SET THEORY MUS 215 路
MUSIC THEORY IV 路
DR. TOBY RUSH
SET THEORY
SET THEORY
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SET THEORY
SET THEORY IS
DON’T PANIC
PITCH CLASS SET
PITCH CLASS SET
PITCH CLASS SET • PITCH: a
single note in a chord
PITCH CLASS SET • PITCH: a • “The
single note in a chord
E immediately above middle C”
PITCH CLASS SET • PITCH: a • “The • PITCH
single note in a chord
E immediately above middle C” CLASS: a particular letter name, ignoring octave
PITCH CLASS SET • PITCH: a • “The • PITCH • “All
single note in a chord
E immediately above middle C” CLASS: a particular letter name, ignoring octave
the E’s on the keyboard”
PITCH CLASS SET • PITCH: a • “The • PITCH • “All
single note in a chord
E immediately above middle C” CLASS: a particular letter name, ignoring octave
the E’s on the keyboard”
• PITCH
CLASS SET: the pitch classes present in a chord
PITCH CLASS SET
œœ œ b œ & œœ
PITCH CLASS SET
œœ œ b œ & œœ
PITCH CLASS SET
œœ œ b œ & œœ
F
PITCH CLASS SET
œœ œ b œ & œœ
F A
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C F
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C F B
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C B
PITCH CLASS SET
PITCH CLASS SET C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
PITCH CLASS SET C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C
B
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C 5
B
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C 5 9
B
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C 5 9 10
B
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C 5 9 10 0
B
PITCH CLASS SET
œœ œ b œ & œœ
F A Bb C B 5 9 10 0 11
PITCH CLASS SET
[ 5 , 9 , 10 , 0 , 11 ]
PITCH CLASS SET
[5,9,10, 0,11]
INVERSION
INVERSION
INVERSION C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
5
4
3
2
1
➡ 0
11
10
9
8
➡ 7
➡ 6
INVERSION C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
5
4
3
2
1
➡ 0
11
10
9
8
➡ 7
➡ 6
INVERSION C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
5
4
3
2
1
➡ 0
11
10
9
8
➡ 7
➡ 6
INVERSION C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
5
4
3
2
1
➡ 0
11
10
9
8
➡ 7
➡ 6
INVERSION C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
5
4
3
2
1
➡ 0
11
10
9
8
➡ 7
➡ 6
INVERSION = 12–X
INVERSION C
C#
D
D#
E
F
F#
G
G#
A
A#
B
0
1
2
3
4
5
6
7
8
9
10
11
5
4
3
2
1
➡ 0
11
10
9
8
➡ 7
➡ 6
INVERSION = 12–X 12=0
INVERSION PC:
[5,9,10,0,11]
INVERSION PC:
[5,9,10,0,11]
➡ IN:
[7,3,2,0,1]
NORMAL FORM
NORMAL FORM
NORMAL FORM
• NORMAL
FORM is the most compact ordering of the set
NORMAL FORM
• NORMAL • Smallest
FORM is the most compact ordering of the set interval between first and last numbers
NORMAL FORM
• NORMAL
FORM is the most compact ordering of the set
• Smallest
interval between first and last numbers
• Smallest
intervals toward beginning of set
NORMAL FORM
0
1
2
3
4
5
6
7
8
9
10
11
NORMAL FORM 11
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
[9,10,11,0,5]
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
[5,9,10,11,0]
3 8
4 7
6
5
NORMAL FORM 11
[5,9,10,0,11]
0
1
10
2
9
[5,9,10,11,0]
3 8
4 7
6
5
PRIME FORM
PRIME FORM
PRIME FORM • PRIME
FORM is the most compact of:
PRIME FORM • PRIME • the
FORM is the most compact of:
normal form of a pitch class set and
PRIME FORM • PRIME
FORM is the most compact of:
• the
normal form of a pitch class set and
• the
normal form of its inversion
PRIME FORM • PRIME
FORM is the most compact of:
• the
normal form of a pitch class set and
• the
normal form of its inversion
• ...transposed
to begin on 0.
