Set Theoretical Analysis   

 copyright © 1997 by Larry Solomon  

Frequently Asked Questions

Q1: What is set theory?
A1: Musical set theory is an application of mathematical set theory to pitch-class sets (groups of pitches) in music. It is (1) a identification and classification of all possible combinations of pitch-classes (pc) in the twelve-tone system, and (2) a system for determining relationships between various pitch sets. These sets are analogous to chords in traditional theory, but are really any group of notes. There are two different branches of set theory: linear (ordered) and nonlinear (unordered). Chords are nonlinear pc sets.

Q2: Why is there a need for it?
A2: In 20th century music there are a large number chords that do not have traditional names, for example, C F# B. Yet, this chord is used throughout Stravinsky's Rite of Spring. In set theory this chord is identified as 016. This provides a way of identifying and discussing the use of this chord and other unconventional chords.

Q3: Does set theory do anything beside label unusual chords?
A3: Yes. It provides ways to determine: (1) the interval (class) content of a set, (2) if one set is contained in another (the subset relation), (3) relationships between sets having the same number of pitch classes (similarity relations). It also embraces all traditional chords under the same theory.

Q4: What do the numbers in set theory refer to?
A4: Each pc (pitch class) in the twelve-tone system is assigned a number. C=0, C#=1, D=2, D#=3, E=4, ..... A#=10, B=11. [If we use hexadecimal numbers the last two (10,11) are represented by A and B, allowing single-digit primes.] Numbers are also used to represent intervals in semitones. A minor third, for example, is 3 semitones, or just 3. A perfect fifth is 7, etc.

Q5: How does set theory label chords?
A5: Chords are tagged by their "prime forms", which is a set of numbers each of which corresponds to a pc (pitch class) or interval measured from a reference that is set to zero. Thus, CEG is 047. The prime form is the most compact representation of a set (normal form) with the first pc set to zero by transposition. Thus, GBDF (7B25), whose normal form is B257, becomes 0368 by transposition. Additionally, Allen Forte has created a system of set nomenclature called set names. See Introduction to Set Theory for more information.

Q6: Why is the Forte Prime different from the real Prime?
A6: Follow this link. (Use your Go Back key to return here.)

Q7: What is an "interval vector"?
A7: It is a tabulation of all the interval classes contained within a particular pc set. A traditional example will serve to illustrate:

A major chord, such as CEG, contains a major third, a perfect fifth, and a minor third. Expressed in set notation this is 473 (semitones). However, although 7 is an interval, it is not an interval-class (ic). There are only six ics, 1-6 (semitones). The last (6) represents the tritone. Anything larger than a tritone is invertible to a smaller interval, which represents the ic. Thus, in our example, 473, 7 must be converted to ic5 by subtracting from 12. The result is 453, or 345 when placed in order. Any interval larger than 6 must go through the same process.

The interval vector (IV) is represented by an array of 6 digits, each of which represents the number of each ic from 1 to 6. An empty table is 000000, with each digit position representing the interval classes 1 to 6 in order. If the set contains any ics, numbers are placed in their proper positions to represent the ic content. So, the IV for a major chord (345) is 001110; i.e., it has 1 ic3, 1 ic4, and 1 ic5.