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Its floating harmonic systems and pastel orchestra are mirror reflections of Debussy, who in 1913 was very much alive.
The lack of forward momentum that was fundamental to his harmonic systems, rhythms and melodies is what sets Messiaen apart not only from his contemporaries but from almost the whole of the western art-music tradition of the previous 500 years.
A general application of matrices in physics is to the description of linearly coupled harmonic systems.
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His complex and idiosyncratic harmonic system was a free adaptation of 12-tone technique, which he called the "mother lode".
Roslavets, an obscure early-20th-century Ukrainian composer, invented his own harmonic system, based on altered scales.
Carter also made use of a highly systematised harmonic system, involving tables of all possible permutations of a given set of intervals.
The Western harmonic system involves keys with increasing numbers of sharps and flats exfoliating out from middle C, or from C major, all white keys on the piano.
Now my feeling is that the next work is going to be more complex, because of the harmonic system being in place.
An opponent of the 12-tone school of composer Arnold Schoenberg, Hindemith formulated the principles of a harmonic system that was based on an enlargement of traditional tonality.
The basic impulse remains the same: the blues is the basic link between Bird and Trane in the way that the major-minor harmonic system of tension and release is the basic link between Bach and Beethoven.
He sought to revitalize tonality the traditional harmonic system that was being challenged by many other composers and also pioneered in the writing of Gebrauchsmusik, or "utility music," compositions for everyday occasions.
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