Paul Hindemith
Most of Hindemith's music employs a unique system that is tonal but non-diatonic. Like most tonal music, it is centered on a tonic and modulates from one tonal center to another, but it uses all 12 notes freely rather than relying on a scale picked as a subset of these notes. Hindemith even rewrote some of his music after developing this system. One of the key features of his system is that he ranks all musical intervals of the 12-tone equally tempered scale from the most consonant to the most dissonant. He classifies chords in six categories, on the basis of how dissonant they are, whether or not they contain a tritone, and whether or not they clearly suggest a root or tonal center. Hindemith's philosophy also encompassed melody—he strove for melodies that do not clearly outline major or minor triads. In the late 1930s, Hindemith wrote a theoretical book The Craft of Musical Composition (vol. 1, Hindemith 1937), which lays out this system in great detail. He also advocated this system as a means of understanding and analyzing the harmonic structure of other music, claiming that it has a broader reach than the traditional Roman numeral approach to chords (an approach that is strongly tied to the diatonic scales). In the final chapter of Book I, Hindemith seeks to illustrate the wide-ranging relevance and applicability of his system in analysis of music examples ranging from the early origins of European music to the contemporary. These analyses include an early Gregorian melody, and compositions by Guillaume de Machaut, J. S. Bach, Richard Wagner, Igor Stravinsky, Arnold Schoenberg, and finally, a composition of his own. His piano work of the early 1940s Ludus Tonalis contains twelve fugues, in the manner of Johann Sebastian Bach, each connected by an interlude during which the music moves from the key of the last fugue to the key of the next one. The order of the keys follows Hindemith's ranking of musical intervals around the tonal center of C. One traditional aspect of classical music that Hindemith retains is the idea of dissonance resolving to consonance. Much of Hindemith's music begins in consonant territory, progresses rather smoothly into dissonance, and resolves at the end in full, consonant chords. This is especially apparent in his Concert Music for Strings and Brass.