Transcript 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)
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created it in 1874,
Although it was more stictly defined over time
 up
until 1897
Mathematical Background
Math and Music - Hand in Hand?

Set Theory began through 12-Tone Music
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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
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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
But, where’s the Set Theory?

The theory of Hexachords was then
presented by Rochberg in 1955
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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
Finally... Set Theory

Proper formulation of set theory of music
only followed the theory of hexachords
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
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
THE TRUE SET THEORY
No actual Analysis of Set Theory was
derived until 1964
 Most significant contribution - Allen Forte

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extended notion of pitch-class set
association of sets
 ‘setcomplexes’,
‘subcomplexes’
 ‘complex’ - the relation by inclusion to central sets
What did this do to Analysis?

These concepts established organization
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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:
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Berg, Schönberg, Stravinsky and Webern
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