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History & Background of Set Theory

History & Background of Set Theory. David Smith 100022110. Origins for Set Theory (MATH). Father of Set Theory & Combinatorics Georg Cantor (German Mathematician) created it in 1874, Although it was more stictly defined over time up until 1897. Mathematical Background.

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History & Background of Set Theory

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  1. History & Backgroundof Set Theory David Smith 100022110

  2. Origins for Set Theory (MATH) • Father of Set Theory & Combinatorics Georg Cantor (German Mathematician) • created it in 1874, • Although it was more stictly defined over time • up until 1897

  3. Mathematical Background

  4. Math and Music - Hand in Hand? • Set Theory began through 12-Tone Music • by Arnold Schönberg’s 12-tone techniques • by Josef Hauer using his theory of ‘tropes’ • Both use Math largely for determining matrices and tone rows, thus the beginning of math in music • by Anton Webern in composition (1930s) • as well as Leibowitz, Rufer, Perle • These are the basic math concepts which are applied to Set Theory later

  5. But, where’s the Set Theory? • The theory of Hexachords was then presented by Rochberg in 1955 • This idea did not cover all of set theory, but more than before • Defined chords by a set • which was a list of the pitch classes in the chord • Boulez also made statements towards the musical techniques with math and music

  6. Finally... Set Theory • Proper formulation of set theory of music only followed the theory of hexachords • Milton Babbitt, Donald Martino, David Lewin and John Rothgeb created it around 1955-1960 • They used the formation of groups, harmony, functions for melodies and rhythms in 12 note music and interaction of these over time. • These were still only compositional methods using set theory

  7. THE TRUE SET THEORY • No actual Analysis of Set Theory was derived until 1964 • Most significant contribution - Allen Forte • extended notion of pitch-class set • association of sets • ‘setcomplexes’, ‘subcomplexes’ • ‘complex’ - the relation by inclusion to central sets

  8. What did this do to Analysis? • These concepts established organization • giving analogies to tonality • Brings tonal coherence to large-scale music structures and the links between sections • With this theory Forte was able to analyze many atonal works: • Berg, Schönberg, Stravinsky and Webern

  9. Computers take it to a New Level • With the use of computers, Forte easily compiled a roster of complexes • Created syntax's for analysis of compositions by use of a computer • each syntax can be formulated for compositional style of composers

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