For the conventional block floating-point quantizer (BFPQ), usually a large block size causes performance degradation and thus small block sizes are preferred, especially when non-uniformly distributed signals are processed. A tunable BFPQ with fractional exponent is proposed in this paper to deal with the problem. We first examine the root cause of degradation through analytic equations and then propose to tune the thresholds for deriving the exponent and fractional exponent of the block so as to strike a good balance between the quantization error and saturation error. An optimal tuning value depending on the block size and mantissa word-length can be obtained. Thus, the tunable BFPQ can achieve better output signal-to-quantization-noise ratio (SQNR) in a wide dynamic range. The analytic equation for the output SQNR of the proposed BFPQ is derived to verify the simulated results. Only one extra multiplication is required for each block to implement the tunable BFPQ. Finally, we show the obvious SQNR improvements compared to the conventional scheme for various settings of block sizes and mantissa word-lengths.