Despite Hindemith's intentions, and the overview of the previous chapter, the practical application of the Unterweisung requires further clarification. This is partly owing to the incomplete fourth volume (on four-part writing) and the dubious fidelity of the posthumously-published third volume. There can be little doubt that Hindemith intended his music theory for teaching and composition purposes: many reports from Hindemith's students recall his active use of Unterweisung material in class. However, the texts themselves – particularly the first, theoretical volume – do not always make the crucial relationship between theory and practice explicit, and the volume on two-part writing, though instructive, does not contain sufficient instruction for denser textures. Theorising the psychoacoustic relationship of harmonic partials to construct a pitch hierarchy offers a fascinating and enriching academic pursuit: but how might it help the enthusiastic student to compose music?

The present chapter cannot answer this question directly. We cannot take Hindemith's theory and apply it to his music without the need for several undermining assumptions: what do we make of music in three or more parts? What if our composition is not contrapuntal and functions without clearly defined voices? How should we use Series 1 and 2 to generate compositions? How does an identification of harmonic fluctuation affect our compositional process? Should we aim for certain harmonic fluctuation models, which Hindemith seemed to be moving towards in Unterweisung III? How does a combination of these factors affect foreground and background structural levels? Too much Unterweisung theory is missing or incomplete and though much of its basis is scientifically conceived, its application appears hazily subjective. However, we may consider those elements of Hindemith's music that are representative of his general style, and offer a mixture of set theory, salience principles and voice-leading reduction to provide an analysis of Hindemith's music. Crucially, in doing so we may find a way back to his music theory; his theory and practice are interwoven. In other words, we may reach for a study of his music to better understand and appreciate the theory that he was in the process of building.

Though it is not claimed as a new theory of Hindemith's music per se, this chapter offers a method for the study of Hindemith's theoretical principles within free composition, which is used throughout the rest of the book.