# Chapter 3 – The Prime Form

## 3.1Similar Pitch Class Sets:  Set Classes & Prime Forms

• Some pitch class sets are very similar, for example:  [0,1,4]  is very similar to  [3,4,7] (transposition),
[8,11,0] (inversion),  [5,8,9] (transposition and inversion), and  [8,9,0] (transposition).
• For example, try playing the following chords. Can you hear that they all have something in common?

• A group of similar PC Sets like these is called a “Pitch Class Set Class”, or more simply, a “Set Class”.
• If two PC Sets differ only by transposition or inversion, then they belong to the same Set Class.
• There are only 208 different Set Classes!
• Each Set Class is represented by a “Prime Form” PC Set. For example:
[0,1,4]; [3,4,7]; [0,3,4]; [5,8,9]; and [8,9,0]  all belong to the Prime Form:  (0,1,4)
• Note that parenthesis are used to denote Prime Forms in this tutorial. However, not everybody agrees on this syntax.

## 3.2Uses for The Prime Form

• The prime form is considered to be the “simplest” version of the pitch class set.
• Generally, the “simplest” version of a PC set means that the pitches in the set are packed as tightly together possible, and as far to the left as possible.
• Once you know the prime form of a PC set, you can look it up in a table of prime forms to get more information about the PC Set, such as its interval vector and fellow related PC Sets (see appendix).
• You can also use the prime form to search for other, related PC Sets using other software tools.
See http://www.ComposerTools.com .
• If you are a composer, you can use this information to help you better control, understand, and manipulate the harmonies in your music.

## 3.3Determining the Prime Form:  The Rigorous Method

• Goal:  To identify the prime form for any PC set.
• Example:  What is the prime form of [8,0,4,6]  ?
• Step 1:  Put the Pitch Classes in numerical order  =>  [0,4,6,8]
• Step 2:  List all of the rotations of the pitch class set. To rotate a PC Set, simply move the first number to the end and add 12 to it (i.e. shift it up an octave). For example, the Rotations of [0,4,6,8] are:

[0, 4, 6, 8]

[4, 6,  8, 12]

[6, 8, 12, 16]

[8, 12, 16, 18]

• Step 3:  Determine which rotation of the PC Set has the minimum distance between the first and last numbers in the Set:
• [0,  4,  6,  8]  =>  ( 8 – 0) =  8
• [4,  6,  8, 12]  =>  (12 – 4) =  8
• [6,  8, 12, 16]  =>  (16 – 6) = 10
• [8, 12, 16, 18]  =>  (18 – 8) = 10

There is a tie!  Versions [0,4,6,8] and [4,6,8,12] both have a minimum distance between first and last of 8

• Step 4:  If there is a tie, choose the rotation which has a minimum distance between the first and second numbers:

Distances between the first and second numbers:

• [0,  4,  6,  8]  =>  ( 4 – 0) =  4
• [4,  6,  8, 12]  =>  ( 6 – 4) =  2

So, in our example, [4,6,8,12] is preferred.

• Step 5:  If there is still a tie, then check the first and third numbers, and so on until the tie is resolved.
The PC Set at this point is in “Normal” form.
• Step 6:  Transpose the pitch class set so that the first number is zero:
[4 – 4, 6 – 4, 8 – 4, 12 – 4]  =>  [0, 2, 4, 8]
• Step 7:  Invert the pitch class set and reduce it using steps 1-5 above.
• Invert [0,2,4,8]  =>  [ 12-0, 12-2, 12-4, 12-8 ]  =>  [12, 10, 8, 4] => [0, 10, 8, 4]
• Put in numerical order:  [0, 4, 8, 10]
• Find the best rotation:
PC Set           (last-first)  (second-first)
[0,  4,  8, 10]       10              4
[4,  8, 10, 12]       8              4
[8, 10, 12, 16]       8              2    << Preferred
[10,12, 16, 20]       10              2
• Transpose down:  [8 – 8, 10 – 8, 12 – 8, 16 – 8]  =>  [0, 2, 4, 8]
• Step 8:  Which form, the original or the inverted, is most packed to the left (has the smallest numbers)?  That will be the Prime Form.
In our example, both forms produced the same Prime Form (this is because the original PC Set was “inversionally symmetric”), and so the Prime Form is (0, 2, 4, 8)

## 3.4Determining the Prime Form:  Easier Methods

• Option 1:  Use an online tool at http://www.composertools.com .
• Option 2:  Figure it out on the piano
• Step 1: Keep rotating your chord until it is as small as possible.
• Step 2:  If there are ties, then use the rotation that has the notes most packed towards the bottom.
• Step 3:  Check to see if the inversion is better packed.
• Option 3:  Use the “Simplified Set List” at the back of Post Tonal Theory by Joseph N. Straus.
• Option 4:  Use a MAX/MSP patch which displays the Prime form of a chord you play on your MIDI keyboard.
• Option 5:  Use the table of all prime forms. For example, 1) Find the interval vector first, then look it up in the table of all PC Sets (see below), or 2) skip steps 6 & 7 above and look up the inversion in the table.
• Option 6:  Visualize the Pitch Class Set on a clock face and locate the prime form visually
• Step 1:  The shortest distance traveled around the clock.
• Step 2:  Numbers packed as close to the starting point as possible.

For example, the prime form of [0,8,6,8] is (0,2,4,8); and the prime form of [2,4,8,9] is (01,15,7) :