Notes
- The binary system constitues a real and basic factor that goes beyond the specific features of modern European music. It is at the root of the trifunctional cadence logic that appears in modern art music on several levels. On the elementary level only the three elements appear in diverse variations always represented by the same chords. This level is also reached by music resulting from generative creativity (e.g., in polyphonic folk music, gypsy music, jazz, hit songs, etc.). In European art music the three functional system is much more complex; it consists (in addition to the basic functions) of principal and secondary degrees, polar oppositions, and contrasts among several sequences of chords. This higher‐level system employs multiple contrasts and oppositions instead of simple ones. The higher level of contrasts is illustrated by the quadratic group (group of four elements) described by Klein. This is a mathematical structure in which each quantity (x) appears together with its negation (−x), with its inverse (lix), and with the negative of its inverse (−l/x). In music the modern tonic‐functional system constitutes an incomplete example of this quadratic structure. Here each tone (e.g., the tonic) has two kinds of “negation,” the dominant and the subdominant. In contemporary music the two kinds of negations also appear in a unified form, as shown in Schonberg's “theory of regions” and Lendvai's analysis of Bart6k's work (Lendvai calls this the “polar opposition” of the tonic, e.g., if C is the tonic, the polar opposition is F‐sharp).
- The above experiments are contained in the full‐length study by one of the authors, Mária Sági, “A zenei generativ készségek kisérleti Vizsgálata” (The Experimentul Investigation of Musical Generative Abilities) originally published by the Institute for Culture, Budapest, 1979; revised edition to be published by the publishing house of the Hungarian Academy of Sciences (currently in preparation).