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Visual elements such as grids, labels, and contour lines act as reference structures that support the primary information being presented. Such structures need to be usefully visible, but not so obtrusive that they clutter the presentation. Visual designers know how to carefully manage transparency and layering in an image to balance these elements. We want the presentation of these structures in complex, dynamic, computer-generated visualizations to reflect the same subtlety and comfort of good design. Our goal is to determine the physical, perceptual, and cognitive characteristics of such structures in a way that enables automatic presentation. Our approach to this problem does not try to characterize "ideal” or "best,” but instead seeks boundary conditions that define a range of visible yet subtle legibility. All presentations that are clearly bad lie outside of this range, and can easily be avoided. In this paper, we report three experiments investigating the effects of grid color and spacing on these boundary conditions, defined by manipulating the transparency (alpha) of thin rectangular grids over scatter plots. Our results show that while there is some variation due to user preference and image properties, bounding alpha allows us to reliably predict a range of usable yet unobtrusive grids over a wide variety of conditions.