We present a new survey of HCN(1-0) emission, a tracer of dense molecular gas, focused on the little-explored regime of normal star-forming galaxy disks. Combining HCN, CO, and infrared (IR) emission, we investigate the role of dense gas in Star Formation (SF), finding systematic variations in both the apparent dense gas fraction and the apparent SF efficiency (SFE) of dense gas. The latter may be unexpected, given the popularity of gas density threshold models to explain SF scaling relations. We used the IRAM 30-m telescope to observe HCN(1-0) across 29 nearby disk galaxies whose CO(2-1) emission has previously been mapped by the HERACLES survey. Because our observations span a range of galactocentric radii, we are able to investigate the properties of the dense gas as a function of local conditions. We focus on how the IR/CO, HCN/CO, and IR/HCN ratios (observational cognates of the SFE, dense gas fraction, and dense gas SFE) depend on the stellar surface density and the molecular/atomic ratio. The HCN/CO ratio correlates tightly with these two parameters across a range of 2.1 dex and increases in the high surface density parts of galaxies. Simultaneously, the IR/HCN ratio decreases systematically with these same parameters and is ~6-8 times lower near galaxy centers than in the outer regions. For fixed line-mass conversion factors, these results are incompatible with a simple model in which SF depends only on the gas mass above some density threshold. Only a specific set of environment-dependent conversion factors can render our observations compatible with such a model. Whole cloud models, such as the theory of turbulence regulated SF, do a better job of matching our data. We explore one such model in which variations in the Mach number and in the mean density would respectively drive the trends within galaxy disks and the differences between disk and merging galaxies (abridged).