A reference instrument has been developed at the National Research Council of Canada for rapid, reproducible specular gloss measurements. The design and validation of this instrument for specular gloss measurements in accordance with standard methods for paints and plastics at 20°, 60°, and 85° geometries [American Society for Testing and Materials (ASTM) D523 and the International Organization for Standards (ISO) 2813] have been recently reported. These standard methods require a collimated beam geometry. Here we present the optical design considerations and characterization of this instrument to extend its gloss measurement capabilities to specular gloss measurements of paper samples at 75° geometry in accordance with standard test methods requiring a converging beam geometry (ASTM D1223 and TAPPI T480). This is, to the best of our knowledge, the first reported reference instrument that provides direct traceability for both types of standard gloss method and applications. The design challenge was to convert from a collimated beam to converging beam geometry while meeting the rigorous requirements of beam uniformity at the sample and receptor apertures specified in the 75° geometry test methods. We describe the innovative design to achieve this degree of functionality and reference instrument performance. The instrument's optical performance has been characterized theoretically and by comparison with measurement results. The light collection and detection systems have been analyzed via Monte Carlo simulation and ray tracing. The instrument validation includes comparison of the measurement results with theoretical gloss values for quartz, black glass, Vitrolite, and mirror gloss working standards, giving agreement of better than 0.32\% . Measurement validation also involved participation in the Collaborative Testing Services program interlaboratory comparison measurements of 75° gloss for white papers.