The difference is reduced inharmonicity in the bass notes.
A sound we hear as a single pitch is made from multiple different sine waves playing simultaneously, called "partials" in general, or "harmonics" when those sine waves are integer[0] multiples of the lowest frequency, which is called the "fundamental".
In an string instrument, the ends of the strings are fixed in place. This means any standing wave in the string must have "nodes" (points where the wave has zero amplitude) at the ends. When you pluck or strike the string, you induce multiple different standing waves at once. In a mathematically ideal string (modeled with zero thickness), these standing waves are sine waves. Because the ends are always nodes, and because nodes and anti-nodes (points of maximum amplitude) of a sine wave are evenly distributed along the length of the wave, there will always be an integer number of anti-nodes. The frequency of the wave is proportional to the number of anti-nodes, so you end up with a harmonic sound, where the upper partials are integer multiples of the fundamental. Music played in standard Western tuning sounds the most consonant when it uses harmonic sounds.
But a real string instrument is not mathematically perfect. The strings have thickness, which makes them resist bending. Higher frequencies cause sharper bending of the string, so they are more affected by the bending resistance. This results in a standing wave that's very close to a sine wave, but with the end nodes shifted slightly towards the center of the string. The higher harmonics are effectively played on shorter strings, so their frequency is increased. They are no longer integer multiples of the fundamental, so the sound is "inharmonic".
Bass strings are typically thicker, so this effect is most noticeable in the bass. It's a big enough problem in a traditional piano that a special tuning technique ("stretched tuning") is used to try to hide it, but it still results in a kind of muddy/blurred sound. The only alternative to making the bass strings thicker is to make them longer, which is what the the Alexander piano does. The result is an unusual clarity in the bass.
[0] In practice people use "harmonics" to mean approximately integer multiples too.
As I understand it from my own tuning career, octave stretching is more about correcting dissonances that result from equal temperament than correcting anything about the dynamics of a physical piano. When you tune with octave stretching, you will also widen the high octaves, which are the correct size to hold a standing wave on almost every piano with standard "piano wire" strings. This helps to bring the 2nd overtone of a string more in line with the note a 12th above at a cost of making the 1st and 3rd overtone (one and two octaves up) slightly imperfect.
On the lower, wound strings, the thickness of the string causes the string to also hold a standing wave at the fundamental frequency in the short length that the string has, but it amplifies the higher-order harmonics and adds some secondary effects that cause the note to be less "clean" sounding. It doesn't cause the frequency to change.
By the way, I am not a fan of octave stretching on my own piano, so I usually ask piano tuners not to do it. It's completely optional.
I don't believe that's it. The problem of tuning higher notes to the stretched harmonics of lower notes would still be present even in a piano tuned using just intonation.
Suppose for instance the 2nd harmonic of C2 is 2 cents sharp of where it "ought" to be (which is twice the frequency of C2 itself). Then if you tune C3 (the C in the next octave) to be similarly 2 cents sharp of where it ought to be, it will not clash against the 2nd harmonic in C2 -- but the result is that the distance from C2 to C3 is a little more than an octave.
(I don't know whether 2 cents is anywhere ballpark -- I've studied the theory but haven't performed measurements.)
Incidentally, if you want to play microtonal music on an equally tuned piano, this is an argument for selecting a sharp system, such as 27-edo or 58-edo -- that is, a system whose approximations to the harmonics you're interested in are sharps. (12-edo gives an approximation 2 cents flat of the 3rd harmonic and 14 cents sharp of the 5th harmonic. It ignores the other harmonics. 27-edo gives a sharp approximation to harmonics 3, 5 and 7. 58-edo gives a sharp approximation to 3, 5, 7, 11, and 13 -- nirvana, as far as I'm concerned.)
While it is true that the harmonics will be off for any tuning system, octave stretching is relatively new and arises only on equal tempered instruments because it needs to be applied equally to every note, so it sounds terrible if you apply it to unequal tunings where (for example) the distance between C2 and G3 can be 20 cents different than the distance between C#2 and G#3.
