on its own for one simple reason - overtones.

A string, pipe, or other resonating component of a musical instrument, if it resonates at some frequency X, will also resonate at frequencies 2X, 3X, 4X, etc. (Picture a taut string: you can get a standing wave where the whole thing vibrates back and forth, or you can get a standing wave where there's a node in the middle and it looks like the left and right sides are vibrating in opposite directions, or you can get a standing wave where there are two nodes and three components, or so forth.)

These frequencies are exact pitches in just intonation. Let's say that you play a C at octave 3. Its overtones are C at octave 4 (twice the pitch), G at octave 4 (3x the pitch, or 3/2 the pitch of C4), C at octave 5 (4x the pitch), E at octave 5 (5x the pitch, or 5/2 the pitch of C5).

When you play a note on any instrument except a pure sine generator, there are natural overtones because of the construction of the instrument. A guitar or a piano basically introduces random noise into the string when you pluck/hammer it, and the standing waves stay around and resonate, producing the "bright" sound of these instruments from their overtones.

If you play a chord on an instrument, the overtones of the various notes in the chord will line up - if you play a C3 major chord, both the C3 and the G3 have overtones at G4, both the C3 and the E3 have overtones at E5, etc. If you're playing an instrument in C-major just intonation, those notes will line up perfectly. (Moreover, for e.g. a piano with the sustain pedal down, the strings for those upper keys will resonate by receiving the vibrations in the air, even if you're not playing them!) If the instrument is in equal temperament, they won't exactly line up. That is perceptible, even if only slightly, to the non-musical listener.

Of course, the tradeoff is that if you play a C# major chord on an instrument tuned to C major just intonation, they'll line up even worse than they would in equal temperament. Which is why "modern" instruments - which is to say, from at least around Bach's time - have avoided tuning to a just intonation.

[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.