All About Set Theory

This document is intended as a primer for those who are new to musical set theory and as a companion to my SetTheory java applet. Instructions specific to my Java applet are written in green text.

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.

Arnold Schoenberg Composers needed a new system to organize their pitches. 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).

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 uniqye 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."

When the applet starts, set class (0,1,6) is provided as a default starting set. To enter a new set, you can either:

  1. Click on the "Define Set..." button, select a set from lists of presets, and click OK

    or

  2. Simply type over the set in the "Pitch Class Set:" field and press the Enter key. (Macintosh users press the Return key.) When typing in a set, be sure to type numbers separated by commas. Any number over 11 will automatically be reduced to its mod-12 equivalent, and white space will be ignored.

Your sets will be graphed on the clockface and on the keyboard automatically. To hear your set, click on the "Play" button that appears below the keyboard after the sounds have loaded over the network. Playing the set as a melody allows you to hear the notes one at a time in the order that they are listed in your set class. Playing them as a simultaneity allows you to hear all of the notes at once, like a chord.

For the sake of melodic analysis, the ordering of pitches in the set is maintained in the applet. You can click the "Rotate" button to shift the notes into the ordering you desire.

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 clockface, 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 clockface, so when you invert a set it looks like it was flipped horizontally.

To invert the set in the applet, click the "Invert" button on the right-hand side of the screen. Notice that the clock-face graph of the inverted set is a mirror image of the original.

You can also obtain the complement of any set by clicking the "Get Complement" button. This returns a new set consisting of any note that was not in the original set.

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."

The applet automatically calculates the Normal Form of each set you enter and displays it in the "Normal Form:" field.

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:

  1. The normal form of A is (2,5,7,8).
  2. The inversion of A is (5,4,10,7).
  3. The normal form of A inverted is (4,5,7,10).
  4. 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).

The applet automatically calculates the Prime Form of each set you enter and displays it in the "Prime Form:" field.

What is a Forte Number?

Allen Forte, perhaps the most important music theorist of our time, catalogued every possible prime form for sets with 3-9 members and ordered them according to their interval content. He gave 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 is a brief list of just 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

Allen Forte is on the music faculty at Yale University. He introduced this system of numbering the prime forms in his 1977 book titled The Structure of Atonal Music.

The applet automatically calculates the Forte Number of each set you enter and displays it in the "Forte Number:" field. To see a full list of all Forte numbers, click the "Define Set..." button and choose "Forte Numbers" as your criteria for obtaining a preset.

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.

The interval class vector is a 6-member tally of the number of occurences of each interval class found in a set. To obtain the tally, you find the interval between every possible pairing of notes in a set and increment the tally of that interval class.

For example, consider the set (2,3,9). There is one occurence of interval class 1 (between the 2 and the 3), one occurence of interval class 6 (between the 3 and the 9) and one occurence 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.

The applet automatically calculates the Interval Class Vector of each set you enter and displays it in the "Interval Class Vector:" field.

What does T(n) and T(n)I mean?

T(n) refers to another set class whose pitches are all transposed up by n semitones from the original. For example, if your original set class is (1,2,7), then T(3) would be (4,5,10).

T(n)I just means that first you invert your original set, and then perform the transposition. So to get T(3)I of (1,2,7) you would first invert (1,2,7) to get (11,10,5), and then transpose (11,10,5) up by 3 to get (2,1,8).

The applet automatically calculates T(n) and T(n)I of each set you enter and displays it in their appropriate fields. To change the value of n, use the scrollbar to the left of the T(n) and T(n)I fields.

To transpose the set itself, use the "<" and ">" buttons below the clock-face graph.

How are Matrices useful?

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."

To generate the matrix for your set, click on the "Matrices..." button on the right-hand side of the screen. Radio buttons will appear that allow you to set the Y-axis to be the same as the X-axis (Normal) or the inversion of the X-axis (Inverted).

Hint: If a set is combinatorial, you should be able to click the "Rotate" button while viewing its matrix until some number appears all the way down the Northeast-Southwest diagonal of the matrix. To borrow from the example above, we could have clicked the "Rotate" 4 times to reveal this matrix with 2's along the diagonal:

9 0 1 2 5
9 6 9 10 11 2
0 9 0 1 2 5
1 10 1 2 3 6
2 11 2 3 4 7
5 2 5 6 7 10