Lesson 43 - Pitch Class Set Theory
Unfortunately, there is no quick way to explain this, so A WALL OF TEXT! The good news is, we're getting to the end of the theory lessons!
Part I - Introduction
Pitch class set theory is a way of analyzing atonal music using the principles of mathematical set theory.
Analyzing tonal music in terms of chordal structure makes sense, but analyzing atonal music in terms of chordal structure doesn't make sense. Pitch class set theory provides a means of studying similarities and differences of pitch content as well as organizational aspects of atonal music within groups of notes called "sets."
This theoretical system was developed by Allen Forte in his book, The Structure of Atonal Music(New Haven:Yale University Press, 1973) Pitch class sets are collections of three or more pitches.
This theoretical system was developed by Allen Forte in his book, The Structure of Atonal Music(New Haven:Yale University Press, 1973) Pitch class sets are collections of three or more pitches.
There are a number of founding premises that must be understood in order to begin working with Pitch class set theory. They are:
1. Octave equivalence. No distinction is made to represent octave/register.
2. Enharmonic equivalence. All spellings of a given pitch are considered equal.
3. Integer notation. pitch classes are represented by numbers instead of letters.
4. Sets are not represented on a staff but instead grouped within brackets and separated with comas as such: [0,1,2]
5. Elements are always listed in ascending numerical order regardless of the order in which they appear as notated.
These five premises are crucial to the beginning of the study of set theory. Pitch class is conveyed in the following system: C=0, C#(or any of it's enharmonic equivalents)=1, D=2, and so on.
1. Octave equivalence. No distinction is made to represent octave/register.
2. Enharmonic equivalence. All spellings of a given pitch are considered equal.
3. Integer notation. pitch classes are represented by numbers instead of letters.
4. Sets are not represented on a staff but instead grouped within brackets and separated with comas as such: [0,1,2]
5. Elements are always listed in ascending numerical order regardless of the order in which they appear as notated.
These five premises are crucial to the beginning of the study of set theory. Pitch class is conveyed in the following system: C=0, C#(or any of it's enharmonic equivalents)=1, D=2, and so on.
Part II - Naming sets
In order to work with sets, it is necessary to have a naming system to distinguish unique sets. Naming sets was developed by Allen Forte and sometimes referred to as "Forte Numbers." A set's name consists of two elements. The first number is the number of elements in a set. The second number describes the ordering of the set.
Here is a chart of some sets as named by Forte:
Part III - Inversion of sets
To start the process of inverting a set, we must understand the principle of inversional equivalence. Just as pitch classes can be represented with integers, intervals can be represented in the same way:
prime (unison) = 0
minor 2nd = 1
major 2nd = 2
minor 3rd = 3
major 3rd = 4
Perfect 4th = 5
tritone = 6
perfect 5th = 7
minor 6th = 8
major 6th = 9
minor 7th = 10
major 7th = 11
octave = 12
Of course, interval inversion is also acceptable.
0 = 12
1 = 11
2 = 10
3 = 9
4 = 8
5 = 7
6 = 6
When a set has been placed in normal order and transposed to start with 0 but still does not appear in the list of Forte Numbers, it is because the set has not been packed so that the smallest intervals appear on the left side of the set and thus must be inverted.
Of course, interval inversion is also acceptable.
0 = 12
1 = 11
2 = 10
3 = 9
4 = 8
5 = 7
6 = 6
When a set has been placed in normal order and transposed to start with 0 but still does not appear in the list of Forte Numbers, it is because the set has not been packed so that the smallest intervals appear on the left side of the set and thus must be inverted.
Part IV - Interval Vectors
Apart from identifying the names of the set, many other types of relationships can be demonstrated through the use of pitch class set theory. A representation of all of the intervals contained within a set is called its interval vector.
The Cardinal 5 set, [0,1,4,5,8] will be used to demonstrate interval vectors. Each interval class is determines through subtraction.
0 1 4 5 8
| ic1 |
| ic4 |
| ic5 |
| ic4 |
| ic3 |
| ic4 |
| ic5 |
| ic1 |
| ic4 |
| ic3 |
Using a six digit string of integers, each of which represents one of the interval classes, it is possible to map the entire interval content of a pc set. The first digit of the string will represent how many times that interval appears in a set. The second number will represent the interval class 2, and so on. Interval class 0 is disregarded.
The interval vector for set [0,1,4,5,8] contains two occurrences of ic 2, two occurrences of ic3, four of ic4, one of ic5, and none of ic6. The interval vector would be represented in this manner: 202420.
Pairs of unique Forte sets that have unique Forte names, but identical interval vectors are described as being Z-related pairs.
The interval vector for set [0,1,4,5,8] contains two occurrences of ic 2, two occurrences of ic3, four of ic4, one of ic5, and none of ic6. The interval vector would be represented in this manner: 202420.
Pairs of unique Forte sets that have unique Forte names, but identical interval vectors are described as being Z-related pairs.
