An Introduction to Pitch-Class Set Theory

1
An Introductionto Pitch-Class Set Theory

• The following slides give a brief introduction to
  some of the concepts and terminology necessary in
  discussing post-tonal music.

2
Pitch-Class Notation

1. Pitch class (abbreviated, pc) The twelve
   different pitches (independent of octave
   displacement) of the equal tempered system.

3
Pitch-Class Notation

• Pitch class (abbreviated, pc) The twelve
  different pitches (independent of octave
  displacement) of the equal tempered system.
• 2. The twelve pitch classes are C (CBD_at__at_
  etc.), C, D, D, E, F, F, G, G, A, A, B)

4
Pitch-Class Notation

• Pitch class (abbreviated, pc) The twelve
  different pitches (independent of octave
  displacement) of the equal tempered system.
• 2. The twelve pitch classes are C (CBD_at__at_
  etc.), C, D, D, E, F, F, G, G, A, A, B)
• 3. Pitch classes may be expressed as integers
  0, 1 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. (0C,
  1C, 2D, . . . 9A, 10B_at_, 11B.)

5
Pitch-Class Notation

• Pitch class (abbreviated, pc) The twelve
  different pitches (independent of octave
  displacement) of the equal tempered system.
• 2. The twelve pitch classes are C (CBD_at__at_
  etc.), C, D, D, E, F, F, G, G, A, A, B)
• 3. Pitch classes may be expressed as integers
  0, 1 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. (0C,
  1C, 2D, . . . 9A, 10B_at_, 11B.)
• Each integer representing the distance from 0 (or
  C) in half steps.

6
Pitch-Class Notation

• Pitch class (abbreviated, pc) The twelve
  different pitches (independent of octave
  displacement) of the equal tempered system.
• 2. The twelve pitch classes are C (CBD_at__at_
  etc.), C, D, D, E, F, F, G, G, A, A, B)
• 3. Pitch classes may be expressed as integers
  0, 1 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. (0C,
  1C, 2D, . . . 9A, 10B_at_, 11B.)
• Each integer representing the distance from 0 (or
  C) in half steps.
• 5. Sometimes A is substituted for 10 and B
  substituted for 11. So the list of pcs
  0123456789AB. (Less frequently T is
  substituted for 10 and E for 11 0123456789TE)

7
Pitch-Class Notation

• Pitch class (abbreviated, pc) The twelve
  different pitches (independent of octave
  displacement) of the equal tempered system.
• 2. The twelve pitch classes are C (CBD_at__at_
  etc.), C, D, D, E, F, F, G, G, A, A, B)
• 3. Pitch classes may be expressed as integers
  0, 1 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. (0C,
  1C, 2D, . . . 9A, 10B_at_, 11B.)
• Each integer representing the distance from 0 (or
  C) in half steps.
• 5. Sometimes A is substituted for 10 and B
  substituted for 11. So the list of pcs
  0123456789AB. (Less frequently T is
  substituted for 10 and E for 11 0123456789TE)

8
Modulo Math

• The modulo operator takes the remainder of an
  integer divided by some other integer, the
  modulo.

9
Modulo Math

• The modulo operator takes the remainder of an
  integer divided by some other integer, the
  modulo.
• For example 14 modulo 12 2. (That is, 12
  goes into 14 once with a remainder of 2.

10
Modulo Math

• The modulo operator takes the remainder of an
  integer divided by some other integer, the
  modulo.
• For example 14 modulo 12 2. (That is, 12
  goes into 14 once with a remainder of 2.
• Hence, we say 14 is 2 mod 12. (That is, 14 is
  equivalent to 2 in a modulo 12 system.)

11
Modulo Math

• The modulo operator takes the remainder of an
  integer divided by some other integer, the
  modulo.
• For example 14 modulo 12 2. (That is, 12
  goes into 14 once with a remainder of 2.
• Hence, we say 14 is 2 mod 12. (That is, 14 is
  equivalent to 2 in a modulo 12 system.)

12
Clock Math

• When doing mod 12 arithmetic we can visualize the
  process easily by using the face of a clock.

