History

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 Schönbergs 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, wheres 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