History & Backgroundof 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

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

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

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

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

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

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