20th Century Analytical Techniques

All graduate music students are expected to demonstrate a competency with the basics of Set Theory and 12-tone theory. We use the text Introduction to Post-tonal Music » by Josef Straus.

Set Theory - Sample Questions

Normal Order

Give the set of pitch classes in the example below in Normal Order.

normal form exercise

Solution

The Normal Order of this PC set is (10 1 2 3)


Give the set of pitch classes in the example below in Normal Order.

normal form exercise

Solution

The Normal Order of this PC set is (8 10 11 0 2)


Prime Form

Give the Prime Form of the set of pitch classes in the example below.

prime form exercise

Solution

The Prime Form for this PC set is (0 1 5 7)


Give the Prime Form of the set of pitch classes in the example below.

prime form exercise

Solution

The Prime Form for this PC set is (0 1 3 4 6 8)


Interval Vectors

Give the interval vector for the set of pitch classes in the example below.

interval vector exercise

Solution

The Interval Vector for this PC set is (2 0 1 2 1 0)


Give the interval vector for the set of pitch classes in the example below.

interval vector exercise

Solution

The Interval Vector for this PC set is (2 1 2 2 2 1)


Symmetry in Unordered Sets

Which of the following sets is transpositionally symmetrical?

  • A. (2 4 5 8)
  • B. (6 9 0 2)
  • C. ( 3 4 9 10)
  • D. (1 3 6 8 11)

Solution

Only set C. (3 4 9 10) is transpositionally symmetrical.


Which of the following sets are inversionally symmetrical but not transpositionally symmetrical?

  • A. (2 4 5 8)
  • B. (6 9 0 2)
  • C. ( 3 4 9 10)
  • D. (1 3 6 8 11)

Solution

Set D. (1 3 6 8 11) is inversionally symmetrical but NOT transpositionally symmetrical. NOTE: Set C. (3 4 9 10) is both inversionally and transpositionally symmetrical.


12-tone Theory - Sample Questions

Use the 12-tone Matrix below to answer the rest of the questions.

12-tone matrix

Basic Row Forms

Name the row that starts with the pitch classes 4 3 11...

Solution

The row beginning with the pitch classes 4 3 11... is P4 (the prime row starting on 4)


Name the row that starts with the pitch classes 8 1 0...

Solution

The row beginning with the pitch classes 8 1 0... is RI5 (the retrograde inversion row ending on 5)


Name the row form that starts with the pitch classes 4 5 9...

Solution

The row beginning with the pitch classes 4 5 9... is I4 (the inversion row starting on 4)


Name the row form that starts with the pitch classes 4 11 0...

Solution

The row beginning with the pitch classes 4 11 0... is R7 (retrograde row ending on 7)


Which row other than R2 can be used combinatorially with the row P2 (2 1 9 10 5 3 4 0 8 7 6 11)?

Solution

The row I7 (7 8 0 4 11 6 5 9 1 2 3 10)


Find a row in the matrix that has two invariant trichords with the row R1 (10 5 6 7 11 3 2 4 9 8 0 1).

One Solution

The row R1 (10 5 6 7 11 3 2 4 9 8 0 1) shares two invariant trichords with three different rows...I6 [(6 7 11) and (8 0 1)], I10 [(11 3 2) and (9 8 0)], and RI1 [(4 9 8) and (10 5 6)


In the example below determine the two row forms used (use the matrix above to help you out). Then do a '12-count' to show how the rows are used in the music.

quartet excerpt

Solution

quartet excerpt key


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