# 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. 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. 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. 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. 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. 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. 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)

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)

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. ### Basic Row Forms

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

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...

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...

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...

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)?

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).

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.  /div>