Mus 420
PITCH-CLASS SETS
Introduction
In the first decade of the twentieth century, a few composers developed an approach
to composition that, in retrospect, was perhaps inevitable. The chromaticism of the
nineteenth century had chipped away at the tonal system so successfully that it was
only a natural outcome for the system to be eventually abandoned altogether. This
new music without a tonal center, and no apparent method of pitch organization (such
as the serial and twelve-tone methods), eventually became known as "atonal"
music.
Atonality is one of the more important aspects of twentieth century music,
and it is a major factor that distinguishes much of this century from any other music
in teh Western tradition.
Characteristics of atonal music
The following are some of the general features shared by most compositions
that are written in an atonal style.
- Absence of a clear tonal "center"
- Avoidance of conventional melodic, harmonic, and
rhythmic patterns
- Use of unresolved dissonances
- Preponderance of mixed-interval chords
- Pitch material derived freely from the chromatic
scale
- Preponderance of contrapuntal textures
- Complex, often ambiguous metrical organization
Pitch-class sets
The pitch aspect of atonal music requires a
new vocabulary if the analysis of this music is ever to be more descriptive. It is
recognized that atonal music often achieves a certain degree of unity through the
recurrent use of a new kind of motive. This new kind of motive has been given
various names, including cell, basic cell, set, pitch set, and pitch-class
set. It can appear melodically, harmonically, or as a combination of both.
The set can also be transposed and/or inverted and its pitches may appear in any
order. Most atonal pieces employ a large number of different kinds of pitch sets,
but only a few of them seem important in unifying a piece. The analysis of atonal
music music has largely become a process ofs identifying these important pitch
sets. This process is called segmentation .
Since an atonal chord or melodic fragment ("segment")
can consist of any combination of pitches, thousands of different sets are possible.
Fortunately, there is a helpful system of atonal analysis (developed primarily by
Allen Forte) that makes it possible to accurately label, distinguish, and reduce
the number of the different sets found in a piece.
Basic Terminology
Pitch
A single note within a certain frequency, e.g. A-440
Pitch class
A group of pitches with the same name, e.g. all A's
Octave equivalence
Pitches separated by one or more octaves are equivalent, e.g. unison=octave, major
3=major 10, etc.
Inversional equivalence
Pitches on either side of the "inversion clock" are equivalent, e.g. major
3=minor 6
Enharmonic equivalence
Pitch classes that are spelled differently are still considered equally, e.g. b-flat=a-sharp
Integer Notation
Pitches
The numbers 0-10 are used to represent chromatic pitches C through B (always upwards)
The integers can be used as a "fixed do" system, in which 0 always represents
C, 1 always represents C-sharp or D-flat, etc., or as a "movable do" system,
in which 0 represents the first note of a collection. When C is the first note of
a collection, the two systems are synonymous.
Example:
| Note name |
C
|
E-flat
|
F
|
A-flat
|
B
|
| Integer name |
0
|
3
|
5
|
8
|
11
|
Intervals
Intervals will be identified by the numbers of half-steps they contain. This identifies
enharmonically equivalent intervals by a single nomenclature, and follows from the
logic of enharmonic equivalence for a single pitch (see above)
Example:
Tonal
Labels |
P1
|
m2
|
M2
|
m3
|
M3
|
P4
|
A4
|
P5
|
m6
|
M6
|
m7
|
M7
|
P8
|
Post-tonal
Labels |
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
0
|
Modulus 12
Generally, only the numbers 0-11 are used in integer notation. Modulus 12
("mod12") is used for this purpose.
An interval encopassing 14 minor seconds (i.e, a major ninth) will be referred to
as 2, not 14.
To determine the mod 12 integer for intervals larger than 11, find the total number
of half-steps, then subtract 12 from that number.
Using a clockface can be a useful shortcut:
1400 hours, fro example, can be easily seen to be 2 o'clock; 1800 hours, 6 o'clock,
etc.
For intervals of two octaves and larger, keep subtracting 12 until the difference
is less than 12.
Interval class
Interval classes are used in situations where intervals are inversionally equivalent.