In equal temperament, the distance is ~2 cents between the 2nd harmonic of C2 and G3 (and every other pair a 12th apart), so stretching an octave by ~1 cent spreads that distance out while being pretty much imperceptible. In a decent quarter-comma tuning system like Werckmeister (a very common Baroque tuning), four of the 5ths are off by about 5 cents, and the others are pure. As a result, the 2nd harmonic differs from the note a 12th above by a variable amount. Stretching every octave by 2.5 cents to balance this becomes very much audible across a keyboard, and sounds very odd on the harmonic series' that are nearly perfect.
Baroque tunings also pay a lot of attention to the major 3rd (the 4th harmonic), which is a notable weakness of equal temperament - it is far too narrow.
Regardless, octave stretching doesn't have to do with piano dynamics, it has to do with tuning systems.
As I understand it, stretched tuning in a piano is specifically intended to compensate for inharmonicity, not to compensate for the weaknesses of equal temperament. You can apply stretch to unevenly tempered pianos too, and it should also make those sound better. A formal description of stretched piano tuning is relatively recent (1938, with Railsback's publication of his eponymous curve), but some kind of stretched tuning likely goes back to the first piano, as it happens automatically if you tune the piano against itself by ear.
12-edo's approximation to the 5th harmonic (a.k.a. 4th overtone) is 14 cents sharp, not flat -- it's at 400 cents plus two octaves, whereas the 5th harmonic is at 386 cents plus two octaves.
I admit to being unfamiliar with unequal temperaments. As a jazz player they seems like a complete bummer to me, as I love to transpose on a whim and don't want the intervallic structure to depend on what key I'm in.
Yeah, I had a sign error in my head there (unequal major thirds are the relatively flat ones).
I know that there is some unequal tempered jazz out there from very early recordings, and I assume they thought the different character of the keys added extra character and possibly some extra "grunge," but it would be interesting to compare the improvisational style to equal tempered jazz.
I know of at least one person who's very fond of a 12-tone JI scale she came up with, which she happily plays in every key. She tattooed it onto her arm.
I've played in 58-edo, which to me is basically indistinguishable from JI. It's just a little too big to use comfortably on the Lumatone. I can use it on the monome comfortably, but the monome doesn't have velocity sensitivity, which for me is a deal killer. On the Lumatone the biggest I feel comfortable playing is 46-edo. Fortunately, 46-edo sounds incredible.
It doesn't have the best approximation to 5/4 -- it's 5 cents sharp -- but that error goes in the same direction as the error in 12-edo that we're used to, so (because we're all used to 12-edo) it sounds world better than, say, the 5/4 that 41-edo gives you, which is 6 cents flat.
A sound we hear as a single pitch is made from multiple different sine waves playing simultaneously, called "partials" in general, or "harmonics" when those sine waves are integer[0] multiples of the lowest frequency, which is called the "fundamental".
In an string instrument, the ends of the strings are fixed in place. This means any standing wave in the string must have "nodes" (points where the wave has zero amplitude) at the ends. When you pluck or strike the string, you induce multiple different standing waves at once. In a mathematically ideal string (modeled with zero thickness), these standing waves are sine waves. Because the ends are always nodes, and because nodes and anti-nodes (points of maximum amplitude) of a sine wave are evenly distributed along the length of the wave, there will always be an integer number of anti-nodes. The frequency of the wave is proportional to the number of anti-nodes, so you end up with a harmonic sound, where the upper partials are integer multiples of the fundamental. Music played in standard Western tuning sounds the most consonant when it uses harmonic sounds.
But a real string instrument is not mathematically perfect. The strings have thickness, which makes them resist bending. Higher frequencies cause sharper bending of the string, so they are more affected by the bending resistance. This results in a standing wave that's very close to a sine wave, but with the end nodes shifted slightly towards the center of the string. The higher harmonics are effectively played on shorter strings, so their frequency is increased. They are no longer integer multiples of the fundamental, so the sound is "inharmonic".
Bass strings are typically thicker, so this effect is most noticeable in the bass. It's a big enough problem in a traditional piano that a special tuning technique ("stretched tuning") is used to try to hide it, but it still results in a kind of muddy/blurred sound. The only alternative to making the bass strings thicker is to make them longer, which is what the the Alexander piano does. The result is an unusual clarity in the bass.
[0] In practice people use "harmonics" to mean approximately integer multiples too.