Heh Hayediya before the letter Chet: Segol or Patach?

Question

In Acharei Mot the verses (Bamidbar 16:20 (end) and 21 (beginning)) read as follows:

וְהִקְרִ֖יב אֶת־הַשָּׂעִ֥יר הֶחָֽי׃ כא וְסָמַ֨ךְ אַהֲרֹ֜ן אֶת־שְׁתֵּ֣י יָדָ֗ו עַ֣ל רֹ֣אשׁ הַשָּׂעִיר֮ הַחַי֒

The first occurrence at the end of a verse 20 is Hechay, and the second occurrence in verse 21 is Hachay.

What is the reason for the difference?

The Minchat Shai addresses this and if I understand correctly, he doesn't explain it grammatically but gives a mnemonic to remember.

את השעיר החי – קדמאה בסגול תניי בפתח וחד פסוק סימן (ר״ל כמו מן הם המדברים בתיבה השנייה ה״א בפתח כן החי תניין בפתח וכמו האנכי הריתי בתיבה הראשונה הה״א בסגול כן החי קדמאה בסגול, ובדעת האחד נודע השני ולא כדוחק המהרש״ד בת״ס שלו, ולי בחרתי הסימן החכם עניו בראשו, והכסיל בחושך הולך.) הם המדברים סימן אחר האנוכי הריתי.‏

Is there despite that a logical explanation?

Answer 1

The change of vowel on the definite article is due to assimilation of its vowel to the vowel of the gutteral. As Joshua Blau puts it (Phonology and Morphology of Biblical Hebrew, p181): "Because of the weak pronunciation of laryngeals/pharyngeals, vowels preceding and following them were in closer contact than vowels divided by other consonants, and thus the laryngeals/pharyngeals promoted vowel assimilation."

In this case, the pataḥ of חַי requires no change of the vowel of הַ. However, the first case is חָי (as it's in pause), and kamats is a higher vowel (in the mouth, it's a open-mid back rounded vowel) than pataḥ (open front unrounded vowel), and segol (a open-mid front rounded vowel) matches the kamats better than pataḥ does. (See the vowel chart on Wikipedia.)

To go a bit beyond your question: different laryngeals/pharyngeals (אהחע and additionally ר) have different amounts and locations (in the mouth) of articulation, and so we see different effects on the vowel of the definite article. Cases where הַ becomes הָ can be understood additionally as "compensatory lengthening", wherein the vowel of הַ gets lengthened to replace the expected dagesh in the following letter (which is omitted in these letters; rhythmically, a closed syllable with a short vowel is similar to an open syllable with a long vowel).