For the calculation of the color gamut volume and the maximum number of mutually discernible colors, an algorithm based on a Riemannian metric and the densest packing of spheres is proposed. With this algorithm, the color gamut volume was calculated for the conditions of experiments reported in literature. Good agreement was found with the experimental findings of the color gamut volume as a function of the peak luminance. Using the new algorithm, the color gamut volume and the maximum number of mutually discernible colors was calculated for various sets of primary colors corresponding to display standards and various dynamic ranges. Comparisons were made with state-of-the-art methods which are based on the Euclidean metric in approximately uniform color spaces and a simple cubic lattice. It was found that the state-of-the-art methods underestimate the maximum number of mutually discernible colors. However, the relative differences decrease as the primary colors are more saturated. Based on the new algorithm the maximum number of mutually discernible colors was calculated for a range of peak retinal illuminance levels and various sets of primary colors. We found that, for a given set of primary colors, the maximum number of mutually discernible colors is proportional to the logarithm of the ratio of the peak retinal illuminance level and a fitting parameter.