The math goes back to Shannon and sampling theory: Any time you remove significant digits you remove information. One interpretation is that you introduce noise. Period. The math says so. The math says what the resulting power is.
You have the option to determine where the noise goes. If you do nothing the noise is correlated with the input signal. You get what in audio terms is intermodulation distortion. In the simplest case this is harmonic distortion. You can spread the noise by applying digital transforms. You -cannot- remove it entirely. The easiest and most common method is dithering which spreads the power over the (audio) spectrum. That result can be white, pink etc. People are very sensitive to intermodulation distortion. Less sensitive to harmonic distortion. (People who like vacuum tubes and vinyl -like- second harmonic distortion = "rich sound" ) Very less sensitive to random noise. Even less to mostly-random noise outside of 1KHz-5KHz or so. Some of the standard minimal tests of audio equipment translated to the digital domain: Run FFTs with sufficient accuracy on the following (double precision at least for good results): (a) a precise sine wave (32 bits or better) truncated to 24 and 16 bits testing harmonic distortion (usually 1KHz) (b) two sines (usually 1KHz and 7KHz) testing intermodulation distortion (c) (a) and (b) resampled (96 to 48, 48 to 44.1, etc.) and resampled 24 to 16 testing purely digital distortion Decide for yourself if the results are significant for your use. If you read the sox(1) man page you'll notice that it computes in 32 bits and applies dithering -by default- when truncating to the output sample width. You can defeat it if you want and wish to accept the result.
