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What is Audio Sample Rate?

TL;DR: Sample rate is how many times per second audio is measured (sampled) during recording or playback. Standard CD quality is 44,100 Hz (44.1 kHz), meaning the audio is captured 44,100 times per second.
Higher sample rates capture more detail in high frequencies. 44.1 kHz captures up to 22 kHz (above human hearing limit). 48 kHz is standard for video production. 96 kHz and 192 kHz are used in professional music production.