ComposerTools.com / Theory / Pitch Class Sets / 5. The Table of All Prime Forms - Description

5 The Table of All Prime Forms - Description

Please refer to the Appendix for a two-page table of all possible prime forms of Pitch Class Sets. This table is an indispensable aid for composers, since it is, essentially, a table of all possible types of chords. Not only does it contain all of the standard chords from tonal harmony such as triads (major, minor, diminished, and augmented) and seventh chords (dominant, major-minor sevenths, major-major sevenths, minor sevenths, etc.), but it also contains all chord types used by modern composers as well. Any chord which can be constructed using a 12-tone equal tempered scale is represented in the table.

5.1 The Columns of Data in the Table

For each prime form in the table, there are five columns of data:

Column 1: The interval vector
Column 2: The count of PC Sets which reduce to the prime form
Column 3: The Forte code (see below)
Column 4: The Prime Form PC Set
Column 5: The inverted form (if different than the Prime Form)

5.2 The Layout of the PC Sets in the Table

The PC Sets are grouped in the table by size, into 13 sections (from 0 pitches to 12 pitches per PC Set).
Within each group the list is sorted by interval vector. Interval vectors with the most half-step intervals are listed first, then vectors with the most whole-step intervals, and so on.
Z-related forms are listed together, one after the other (see section 5.4)
Commonly known pitch class sets (e.g. well-known chord qualities, types of scales, etc.) are labeled with {curly braces}. For example, (0, 3, 7) is labeled as {min} because it is a minor triad.
With the exception of the sets of 6 Pitch Classes, each set is listed opposite of its "complement". For example, set 4-16, (0,1,5,7) is listed to the left of set 8-16, (0,1,2,3,5,7,8,9). A set and its complement share many similar properties (see below for a discussion of Pitch Class Set complements).
To conserve space, the table uses the letters A, B, and C for the numbers 10, 11, and 12.

5.3 Forte Names

Allen Forte's book, The Structure of Atonal Music, published the first version of this table. In his table, he labeled each prime form of the PC Set with a unique designation, such as 5-20.
The first number (5-) specifies the number of pitches in the pitch class set.
The second number (20) is a unique number given to the prime form, which was sequentially assigned by Dr. Forte when he first created the table.
When analyzing PC Sets, many music theorists will label them using the Forte designation, although simply using the prime form (e.g. (0,1,3,7,8) or (01378) ) is becoming more common.

5.4 Z-Related Sets

When two prime forms produce the same interval vector, and when one can not be reduced to the other (by inversion or transposition), they are said to be "Z-Related", or "Z Correspondents".
The Forte Code for all PC Sets which are Z related contains a 'Z' in the PC Set ID. For example, 6-Z25.
'Z' doesn't stand for anything, it is just an identifier chosen by Dr. Forte when the table was first created.
Z-related sets are "close cousins" to one another. They sound similar to each other, but not as similar as sets related by (say) transposition or inversion. For example, try playing the following PC Sets on the piano. Listen for the intervals they contain. Since the Z-related sets contain the same intervals, do they not sound at least somewhat similar?

5.5 Other Comments on the Table

When I first encountered the table, I was surprised that it contained so few interval vectors (200), prime forms (208) and chord qualities (351). For some reason, in my mind, I had always thought that the complete list of possible chord types was much much larger.
Along the same lines, the number of chord types used by in common practice music is quite small, as few as a dozen different types chords, perhaps as many as 20 if you include Jazz chords.

This implies that there is a very number of chords yet to be thoroughly explored!

The following music shows some very famous chords. With out PC Sets, how could the types of these chords be specified?