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I was just reading the Wikipedia page on the note F (as I do every evening) and was confused by this part where it says that even though F♮ and E♯ are enharmonic they “do not sound the same”:

E♯ is a common enharmonic equivalent of F, but is not regarded as the same note. E♯ is commonly found before F♯ in the same measure in pieces where F♯ is in the key signature, in order to represent a diatonic, rather than a chromatic semitone; writing an F♮ with a following F♯ is regarded as a chromatic alteration of one scale degree (E♯ and F♮ do not sound the same, except in some tunings that define the notes in that way).

What does the author of this sentence mean? Do they not by definition sound the same?

The thing is that the "some tunings that define the notes in that way" in the Wikipedia quote include the most common tuning today, 12-tone equal temperament (12-TET). So, E# and F natural do usually sound the same.

...But not always. Change the tuning system and you can easily have an E# and an F natural that sound slightly different. Just intonation will likely do it, since its perfect fifths are slightly larger than 12-TET's. (Just intonation is a mess the more of the chromatic scale you want to tune with it.)

I think this particular phrasing is rather confusing, as it is trying to talk about two concepts at the same time: enharmonic equivalence, and intonation.

The concept of intonation (and temperament, which relates to systems of intonation) deals with the fact that even given a certain reference pitch (such as A4=440), there is no one absolutely correct frequency for the other notes to be sounded at. The exact frequencies of notes might be selected to make a certain key sound harmonious, or to be a good compromise that allows a range of keys to sound good (such as 12-tone equal temperament).

On instruments that allow the intonation to be varied by the player (such as fretless stringed instruments), the very same note - even with the same name - might be sounded at a slightly different pitch to make it sound better in a certain chord or melodic phrase. So even two notes notated as E4 might not be at the same pitch; following the logic in the quote from Wikipedia, one could go so far as to say "E and E do not sound the same".

So when the article says "E♯ and F♮ do not sound the same, except in some tunings that define the notes in that way", the fact that the note might be called both 'E♯' and 'F♮' is a little bit of a red herring; a note's intonation might vary regardless of variations in how it is named. Nevertheless, there might be some contexts in which the note notated 'F♮' tends towards one pitch, and 'E#' tends towards another.

Some tunings are designed so that, whenever possible, two notes which are separated by a perfect fifth will have a precise 3:2 frequency ratio.

If that 3:2 relationship holds between A#->E#, then D#->A#, G#->D#, C#->G#, F#->C#, and B->F#, that would suggest that the frequency ratio between B and the E# above it would be 729:512 (about 1.42).

On the other hand, if that 3:2 relationship holds between F and C, C and G, G and D, D and A, A and E, and E and B, then the frequency relationship between the B and the F above it would be 1024:729 (about 1.40).

It would be possible for all the 3:2 relationships to hold if E# and F were recognized as different notes with slightly different pitches, but if E# and F are the same pitch then at least one of the perfect-fifths relationships much involve something other than a perfect 3:2 frequency ratio.

Totally disagree. This paragraph is not about whether the two notes sound the same melodically, but whether they sound the same harmonically. Depending on key and counterpoint there are times when it is clearer to label a note Fnatural instead of Esharp. This also leads to double flats, double sharps, etc. The end result is purely academic, but makes compositional intent clearer to people who are well versed on the academics. The big hint here are the terms diatonic, chromatic, and key signature which have little or no meaning in atonal music.