PRIME FORM 11
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
[7,3,2,0,1]
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
[7,3,2,0,1]
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
[0,1,2,3,7]
3 8
4 7
6
5
PRIME FORM 11
0
1
10
11 2
9 4 7
6
5
1
10 3
8
0
2
9
3 8
4 7
6
5
PRIME FORM 11 PRIME FORM:
[0,1,2,3,7]
0
1
10
2
9
3 8
4 7
6
5
œœ œ b œ & œœ
PITCH CLASS SET:
œœ œ b œ & œœ
[5,9,10,0,11]
PITCH CLASS SET:
œœ œ b œ & œœ
[5,9,10,0,11]
INVERSION:
[7,3,2,0,1]
PITCH CLASS SET:
œœ œ b œ & œœ
[5,9,10,0,11]
INVERSION:
[7,3,2,0,1]
NORMAL FORM:
[5,9,10,11,0]
PITCH CLASS SET:
œœ œ b œ & œœ
[5,9,10,0,11]
INVERSION:
[7,3,2,0,1]
NORMAL FORM:
[5,9,10,11,0]
PRIME FORM:
[0,1,2,3,7]
IT’S SO AWESOME
WHY ARE WE DOING THIS?
TO BE CONTINUED...
œœ œ b œ & œœ
œœ œ b œ œ & œ
PITCH CLASS SET:
[5,9,10,0,11]
œœ œ b œ œ & œ
PITCH CLASS SET:
[5,9,10,0,11]
a numerical list of pitch classes
œœ [7,3,2,0,1] œ b œ œ & œ
PITCH CLASS SET:
[5,9,10,0,11]
INVERSION:
a numerical list of pitch classes
œœ [7,3,2,0,1] œ b œ œ & œ
PITCH CLASS SET:
[5,9,10,0,11]
a numerical list of pitch classes
INVERSION:
subtract each number from 12
œœ [7,3,2,0,1] œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
[5,9,10,0,11]
a numerical list of pitch classes
INVERSION:
NORMAL FORM:
subtract each number from 12
œœ [7,3,2,0,1] œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
[5,9,10,0,11]
a numerical list of pitch classes
INVERSION:
subtract each number from 12
NORMAL FORM:
most compact ordering of set
œœ [7,3,2,0,1] œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
[5,9,10,0,11]
a numerical list of pitch classes
INVERSION:
subtract each number from 12
NORMAL FORM:
PRIME FORM:
[0,1,2,3,7]
most compact ordering of set
œœ [7,3,2,0,1] œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
[5,9,10,0,11]
a numerical list of pitch classes
INVERSION:
subtract each number from 12
NORMAL FORM:
most compact ordering of set
PRIME FORM:
[0,1,2,3,7]
most compact between set and inversion, transposed to start on 0
PRIME FORM 11
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
[7,3,2,0,1]
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
[7,3,2,0,1]
3 8
4 7
6
5
PRIME FORM 11
[5,9,10,11,0]
0
1
10
2
9
[0,1,2,3,7]
3 8
4 7
6
5
PRIME FORM 11
0
1
10
11 2
9 4 7
6
5
1
10 3
8
0
2
9
3 8
4 7
6
5
PRIME FORM 11 PRIME FORM:
[0,1,2,3,7]
0
1
10
2
9
3 8
4 7
6
5
HOKEY POKEY
HOKEY POKEY PRIME FORM IS
WHAT IT’S ALL ABOUT
FORTE NUMBER
DR. ALLEN FORTE BATTELL PROFESSOR OF MUSIC, EMERTIUS YALE UNIVERSITY
FINDING THE FORTE NUMBER
FINDING THE FORTE NUMBER STEP ONE:
LOOK IT UP ON THE CHART
