What is Musical Set Theory?
After Brahms, tonality in Western music began to break down. Whereas before composers had relied upon a specific key area to organize the notes they wrote (e.g. a concerto in C-Sharp Minor), the idea of having such a tonal “home-base” had grown stale by the turn of the 20th century. Composers needed a new system to organize their pitches. In this article, we’ll introduce the system used by post-tonal composers known as Set Theory.
Arnold Schoenberg
Arnold Schoenberg spearheaded the move away from tonality and began writing atonal music around 1908. By 1923, he had fully developed a “12-tone” system of pitch organization, in which the composer arranges all twelve unique pitches into an ordered row and performs various manipulations on that row to generate pitch content for a composition. This system is usually referred to as ‘serialism.’
Set theory is not the same as serialism, but the two share many of the same methods and ideas. Set theory encompasses the notion of defining sets of pitches and organizing music around those sets and their various manipulations. Set class analysis refers to the efforts of music theorists to reveal the systems that composers like Schoenberg and his followers used to organize the pitch content of their works. Keep in mind that sets and set classes determined pitch content only; the composers remained free to fashion all other aspects of the music according to their artistic desires (at least until super-serialism, a philosophy of subjecting every aspect of the music to serial techniques, came into fashion in the 1950s).
Second Viennese School
In their day, Mozart, Haydn, and Beethoven were collectively referred to as “The Viennese School” of composers. Schoenberg’s ideas about music were so unorthodox and so radically changed the face of music history, that together with two of his students from Vienna, Alban Berg and Anton Webern, they are called “The Second Viennese School.”
What is a Pitch Class Set?
A Pitch Class Set is simply an unordered collection of pitches. The 12 unique pitches on the keyboard, or pitch classes, are numbered from 0 to 11, starting with ‘C’. For example, the pitch class set consisting of the notes C, E, and G would be written as (0,4,7). Composers treat sets with varying amounts of freedom when applying the set-class method to their atonal music.
Set class (0,1,6) was so popular with Schoenberg and his disciples that it has been nicknamed “The Viennese Trichord.”
What does it mean to invert a set?
A melody is inverted by swapping the direction of its intervals. If the original goes up a minor third, the inversion goes down a minor third. In set theory, any note can be inverted by subtracting its value from 12. (The inversion of 1 is 11, the inversion of 2 is 10, etc. 0 and 6 invert onto themselves.)
If you map a set onto a clock face, the inversion of that set is its mirror image on the clock. The axis of inversion lies on the line between the 0 and the 6 on the clock face, so when you invert a set it looks like it was flipped horizontally.
What is Normal Form?
Pitch sets can be put into Normal Form, which is an ordering of the pitches in the set which is deemed the most “compact.” Compact ordering means that the largest of the intervals between any two consecutive pitches is between the first and last pitch listed. If you look at a pitch set graphed on a clock face, the normal form will be the clockwise spelling of the set that traverses the smallest distance on the circumference of the circle.
The set (2,9,10), for example, is not in normal form because the interval between 2 and 9 (7) is larger than the intervals between 9 and 10 (1) or between 10 and 2 (4). To put the set (2,9,10) into normal form, you would spell it (9,10,2). That way the largest interval is “on the outside.”
If there is no single interval that is larger than all the others, then the normal form is the representation of the set that is “packed most tightly to the left,” that is, the representation where smaller intervals are closer to the beginning of the set and larger intervals are nearer to the end.
For example, (0,2,3,7) is packed more tightly to the left than (0,4,5,7) because the largest interval on the inside of (0,2,3,7) is between the 3 and the 7 (or “to the right”), whereas the largest interval on the inside of (0,4,5,7) is between the 0 and the 4, closer to the left. Both of these sets are in normal form, but the first is “packed more tightly to the left.”
What is Prime Form?
If you obtain the normal form of a set and the normal form of its inversion, then its prime form would be the more tightly packed of the two normal forms, transposed to begin on zero.
For example, consider the set (7,8,2,5), which we’ll call set A. Here is how we would calculate its prime form:
- The normal form of A is (2,5,7,8).
- The inversion of A is (5,4,10,7).
- The normal form of A inverted is (4,5,7,10).
- Since (4,5,7,10) is packed more tightly to the left than (2,5,7,8), we transpose (4,5,7,10) to begin on zero and get (0,1,3,6) as the prime form.
Why is this useful?
