We analyze the tradeoff between factuality and abstractiveness of summaries. We introduce abstractiveness constraints to control the degree of abstractiveness at decoding time, and we apply this technique to characterize the abstractiveness-factuality tradeoff across multiple widely-studied datasets, using extensive human evaluations. We train a neural summarization model on each dataset and visualize the rates of change in factuality as we gradually increase abstractiveness using our abstractiveness constraints. We observe that, while factuality generally drops with increased abstractiveness, different datasets lead to different rates of factuality decay. We propose new measures to quantify the tradeoff between factuality and abstractiveness, incl. muQAGS, which balances factuality with abstractiveness. We also quantify this tradeoff in previous works, aiming to establish baselines for the abstractiveness-factuality tradeoff that future publications can compare against.