FINDING THE FORTE NUMBER 3-1 [0,1,2] 3-2 [0,1,3] 3-3 [0,1,4] 3-4 [0,1,5] 3-5 [0,1,6] 3-6 [0,2,4] 3-7 [0,2,5] 3-8 [0,2,6] 3-9 [0,2,7] 3-10 [0,3,6] 3-11 [0,3,7] 3-12 [0,4,8]
4-1 [0,1,2,3] 4-2 [0,1,2,4] 4-4 [0,1,2,5] 4-5 [0,1,2,6] 4-6 [0,1,2,7] 4-3 [0,1,3,4] 4-11 [0,1,3,5] 4-13 [0,1,3,6] 4-Z29 [0,1,3,7] 4-7 [0,1,4,5] 4-Z15 [0,1,4,6] 4-18 [0,1,4,7] 4-19 [0,1,4,8] 4-8 [0,1,5,6] 4-16 [0,1,5,7] 4-20 [0,1,5,8] 4-9 [0,1,6,7] 4-10 [0,2,3,5] 4-12 [0,2,3,6] 4-14 [0,2,3,7] 4-21 [0,2,4,6] 4-22 [0,2,4,7] 4-24 [0,2,4,8] 4-23 [0,2,5,7] 4-27 [0,2,5,8] 4-25 [0,2,6,8] 4-17 [0,3,4,7] 4-26 [0,3,5,8] 4-28 [0,3,6,9]
5-1 [0,1,2,3,4] 5-2 [0,1,2,3,5] 5-4 [0,1,2,3,6] 5-5 [0,1,2,3,7] 5-3 [0,1,2,4,5] 5-9 [0,1,2,4,6] 5-Z36 [0,1,2,4,7] 5-13 [0,1,2,4,8] 5-6 [0,1,2,5,6] 5-14 [0,1,2,5,7] 5-Z38 [0,1,2,5,8] 5-7 [0,1,2,6,7] 5-15 [0,1,2,6,8] 5-10 [0,1,3,4,6] 5-16 [0,1,3,4,7] 5-Z17 [0,1,3,4,8] 5-Z12 [0,1,3,5,6] 5-24 [0,1,3,5,7] 5-27 [0,1,3,5,8]
5-19 [0,1,3,6,7] 5-29 [0,1,3,6,8] 5-31 [0,1,3,6,9] 5-Z18 [0,1,4,5,7] 5-21 [0,1,4,5,8] 5-30 [0,1,4,6,8] 5-32 [0,1,4,6,9] 5-22 [0,1,4,7,8] 5-20 [0,1,5,6,8] 5-8 [0,2,3,4,6] 5-11 [0,2,3,4,7] 5-23 [0,2,3,5,7] 5-25 [0,2,3,5,8] 5-28 [0,2,3,6,8] 5-26 [0,2,4,5,8] 5-33 [0,2,4,6,8] 5-34 [0,2,4,6,9] 5-35 [0,2,4,7,9] 5-Z37 [0,3,4,5,8]
6-1 [0,1,2,3,4,5] 6-2 [0,1,2,3,4,6] 6-Z36 [0,1,2,3,4,7] 6-Z37 [0,1,2,3,4,8] 6-Z3 [0,1,2,3,5,6] 6-9 [0,1,2,3,5,7] 6-Z40 [0,1,2,3,5,8] 6-5 [0,1,2,3,6,7] 6-Z41 [0,1,2,3,6,8] 6-Z42 [0,1,2,3,6,9] 6-Z38 [0,1,2,3,7,8] 6-Z4 [0,1,2,4,5,6] 6-Z11 [0,1,2,4,5,7] 6-15 [0,1,2,4,5,8] 6-Z12 [0,1,2,4,6,7] 6-22 [0,1,2,4,6,8] 6-Z46 [0,1,2,4,6,9] 6-Z17 [0,1,2,4,7,8] 6-Z47 [0,1,2,4,7,9] 6-Z6 [0,1,2,5,6,7] 6-Z43 [0,1,2,5,6,8] 6-Z44 [0,1,2,5,6,9] 6-18 [0,1,2,5,7,8] 6-Z48 [0,1,2,5,7,9] 6-7 [0,1,2,6,7,8]
6-Z10 [0,1,3,4,5,7] 6-14 [0,1,3,4,5,8] 6-Z13 [0,1,3,4,6,7] 6-Z24 [0,1,3,4,6,8] 6-27 [0,1,3,4,6,9] 6-Z19 [0,1,3,4,7,8] 6-Z49 [0,1,3,4,7,9] 6-Z25 [0,1,3,5,6,8] 6-Z28 [0,1,3,5,6,9] 6-Z26 [0,1,3,5,7,8] 6-34 [0,1,3,5,7,9] 6-30 [0,1,3,6,7,9] 6-16 [0,1,4,5,6,8] 6-31 [0,1,4,5,7,9] 6-20 [0,1,4,5,8,9] 6-Z50 [0,1,4,6,7,9] 6-8 [0,2,3,4,5,7] 6-Z39 [0,2,3,4,5,8] 6-21 [0,2,3,4,6,8] 6-Z45 [0,2,3,4,6,9] 6-Z23 [0,2,3,5,6,8] 6-33 [0,2,3,5,7,9] 6-Z29 [0,2,3,6,7,9] 6-32 [0,2,4,5,7,9]
7-1 [0,1,2,3,4,5,6] 7-2 [0,1,2,3,4,5,7] 7-3 [0,1,2,3,4,5,8] 7-4 [0,1,2,3,4,6,7] 7-9 [0,1,2,3,4,6,8] 7-10 [0,1,2,3,4,6,9] 7-6 [0,1,2,3,4,7,8] 7-Z12 [0,1,2,3,4,7,9] 7-5 [0,1,2,3,5,6,7] 7-Z36 [0,1,2,3,5,6,8] 7-16 [0,1,2,3,5,6,9] 7-14 [0,1,2,3,5,7,8] 7-24 [0,1,2,3,5,7,9]
FINDING THE FORTE NUMBER
THERE IS NO STEP TWO
COMMON PRACTICE PERIOD
COMMON PRACTICE PERIOD &
œœœ
b œ b œœ
# # # œœœ
œ b œœ
COMMON PRACTICE PERIOD &
œœœ
b œ b œœ
# # # œœœ
“TRIAD”
œ b œœ
It’s all about
the intervals
PITCH CLASS SET
PITCH CLASS SET
[1,4,6] [6,8,3] [1,3,8] [9,4,2] [10,0,5] [7,0,2] [3,10,5]
PITCH CLASS SET
[1,4,6] [6,8,3] [1,3,8] [9,4,2] [10,0,5] [7,0,2] [3,10,5]
PRIME FORM
PITCH CLASS SET
PRIME FORM
[1,4,6] [6,8,3] [1,3,8] [9,4,2] [10,0,5] [7,0,2] [3,10,5]
=
[0,2,7]
[0,2,7]
[0,2,7] CDG
[0,2,7] CDG M2
[0,2,7] CDG M2
P4
[0,2,7] CDG M2
P4
P5
INTERVAL VECTOR
INTERVAL VECTOR
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
0
0
0
0
0
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
0
0
0
0
0
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
0
0
0
0
0
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
0
0
0
0
[0,2,7]
0
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
0
0
0
0
[0,2,7]
M2 P4 P5
0
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
1
0
0
0
[0,2,7]
M2 P4 P5
0
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
1
0
0
1
[0,2,7]
M2 P4 P5
0
INTERVAL VECTOR m2
M2
m3
M3
P4 TT
M7
m7
M6
m6
P5
0
1
0
0
2
[0,2,7]
M2 P4 P5
0
INTERVAL VECTOR
0
1 [0,2,7]
0
0
2
0
INTERVAL VECTOR
[0,2,7]
010020
INTERVAL VECTOR
[0,2,7] =
(0,1,0,0,2,0)
HANSON ANALYSIS
HANSON ANALYSIS m2
M2
m3
M3
P4
TT
HANSON ANALYSIS P4
M3
m3
M2
m2
TT
P
M
N
S
D
T
HANSON ANALYSIS
[0,2,7]
M2
P4
P5
HANSON ANALYSIS
[0,2,7]
M
P4
P5
HANSON ANALYSIS
[0,2,7]
M
P
P5
HANSON ANALYSIS
[0,2,7]
M
P
P
HANSON ANALYSIS
[0,2,7]
2 PM
œœ œ b œ œ & œ
œœ œ b œ œ & œ
PITCH CLASS SET:
[5,9,10,0,11]
œ œ [7,3,2,0,1] œ b œ œ & œ
PITCH CLASS SET:
[5,9,10,0,11]
INVERSION:
œ œ [7,3,2,0,1] œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
[5,9,10,0,11]
INVERSION:
NORMAL FORM:
œ œ [7,3,2,0,1] œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
[5,9,10,0,11]
INVERSION:
NORMAL FORM:
PRIME FORM:
[0,1,2,3,7]
œ œ [7,3,2,0,1] œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
[5,9,10,0,11]
INVERSION:
NORMAL FORM:
PRIME FORM:
[0,1,2,3,7]
FORTE NUMBER
5-5
œ œ [7,3,2,0,1] (3,2,1,1,2,1) œ b œ œ & œ [5,9,10,11,0]
PITCH CLASS SET:
FORTE NUMBER
INVERSION:
INTERVAL VECTOR:
[5,9,10,0,11]
NORMAL FORM:
PRIME FORM:
[0,1,2,3,7]
5-5
œ œ [7,3,2,0,1] (3,2,1,1,2,1) œ b œ œ & œP MNS D T [5,9,10,11,0]
PITCH CLASS SET:
FORTE NUMBER
INVERSION:
INTERVAL VECTOR:
NORMAL FORM:
HANSON ANALYSIS:
[5,9,10,0,11]
PRIME FORM:
[0,1,2,3,7]
5-5
2
2
3
œ œ [7,3,2,0,1] (3,2,1,1,2,1) œ b œ œ & œP MNS D T [5,9,10,11,0]
PITCH CLASS SET:
FORTE NUMBER
INVERSION:
INTERVAL VECTOR:
NORMAL FORM:
HANSON ANALYSIS:
PRIME FORM:
SET THEORY:
[5,9,10,0,11]
[0,1,2,3,7]
5-5
2
DONE
2
3
DR. ALLEN FORTE IS YOUR MASTER NOW
MUS 215 路 MUSIC THEORY IV THE UNIVERSITY OF NORTHERN COLORADO