13
Pitch Class Sets (PC Sets)

• A pc set is simply a list of pcs between
  brackets.
• C major triad 0,4,7
• F minor triad 5,8,0

14
Pitch Class Sets (PC Sets)

• A pc set is simply a list of pcs between
  brackets.
• C major triad 0,4,7
• F minor triad 5,8,0
• Note that only one representation of a pc is
  necessary in describing a pc set octave
  doublings and displacements are ignored
• 0,4,7,12,19 ? 0,4,7 (Since 120 mod 12 and
  197 mod 12, they are unnecessary in describing
  the pc set 0,4,7 is sufficient.)

15
Pitch Class Sets (PC Sets)

• A pc set is simply a list of pcs between
  brackets.
• C major triad 0,4,7
• F minor triad 5,8,0
• Note that only one representation of a pc is
  necessary in describing a pc set octave
  doublings and displacements are ignored
• 0,4,7,12,19 ? 0,4,7 (Since 120 mod 12 and
  197 mod 12, they are unnecessary in describing
  the pc set 0,4,7 is sufficient.)
• Also note that we always use the mod 12
  designation of the pc 5,8,12 ? 5,8,0

16
Pc sets, cont.

• Each one of the follow is merely one expression
  of a 0,5,6 pc set

0,5,6 0,5,6 0,5,6 0,5,6

• Sometimes the commas are left out. This allows
  for a very compact notation using A10 and B 11,
  or T10 and E11. Thus 9, 10, 11 can be
  expressed 9AB or 9TE. (Well use the former
  in this class.)

17
Transposing PC Sets

• To transpose a pc set simply add or subtract the
  same integer (number of half steps) from each
  member of the set.
• Example Transpose 158 by a minor third up
  (3) 13, 53, 83 4,8,11 48B

18
Transposing PC Sets

• To transpose a pc set simply add or subtract the
  same integer (number of half steps) from each
  member of the set.
• Example Transpose 158 by a minor third up
  (3) 13, 53, 83 4,8,11 48B
• Addition/subtraction must be by modulo 12 (mod
  12)
• Example Transpose 158 down by a perfect
  fifth (-7) 1-7, 5-7, 8-7 mod 12 6, 10, 11
  6AB

19
Transposing PC Sets

• To transpose a pc set simply add or subtract the
  same integer (number of half steps) from each
  member of the set.
• Example Transpose 158 by a minor third up
  (3) 13, 53, 83 4,8,11 48B
• Addition/subtraction must be by modulo 12 (mod
  12)
• Example Transpose 158 down by a perfect
  fifth (-7) 1-7, 5-7, 8-7 mod 12 6, 10, 11
  6AB
• Note that in mod 12 terms, transposition down by
  7 (-7) is the same as transposing up by 5 (5).
  Both yield the same pc set.

20
Transposing PC Sets

• To transpose a pc set simply add or subtract the
  same integer (number of half steps) from each
  member of the set.
• Example Transpose 158 by a minor third up
  (3) 13, 53, 83 4,8,11 48B
• Addition/subtraction must be by modulo 12 (mod
  12)
• Example Transpose 158 down by a perfect
  fifth (-7) 1-7, 5-7, 8-7 mod 12 6, 10, 11
  6AB
• Note that in mod 12 terms, transposition down by
  7 (-7) is the same as transposing up by 5 (5).
  Both yield the same pc set.

21
Normal Form

1. Arrange the pcs in ascending numerical order

22
Normal Form

1. Arrange the pcs in ascending numerical order
2. List all of the sets rotations.

23
Normal Form

1. Arrange the pcs in ascending numerical order
2. List all of the sets rotations.
3. Compute the ordered pc interval between the first
   and last pcs of each rotation. Compare these
   intervals. If one rotation has the smallest
   outside interval, it is the normal form.

24
Normal Form

1. Arrange the pcs in ascending numerical order
2. List all of the sets rotations.
3. Compute the ordered pc interval between the first
   and last pcs of each rotation. Compare these
   intervals. If one rotation has the smallest
   outside interval, it is the normal form.
4. If the smallest outside interval is shared by
   more than one rotation, compute and compare the
   interval between the first and next-to-last pcs
   for only those rotations.

25
Normal Form

1. Arrange the pcs in ascending numerical order
2. List all of the sets rotations.
3. Compute the ordered pc interval between the first
   and last pcs of each rotation. Compare these
   intervals. If one rotation has the smallest
   outside interval, it is the normal form.
4. If the smallest outside interval is shared by
   more than one rotation, compute and compare the
   interval between the first and next-to-last pcs
   for only those rotations.
5. If a tie still exists, compute and compare the
   interval between the first and next-to-the-next-to
   -the-last pcs of only those rotations that are
   still in contention.

