How did "de les" end up becoming "des", phonetically?
I want to know is what was the phonetic evolution process that made "de les" end up becoming "des" through time. If possible explain "du".
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It wasn't "de les" that became des, but "dē illōs" and "dē illās". In Latin, those were independent words, with their own stress, but in the transition into French (and the other Romance Languages) they lost their stress and came to evolve in unusual ways. (For example, ille and illī yelded il while illōs and illās yelded les, despite being initially stressed on the first syllable - the expected form are el for the subjects pronouns, singular and plural both, and els then eux for the plural determiner).
For simplicity's sake, I'll just track the evolution of "dē illum" (masc sg) and "dē illōs" (masc pl). The feminine forms are slightly more puzzling and become homophonous in the plural after stage 2.
In Classical Latin, "dē illum" would have been pronounced /'deː 'ɪlːʊ̃/ and "dē illōs" /'deː 'ɪlːoːs/.
The transition from Latin to Proto-Gallo-Romance involved the loss of all vowel and most consonant length distinctions (the colon marks in my IPA rendition), yelding */'de 'elo/ and */'de 'elos/
From Proto-Gallo-Romance to Old French, post-tonic vowels were lost (the /o/s in other words) and the determiner became a clitic, losing its stress and attaching to a nearby word. When the following word started with a vowel, the singular determiner attached to it: *"de elo arbre" became "de l'arbre". In the plural, or if the following word started with a consonant, the determiner attached to the preposition: *"de elo grande arbre" became "del grant arbre" *"de elos grandes arbres" became "dels granz arbres". The loss of the initial vowel of "elo(s)" isn't a regular process, but can be explained as the normal reduction of an unstressed word.
The three forms (de l', del, dels) had an influence on the pronunciation of the vowel of the preposition. Where it ended up in an open syllable, it lenited to schwa ("de l'arbre" -> /də.lar.brə/), while it retained its quality when it was in a closed syllable ("del grant" -> /dɛɫ.grant/; "dels granz" -> /dɛɫs.grants/). I'm not sure whether this quality was /e/ or /ɛ/ at the time. We've reached the modern condition for "de l'" so i'll stop tracking it here, the only change is that /ə/ started dropping more and more often with the centuries.
In the plural, a common sound change in Old French is that consonant clusters ending in /s/ started simplifying, dropping the preceding consonant (a remnant of this change in Contemporary French is the alternation between bœuf and bœufs, or œuil and yeux, where the final consonant is present in the singular but not in the plural). This affected all the plural contractions, als becoming as, els becoming ès and dels becoming des. As and ès were eventually affected by their singular into becoming aux, but des retained its form into Contemporary French, eventually losing its final /s/ before consonants.
In the singular, "del" was affected by a sound change that should be familiar from Modern English: syllable final /l/ was pronounced differently from syllable initial /l/, by raising the back of the tongue toward the soft palate. This made the pronunciation of syllable final /l/ closer and closer to that of /w/ or to a vowel like /o/ or /u/. In Late Old French (or in modern London English), this eventually transformed groups of a vowel + /l/ into diphthongs: al became au, el become eu (also often spelled "ou") and del became deu (also often spelled "dou" or "du"). The evolution of del /dɛɫ/ into modern /dy/ is irregular, you'd expect /dœ/ or /dø/.
/'deː 'ɪlːʊ̃/ -> */'de 'elo/ -> /dɛɫ/ - > /dɛʊ̯/ -> /dy/
/'deː 'ɪlːoːs/ -> */'de 'elos/ -> /dɛɫs/ - > /dɛs/ -> /dɛ(z)/
In french langage (as in most langages), some elementary rules must be observed often called "règles d'usage".
When constructing a sentence, if a "préposition" is followed by a definite article "article défini", then "préposition + article" are contracted to "produce" a new "article". For example in this plural form ("forme plurielle"):
Je mange des légumes
des is a contraction of "préposition: de" + "article: les" because in the french practice, one cannot say:
Je mange de les légumes
Je mange des légumes
des is pronouced "dé".
But the above contraction rule observe a slight change when plural form is not used. In the singular/feminine genre the rule is not applied. It is indeed correct to say:
Je bois de la limonade
Je mange de la viande
but in singular/masculine genre the rule is applied as in:
Je bois du jus de fruit
Je mange du poisson