Prime form is an abstraction of set classes that gives a unique “picture” of that particular collection of notes. If two sets have the same prime form, we can be assured that they will sound similar to one another. Sets with the same prime form contain the same number of pitches and the same collection of intervals between its pitches, hence they are in some sense aurally “equivalent,” in much the same way that all major chords are aurally equivalent in tonal music.
Prime form representations are also referred to as “Set classes.” Sets whose prime forms are identical are said to belong to the same set class. For example, the pitch class sets (1,2,7), (8,2,3), and (0,11,6) all belong to set class (0,1,6).
What is a Forte Number?
Allen Forte (1926-2014) was a musicologist and theory professor at Yale University. He catalogued every possible prime form for sets with 3-9 members and ordered them according to their interval content. He then assigned each of these prime forms a name, like “5-35.” The first number is an index of how many pitches are in the set, the second number was assigned by Dr. Forte.
The complement of a set consists of all notes not in the set. Complement sets share the same catalog number in Forte’s classification system (e.g., the complement of 5-35 is 7-35).
Here are a few popular Forte numbers:
| Prime Form | Forte Number | |
|---|---|---|
| Viennese trichord | (0,1,6) | 3-5 |
| Major and minor triads | (0,3,7) | 3-11 |
| Major and minor scales | (0,1,3,5,6,8,10) | 7-35 |
| The octatonic scale | (0,1,3,4,6,7,9,10) | 8-28 |
What is an Interval Class Vector?
Intervals that are inverted onto one another are in the same “interval class.” (Intervals 1 and 11 are in interval class 1; 2 and 10 are in interval class 2; 3 and 9 are in interval class 3, and so on.) There are 6 unique interval classes, ranging from 1 to 6. Note that intervals are not the same as pitches! For example, the interval between pitches 2 and 9 is 7, which belongs to interval class 5.
Intervals that are inverted onto one another are in the same “interval class.” (Intervals 1 and 11 are in interval class 1; 2 and 10 are in interval class 2; 3 and 9 are in interval class 3, and so on.) There are 6 unique interval classes, ranging from 1 to 6. Note that intervals are not the same as pitches! For example, the interval between pitches 2 and 9 is 7, which belongs to interval class 5.
For example, consider the set (2,3,9). There is one occurrence of interval class 1 (between the 2 and the 3), one occurrence of interval class 6 (between the 3 and the 9) and one occurrence of interval class 5 (between the 2 and the 9). Therefore the interval class vector for set (2,3,9) is <1,0,0,0,1,1>.
Why is this useful?
The interval class vector provides at a glance the interval content of a set, and hence gives a reliable indication of its sound.
What do T(n) and T(n)I mean?
To generate a normal matrix for any set, write the pitch numbers of the set across the top of a page, and then write the same sequence of numbers down the side. For each cell in the matrix, add its corresponding pitches on each axis and adjust the result to mod-12. Here’s a simple example, the normal matrix for set class (2,3,9):
| 2 | 3 | 9 | |
| 2 | 4 | 5 | 11 |
| 3 | 5 | 6 | 0 |
| 9 | 11 | 0 | 6 |
You can then use this matrix to determine whether or not the set “inverts onto itself,” and if so, where. By “inverting onto itself,” I refer to the property inherent in some sets that for some transposition n, T(n)I returns the exact same pitches as the original set.
For a set class with x number of pitches, if any number n appears x times in the body of that set’s matrix, then T(n)I will contain the same notes as the original set. Take as an example the set (0,1,2,5,9):
| 0 | 1 | 2 | 5 | 9 | |
| 0 | 0 | 1 | 2 | 5 | 9 |
| 1 | 1 | 2 | 3 | 6 | 10 |
| 2 | 2 | 3 | 4 | 7 | 11 |
| 5 | 5 | 6 | 7 | 10 | 2 |
| 9 | 9 | 10 | 11 | 2 | 6 |
Since (0,1,2,5,9) has 5 members, we look for any number that appears 5 times in the body of the matrix. In this case, there is only one such number: 2. that means that T(2)I should consist of the same pitches as the original. And it does: T(2)I of (0,1,2,5,9) is (2,1,0,9,5). Composers and theorists refer to this property as “combinatoriality.”
Post-Tonal Theory Calculator
To get help defining and manipulating pitch class sets, and to determine their normal form, prime form, Forte number and interval class vectors, you can use the Post-Tonal Theory Calculator for iPhone. The app also includes tools for generating T(n) and T(n)I matrices and lets you hear the sets as they would sound played on a piano or violin:
