Flats and sharps in a melody

[walternewton]

I didn't understood it either.

Looking at it again, I think he was basically saying that B# is enharmonic to C.

True enough, though not really relevant to what you asked (and there are cases where B# rather than C is the correct notation).

[JonR]

I was thinking that in the C Key signature, because there are no sharps or flats, it would be enough to place the accidentals in the notes affected by them (and the key signature would take care of the function of the natural). This means that I didn't knew that once you affect a note with an accidental, all the others after would keep it, regardless of key signature used.

Only in the same bar - and only on the same line or space - as shown in walter's example. The key sig is re-established in the next bar.

[JonR]

So what I tought was correct. It's not needed to put an accidental sign before a note already affect by it via the key signature.

Correct.
In fact, the word "accidental" (strictly speaking) only applies to sharps, flats or naturals appearing in the music itself, not the sharps or flats in the key signature. (Though to be honest I don't know what name they have...)

Wouldn't it be simpler if, instead of a natural, let the key signature take care of that?

The key signature does take care of it, in subsequent bars.
I guess it's evolved that way because it was common for an altered note to apply for a whole bar, but not for later bars - so the simplest rule to cover that was to let an accidental refer to a whole bar, but not to later ones.

Of course, there are exceptions. But with notation, the best explanation of the rules is that they are designed to be as economical as possible (saving ink, time and space, for writers as well as readers), given the musical conventions of the time.

[JonR]

So, the rule is:

1 - The only signals that have the "power" to put an accidental on a note are the signals themselves or the key signatures.

Er, yes I think so, if I understand you right. (But what other "signals" could there be?)

2 - The only signal that have the "power" to modify any accidental into natural is the natural sign.

Yes.

[rbarata]

(But what other "signals" could there be?)

I was making a distinction (an erroneous one) between flats/sharps and naturals.