26
Normal Form

1. Arrange the pcs in ascending numerical order
2. List all of the sets rotations.
3. Compute the ordered pc interval between the first
   and last pcs of each rotation. Compare these
   intervals. If one rotation has the smallest
   outside interval, it is the normal form.
4. If the smallest outside interval is shared by
   more than one rotation, compute and compare the
   interval between the first and next-to-last pcs
   for only those rotations.
5. If a tie still exists, compute and compare the
   interval between the first and next-to-the-next-to
   -the-last pcs of only those rotations that are
   still in contention.
6. Continue this process as long as necessary to
   break a tie. If you have computed all possible
   intervals and a tie still exists, select the
   rotation that begins with the lowest pc number.

27
Normal Form

1. Arrange the pcs in ascending numerical order
2. List all of the sets rotations.
3. Compute the ordered pc interval between the first
   and last pcs of each rotation. Compare these
   intervals. If one rotation has the smallest
   outside interval, it is the normal form.
4. If the smallest outside interval is shared by
   more than one rotation, compute and compare the
   interval between the first and next-to-last pcs
   for only those rotations.
5. If a tie still exists, compute and compare the
   interval between the first and next-to-the-next-to
   -the-last pcs of only those rotations that are
   still in contention.
6. Continue this process as long as necessary to
   break a tie. If you have computed all possible
   intervals and a tie still exists, select the
   rotation that begins with the lowest pc number.
7. Transpose the winner so that the first pc becomes
   0.

28
Normal Form

1. Arrange the pcs in ascending numerical order
2. List all of the sets rotations.
3. Compute the ordered pc interval between the first
   and last pcs of each rotation. Compare these
   intervals. If one rotation has the smallest
   outside interval, it is the normal form.
4. If the smallest outside interval is shared by
   more than one rotation, compute and compare the
   interval between the first and next-to-last pcs
   for only those rotations.
5. If a tie still exists, compute and compare the
   interval between the first and next-to-the-next-to
   -the-last pcs of only those rotations that are
   still in contention.
6. Continue this process as long as necessary to
   break a tie. If you have computed all possible
   intervals and a tie still exists, select the
   rotation that begins with the lowest pc number.
7. Transpose the winner so that the first pc becomes
   0.

29
Inverting PC Sets

• To invert a pc set, simply subtract each element
  from 0 mod 12. (Subtracting from 0 mod 12 is the
  same as subtracting from 12.)
• Example Invert 89B 0-8, 0-9, 0-11
  4,3,1

30
Inverting PC Sets

• To invert a pc set, simply subtract each element
  from 0 mod 12. (Subtracting from 0 mod 12 is the
  same as subtracting from 12.)
• Example Invert 89B 0-8, 0-9, 0-11
  4,3,1
• 2. This creates a mirror image (that is,
  bi-laterally symmetry) around 0. (See clock face
  to the right.)

31
Inverting PC Sets

• To invert a pc set, simply subtract each element
  from 0 mod 12. (Subtracting from 0 mod 12 is the
  same as subtracting from 12.)
• Example Invert 89B 0-8, 0-9, 0-11
  4,3,1
• 2. This creates a mirror image (that is,
  bi-laterally symmetry) around 0. (See clock face
  to the right.)
• A real inversion, however, should be based upon
  the first pc of the original set. Thus we need to
  transpose the result we obtained in 1., above, so
  that the inverted set begins on pc 8 44,
  34,148,7,5875.

32
Inverting PC Sets

• To invert a pc set, simply subtract each element
  from 0 mod 12. (Subtracting from 0 mod 12 is the
  same as subtracting from 12.)
• Example Invert 89B 0-8, 0-9, 0-11
  4,3,1
• 2. This creates a mirror image (that is,
  bi-laterally symmetry) around 0. (See clock face
  to the right.)
• A real inversion, however, should be based upon
  the first pc of the original set. Thus we need to
  transpose the result we obtained in 1., above, so
  that the inverted set begins on pc 8 44,
  34,148,7,5875.
• 4. So, the inversion of 89B is 875, a mirror
  image around the first pc of the original
  set--here pc 8. (See clock face to the right

