1. History & Background of Set Theory David Smith 100022110

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

3. Mathematical Background

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

5. But, where’s the Set Theory? n 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 n which was a list of the pitch classes in the chord n Boulez also made statements towards the musical techniques with math and music

6. Finally... Set Theory n 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. n These were still only compositional methods using set theory

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

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

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