Schenker's Theory of Harmony
Schenker's book on harmony is the first of a three part series called New Musical Theories and Fantasies. Harmony, written in 1906, was one of many books on the subject, but the title of the series points to what makes Schenker's approach to the subject particularly distinctive. Although Schenker does speculate widely on theoretical points, the 'Fantasies' part of the title is also fully justified.
Most theories of harmony before (and since) Schenker's could be characterised as static - they are concerned with the relationship between chords and keys. Probably the best known of these is Hugo Riemann's Harmony Simplified (1895). Riemann is interested in the function of chords and their relationships and borrows from mathematical theories to help explain these. He understood the tonal system as a balanced one, with the tonic in the middle:
|IV (fifth below I)||I||V (fifth above I)
Schenker was not concerned with functional relationships but in a dynamic model of harmony. He was interested in showing how the tonal system was derived from nature and (although modern psychology and acoustics refutes some of his claims) the metaphors he used are still a very powerful and compelling way of understanding the tonal system.