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While I've heard that story many times, I don't think I knew what phrase in particular was problematic. White says it's "When love runs high." And that got me thinking that 1) "high" has a melisma and 2) the second syllable of "high" is a higher pitch than the first, so the meaning is reflected in how it's sung. I figured out the notes for that part, it's only a whole step (A to B), but it's still a higher pitch.
I figured out a bit more of the vocal melody, and I discovered that there's a Picardy third at the end of the verses (it's the last syllable of "loving" in "It's the time of the season for loving"). "Time of the Season" is in E minor, and the "loving" is sung to the phrase G, G#. So the Picardy third (that change from minor to major by raising the middle note of the triad) corresponds to the "loving," as if love is changing the key of the song.