Two Axioms of European Classical Music and Maximum Number of Polytonalities in Music

Article formulates two axioms of European classical music “Every musical piece should be within 24 tonalities, specifically 12 major and 12 minor tonalities, with some temporary slight variations allowed” and "Every musical piece should be written so that all instruments play in the same tonality at any given moment." These axioms stayed unchallenged until the approximately end of 19th century. First axiom was challenged by impressionists’ composers Claude Debussy, Maurice Ravel and other at the end of 19 century and later in the first half of 20 century by atonal, twelve-tone and other kinds of musical systems created by Arnold Schoenberg, Alban Berg, Anton Webern and other. The second axiom was challenged by creation of polytonal music. One of the early examples of polytonality can be found in the ballet of Igor Stravinsky "The Rite of Spring" (1913), where different instruments play in different keys simultaneously. Another influential figure in the development of polytonality was the French composer Charles Ives. The use of polytonal technique raised the theoretical question of how rich a contemporary composer's palette can be when the single-tonality axiom is set aside. In other words, what is the theoretical maximum number of tonalities that a composer can explore within a polytonality approach? Simple calculations show that total palette of polytonal music contains 24*23*22*21*…*1=24! (twenty-four factorial) or 620,448,401,733,239,439,360,000 of theoretically available combinations of tonalities. While obviously most of these combinations will result in cacophony, some tonal combinations have the potential to yield fresh and pleasant impressions that have not been explored in the past. If we consider the complete list of 9 modes per note (Major, Natural Minor, Harmonic Minor, Melodic Minor, Dorian mode, Phrygian mode, Lydian mode, Mixolydian mode, Locrian mode), we obtain 12 notes * 9 modes = 108 base tonalities/modes. Correspondingly in that case use of polytonalities produces 108! (108 factorial) combinations of tonalities/modes. This number appears as the theoretical upper limit of tonal palette in polytonality approach.

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