Chinese · Pronunciation Guide
sahn — an open "ah", a front n, and a pitch held flat at the top. The vowel is broader than English speakers expect.
sān
The pitch here is fixed and never negotiates with its neighbours. Where 三 does interact with its surroundings is in following words: in a phrase like sān diǎn the high level tone runs straight into a low dip, and the size of that drop is what makes the number audible across a noisy room.
Listen: 三
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现在三点。
Xiànzài sān diǎn.
It's three o'clock.
Listen: 现在三点。
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现在三点。 sets two falling tones against the level sān and then the low diǎn. So the pitch pattern is drop, drop, high, low — four syllables and four different pitch heights. Xiànzài in particular should not be smeared into one long descent; each of its syllables restarts high.
English speakers commonly produce "san" with the vowel of "cat", which sounds noticeably off to a Mandarin ear. Open the mouth wider and pull the vowel toward "ah". The second issue is the finish: -an ends with the tongue tip forward, and letting it drift back turns the number into sāng, a different syllable altogether.