1. Musical Numerology in Post-Tonal Theory
MATH 105 Term Project
Audrey Yim, Maria Flores, Kailee Whitten and Theresa Ellsworth

2. Background & Terms Twelve-tone technique in post-tonal theory
Basic elements of music in mathematical terms
Pitch
Accidentals
Musical Intervals

3. Tonal Theory and Post Tonal Theory
Tonal theory = defines a note as the primary or “tonic”, which all other notes functioning relative to their tonic in a strict order
Post-tonal theory = involves the breakdown of hierarchy where all notes and harmonies function according to their unique relationships
VS.

4. Post-Tonal Theory
Pitch = a tone with a specific frequency (a standard piano can play up to 88 pitches)
Pitch class = a group of pitches with the same name
EXAMPLE:
Due to enharmonic and octave equivalence, any pitch named “G” is a member of the pitch class G.

5. Ordered Pitch Class Intervals
Created when moving pitch class in modular pitch class space
It can never be larger than 11 semitones
EXAMPLE:
The ordered interval from C to G is +7 or -5.
From pitch-class C, one can go either up seven semitones or down 5 semitones to get to pitch-class G. This is because +7 and -5 are equivalent by mod 12.

6. 12 Notes of Music It’s Really That Easy
The Piano (for example) is just the same notes repeating!!!!!

7. Mod-12 System Uses a modulus of 12 respectively called “modulo twelve”
figures are calculated in multiples of 12
-12=0=12=24
12 Pitches/12 semitones = {0,1,2,3,4,5,6,7,8,9,10,11}
Octave Equivalence:
Two pitches share the same pitch-class an octave above or below
ex: C =0, C2=0, C3=0, C4=0
Octave equivalence means that these pitches will be very
resonant with each other and will share similar qualities/overtones
Going clockwise around the circle once will land on the same
note, but at a higher octave.
Going around counter-clockwise once goes back down
an octave.

8. Integer Notation
Notes become numbers! (Musical notes are written and understood as numbers)
Accidentals alter notes by adding or subtracting semitones and make them equal to each other at times……
Enharmonic Equivalents:
Pitches that sound the same but
have different names/ notation

9. Here’s A Little Fun Example...
…..AMAZING

10. Arnold Schoenberg: Father of 12-tone row
Written around World War I
Between 1912 and 1922, Schoenberg came to realize that he was searching for a new method of composition that would provide a new basis for musical structure to replace the old basis of tonality, which he thought was being stretched and distorted too much to remain a unifying structural principle
Instead of using 1 or 2 tones as main points of focus for a composition, Schoenberg suggested using all 12 tones “related only to one another”
No notes would predominate as focal points, nor would any hierarchy of importance be assigned to individual tones

11. 12-tone composition
The specific ordering of twelve pitch-classes of a chromatic scale
tone row; set; series.
“pre-compositional”

12. Forms Prime (original) Inversion Retrograde Retrograde inversion
integer notation!!

13. Inversion operation Prime 3 5 6 1 4 t e 9 8 0 2 7 Inversion
+2,+1,-5...
Inversion
t 6 4 e
-2,-1,+5...
Horizontal reflection

14. Retrograde operation Prime 3 5 6 1 4 t e 9 8 0 2 7 Retrograde
Vertical Reflection

15. Retrograde-inversion operation
Prime
t e
Inversion
t 6 4 e
Rotate 180* clockwise
Retrograde-inversion
e 4 6 t

16. 12×12
organization of 48 possible row permutations
1950s-
Milton Babbitt, not Schoenberg

17. Matrix ex.)