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To some extent, these accents represent different sounds, and so they're used "when a word has that sound"-- there isn't always a rule as such as to why a word is composed of some particular sequence of sounds rather than other sounds.
However, that said, here are some generalities that might help:
- the "close" e represented by the acute occurs at the ends of words or at the ends of syllables (in so-called "open" syllables: where the syllable doesn't end in a consonant)
- conversely, the "open e" represented by the grave accent tends to occur inside syllables, or in other words before another consonant that is in the same syllable as the 'e'; in particular, this means that it can't come right at the end of a word.
So this is why you get e.g. préférons (syllabified pré - fé - rons), but (je) préfère (syllable pré - fère) [remembering that a word-final -e, written with no accent, generally isn't pronounced].
- the circumflex essentially occurs for historical reasons, and usually represents another letter (often s) which to cut a long story short has now "merged" with the vowel. Note that you may still see traces of this historic letter in derived forms of a vowel, so e.g. in fête, the circumflex is a historic remnant of an "s" which you still see in e.g. festif, festival.
The tréma is essentially used to indicate that you pronounce an "e" vowel as a vowel in itself whereas normally the spelling would be confusing and possibly indicate something else. For example, in "j'argüe", the tréma indicates that you pronounce the "u" rather than that this indicates a "hard g".
But note that what spelling you get where is a complex combination of historic decisions by particular authors and dictionary editors, sound changes etc. Sometimes you just have to learn "what is what where" rather than trying to apply rules in every case.
The cédille is used to change the pronunciation of the consonant "c" before the vowels "a", "o" and "u"
ca, co, cu the "c' is pronounced as a "k"
ça, çu and ço the "ç" is pronounced as "ss"
I came across these rules about using either accent grave or accent aigu. I would be grateful if you read and tell me if you agree with them.
OK, why isn't préférer spelt "préféré"? -- well, essentially for historic reasons: the spelling became adopted at a time when the "r" was pronounced (as it is still in e.g. "finir"), the language has changed over time so that the "r" is no longer pronounced, but he spelling hasn't been revised to reflect this change. (Actually, very occasionally you can still have 'liaison' with infinitives, but it's very rare: to all intents and purposes, you can assume that the "r" is not pronounced on the end of -er verbs.)
And yes , it is predominantly true that the accent grave is used in a closed syllable followed by a mute 'e', with occasional exceptions and complications (I was trying to avoid too much jargon in my previous answer, but a "closed syllable" is one of the type I describe: where the syllable is 'closed' with a final consonant).
Notable common exceptions: très, après.
[This analysis requires you to assume that the mute 'e' doesn't change the syllabification. In actual pronunciation, if the mute 'e' was pronounced, you could get re-syllabification so that "mène" is pronounced "mè-ne" and the "è" actually ends up being pronounced in an open syllable. And in pronunciation, either e.g. "cher" or "chère" could be pronounced with a mute 'e' in reality. So we're looking at the mute 'e' from the point of view of the spelling only. It's just worth mentioning that.]
Note that this is often a necessary condition, but it's not a defining condition for è being used. For example, in the words "fête" and "pêche" you have an "e" vowel in a closed syllable followed by a mute e, but for historical reasons these words are arbitrarily written with a ê, not è.
Thank you very much for your explanations.