The mnemonic is only useful if you understand what it means, so probably worth translating it

Would that rule correlate to the vowel under the prefix letter?

@rosends I don’t understand your question

@JoelK does it have to do with the vowel sound under the prefix (e.g. a patach would cause a combined syllable while a segol, because it is distinct from the opening vowel sound of A-do-nai, would create 2 separate syllables)?

@rosends I see. The problem is that it’s a bit circular - the vowel on the prefix can sometimes change depending on the vowel on the first letter of the following word.

@JoelK does the vowel under the first letter of Hashem's name change?

@rosends Well you’ve already noted that sometimes it disappears entirely. But other than that I don’t believe so

@JoelK would it be accurate to say "when the name is preceded by a patach, the two sounds become coalesced into one syllable (va/do)"?

@rosends I can’t think of any exceptions to such a rule when it comes to the name pronounced as אדני. But see eg Devarim 11:17 where ואדני האדונים has a patach on the vav and a chataf patach on the aleph

@JoelK yes, but there the word is adonei (יז) כִּי יְקֹוָק אֱלֹהֵיכֶם הוּא אֱלֹהֵי הָאֱלֹהִים וַאֲדֹנֵי הָאֲדֹנִים so I'm wondering if there is a rule that can be applied just to Hashem's name (the question of why there would be different grammar for the name then arises)

@rosends. As I said, as far as I’m aware the rule you propose would always work for Hashem’s Name. But I think the rule is meant to have broader applications. Consider Bereishit 18:12 ואדני זקן

What about when it's a dalet? "בְּרִיךְ יְקָרָא דַיְיָ מֵאֲתַר" בֵּית שְׁכִנְתֵּיהּ