To answer your question: Vietnamese has 6 tones (in the northern dialect like in Hanoi), while Mandarin has 4 main tones (plus the neutral tone). So yes, it’s technically “more,” but since you’ve already studied Mandarin, you’re actually ahead of the game!
In terms of comparison:
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Vietnamese tones are more pitch + voice quality based—some tones sound “creaky” or “breathy,” not just rising or falling.
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There’s some overlap with Mandarin’s tones, like rising (Vietnamese “sắc” ≈ Mandarin 2nd tone), and falling (Vietnamese “huyền” ≈ Mandarin 4th tone), but Vietnamese has additional tones with glottal stops or “heavy” endings that Mandarin doesn’t use.