Rosser's algorithm is typically invoked during discussions of equal temperament scales, and is a way to obtain good approximations for multiple rational irrational numbers simultaneously. Is there a nice, accessible modern treatment? Rosser's paper was published in 1950.
In an equal temperament scale, the ratio of frequencies of adjacent notes is a constant. This is a desirable feature, in that it allows you to transpose a piece from one key into any other. Now, for various reasons (psychological, historical) we'd like have have certain ratios of frequencies, or close approximations. If we require the unison ratio (2:1) to be exact, then we require that our frequency ratio R satisfies R^n = 2 for some integer n. Thus, R is an integral root of 2. Next, if we want a good approximation to the perfect 5th (3:2), we conclude that we need R^r to be a good approximation to log2(3), for some integer r. This leads us into continued fractions. Now, if we'd also like a good approximation to the perfect 3rd (5:4), we would like R^s to be a good approximation to log2(5) for some integer s. We've now left the realm of continued fractions, as we're seeking to approximate two irrational numbers with powers of the same root of 2.