33
Inverting PC Sets

• To invert a pc set, simply subtract each element
  from 0 mod 12. (Subtracting from 0 mod 12 is the
  same as subtracting from 12.)
• Example Invert 89B 0-8, 0-9, 0-11
  4,3,1
• 2. This creates a mirror image (that is,
  bi-laterally symmetry) around 0. (See clock face
  to the right.)
• A real inversion, however, should be based upon
  the first pc of the original set. Thus we need to
  transpose the result we obtained in 1., above, so
  that the inverted set begins on pc 8 44,
  34,148,7,5875.
• 4. So, the inversion of 89B is 875, a mirror
  image around the first pc of the original
  set--here pc 8. (See clock face to the right

34
Prime Form

• Determine the normal form of the pc set.

35
Prime Form

• Determine the normal form of the pc set.
• Determine its inversion.

36
Prime Form

• Determine the normal form of the pc set.
• Determine its inversion.
• Arrange the inversion in ascending numerical
  order.

37
Prime Form

• Determine the normal form of the pc set.
• Determine its inversion.
• Arrange the inversion in ascending numerical
  order.
• Transpose the inversion so that the first pc of
  the inversion is 0

38
Prime Form

• Determine the normal form of the pc set.
• Determine its inversion.
• Arrange the inversion in ascending numerical
  order.
• Transpose the inversion so that the first pc of
  the inversion is 0
• Compare the original and the inversion beginning
  with the last pcs and working backwards toward
  the first pcs. When you find that one set has a
  lower number in a give place than the other set,
  that set is the prime form of the set.

39
Prime Form

• Determine the normal form of the pc set.
• Determine its inversion.
• Arrange the inversion in ascending numerical
  order.
• Transpose the inversion so that the first pc of
  the inversion is 0
• Compare the original and the inversion beginning
  with the last pcs and working backwards toward
  the first pcs. When you find that one set has a
  lower number in a give place than the other set,
  that set is the prime form of the set.
• Obviously, if the two forms are identical no
  comparison is necessary.

40
Prime Form

• Determine the normal form of the pc set.
• Determine its inversion.
• Arrange the inversion in ascending numerical
  order.
• Transpose the inversion so that the first pc of
  the inversion is 0
• Compare the original and the inversion beginning
  with the last pcs and working backwards toward
  the first pcs. When you find that one set has a
  lower number in a give place than the other set,
  that set is the prime form of the set.
• Obviously, if the two forms are identical no
  comparison is necessary.

41
PITCH INTERVALS

• A Pitch Interval is the standard definition of
  an interval between any two pitches
• Ordered Intervals
• A3 to D5 is a Perfect 11th (ascending), also 17
  half steps
• D5 to A3 is a Perfect 11th (descending also -17
  half steps
• Unordered Intervals
• Is strictly a measure of distance.
• The distance between A3 and D5 is a perfect 11th
  (17 half steps)

42
Pitch-Class Interval Classes

• Ordered Pitch Class Intervals
• Can only be from 0 to 11. Since we are dealing
  with pcs, octave placement is irrelevant, so
  there is no need for negative numbers
• Unordered Pitch Class Intervals is the smallest
  distance between two pcs no matter what
  direction you are measuring. Informally referred
  to as Interval Classes or ics.
• There are 7 0, 1, 2, 3, 4, 5, and 6.

43
Interval Vector

• . . . is the measure of the interval content of a
  pc set.
• To create an interval vector for a set we list
  the number of instances of each ic (pitch-class
  interval class) in order of size.

44
Interval Vector

• . . . is the measure of the interval content of a
  pc set.
• To create an interval vector for a set we list
  the number of instances of each ic (pitch-class
  interval class) in order of size.
• For Instance, given the pc set (0,1,6), what is
  its interval vector?
• It contains a single instance of the ic 1 (01)
• It contains no instances of the ic 2.
• It contains no instances of the ic 3.
• It contains no instances of the ic 4
• It contains one instance of the ic 5 (16).
• It contains one instance of the ic 6 (06)
• The interval vector for (016), then, is lt100011gt.
  The following table illustrates how to read an
  interval vector

Interval Classes 1 2 3 4 5 6
1 0 0 0 1 1
45
Interval Vectors, cont.

• A set that contains N pcs, will contain
  (N(N-1))/2 ics.

No. PCs No. ICs
2 1
3 3
4 6
5 10
6 15
7 21
8 28
9 36
10 45
11 55