ComposerTools.com
/ Theory / Pitch Class Sets / 3. The Prime
Form
3
The Prime Form
3.1 Similar
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.
3.2 Uses
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.
3.3 Determining
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)
3.4 Determining
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. See the URL: http://www.euph0r1a.net/projects/?handler=etrof
.
- 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) :
Copyright © 2004 by Paul Nelson, all rights reserved.