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(abridged) In the last decade, the advent of enormous galaxy surveys has motivated the development of automated morphological classification schemes to deal with large data volumes. Existing automated schemes can successfully distinguish between early and late type galaxies and identify merger candidates, but are inadequate for studying detailed morphologies of red sequence galaxies. To fill this need, we present a new automated classification scheme that focuses on making finer distinctions between early types roughly corresponding to Hubble types E, S0, and Sa. We visually classify a sample of 984 non-starforming SDSS galaxies with apparent sizes >14". We then develop an automated method to closely reproduce the visual classifications, which both provides a check on the visual results and makes it possible to extend morphological analysis to much larger samples. We visually classify the galaxies into three bulge classes (BC) by the shape of the light profile in the outer regions: discs have sharp edges and bulges do not, while some galaxies are intermediate. We separately identify galaxies with features: spiral arms, bars, clumps, rings, and dust. We find general agreement between BC and the bulge fraction B/T measured by the galaxy modeling package GIM2D, but many visual discs have B/T>0.5. Three additional automated parameters -- smoothness, axis ratio, and concentration -- can identify many of these high-B/T discs to yield automated classifications that agree ~70% with the visual classifications (>90% within one BC). Both methods are used to study the bulge vs. disc frequency as a function of four measures of galaxy 'size': luminosity, stellar mass, velocity dispersion, and radius. All size indicators show a fall in disc fraction and a rise in bulge fraction among larger galaxies.