In this tour de force, the authors introduce two easily-defined proper subsets of the set PER of all finite words w on {a, b} having two relatively prime periods p and q, such that |w| = p + q - 2. These subsets, Harm and Gold, and the sets of “harmonic words” and “gold words,” respectively, share PER’s property that their factor sets equal the set of finite factors of all Sturmian words. The work contains proofs of numerous interesting results about Gold and Harm, including simple formulas for the number of harmonic words of any given length. It also contains several intriguing conjectures, including an amusing one that, if true, implies that Goldbach’s conjecture is true, and another that is true if there are infinitely many twin primes.

The work is exceptionally well written. It is replete with examples, and directs the reader clearly to references that define or explain those few necessary terms and concepts that it does not. Its proofs are complete, and easy to follow. In summary, the work simultaneously represents a real advance in the art, and whets the appetite of those unfamiliar with the subject to learn more about this fascinating field.