There are seven interval classes, numbered 0 through 6. Each class contains an interval
of a given size an its inversion.
|
Class
|
Integer notation
|
Tonal notation
|
| 0 |
0 (12) |
P1, d2, P8, A7 |
| 1 |
1, 11 |
m2, A1, M7, d8 |
| 2 |
2, 10 |
M2, d3, m7, A6 |
| 3 |
3, 9 |
m3, A2, M6, d7 |
| 4 |
4, 8 |
M3, d4, m6, A5 |
| 5 |
5, 7 |
P5, P4, A3, d6 |
| 6 |
6 |
A4, d5 |
Pitch-class (PC) sets
Definition
A group of two or more unique pitch classes, e.g., [C, E, G-sharp, A, B-flat]
Location in the music
PC sets may be present in the music horizontally and/or vertically. Note that unlike
the twelve-tone technique, music written using PC sets does not require a specific
ordering of pitches.
Notation of PC sets
PC sets are written within parentheses; commas separate the pitch classes.
Normal form
Normal form is the PC set in its most compressed from.
To find a set's normal form:
- Eliminate duplicates and write in ascending form
within an octave
- Using rotation, write all forms of the group
- Choose the form with the smallest outside interval
- In case of a "tie", choose the version
that is the most compact toward the bottom.
When it is necessary to show that a PC set is in
normal form, or if we wish to distinguish a set in normal form from a set that is
not, we write the set in brackets.
Example
Normal form of C, E, G-sharp, A, B-flat is G-sharp, A, B-flat, C, E or [8,9,10,0,4]
Transposition of PC sets
In addition to transposing in the usual manner, PC sets can be transposed by adding
or subtracting the transposition interval number to each element of the set. Mod
12 is used at all times.
Example
Transpose [8,9,10,0,4] up a M6 [9]
or
Show T9 of [8,9,10,0,4]
or
Map [8,9,10,0,4] through T9 |
|
|
8
|
9
|
10
|
0
|
4
|
|
+
|
9
|
9
|
9
|
9
|
9
|
|
_____________________
|
|
5
|
6
|
7
|
9
|
1
|
|
(mod 12!!)
|
|
Inversion of PC sets
In tonal music, inversion can be applied to
a chord, an interval, or a melodic line, but inverting a single pitch is meaningless.
In post-tonal music, any single pitch can be inverted, using pitch 0 as the mirror
point.
Examples
If C=0, then A [3 below 0] inverts to E-flat [3
above 0]
If b-flat = 0, then D (4 above 0) inverts to G-flat [4 below 0]
One alternate way to invert PC sets is to subtract 12 from each PC number
Example
If C=0, invert PCs F and A-flat
12 - 5 [F] = 7 [G]
12 - 8 [A flat] = 4 [E]
If G=0, invert PCs F and A-flat
12 - 10 [F] = 2 [A]
12 - 1 [A-flat] = 11 [F-sharp]
Prime form (or best normal order)
In order to have a single classification for any PC set and its inversion, we need
to carry the concept of normal order a step further, to something called prime
form or best normal order.
This concept is important because the best normal order is the generic representation
of all the possible transpositions and inversions of a set.
In order to find the best normal order of a set, do the following:
- Put the set into normal form
- Transpose it to start on 0
- Invert the normal form, and check to be
sure the result is also in normal form
- Transpose it to start on 0
- Choose the most compact ("best")
identification.
The prime form is then written in parenthesis with
no commas or spaces, ten is expressed as "T" and eleven as "E"
Example
|
Given the set [9,11,5,6]
|
|
1
|
Normal form |
[5,6,9,11]
|
|
2
|
Transposition to start on 0 |
[0,1,4,6]
|
|
3
|
Inversion of normal form |
[6,8,11,0]
|
|
4
|
Transposition of inversion to start on 0 |
[0,2,5,6]
|
|
5
|
Most compact (prime) form |
[0146]
|
Set types and prime forms
By adopting the concepts of transpositional and inversional equivalence, the thousands
of possible pitch combinations have been reduced to a manageable number of prime
forms (or set types):
|
Combinations of pitch classes (PCs)
|
Possible set types (prime forms)
|
| 2 PCs (Dyads) |
6
|
| 3 PCs (Trichords) |
12
|
| 4 PCsn (Tetrachords) |
29
|
| 5 PCs (Pentachords) |
38
|
| 6 PCs (Hexachords) |
50
|
| 7 PCs (Septachords) |
38
|
| 8 PCs (Octachords) |
29
|
| 9 PCs (Nonachords) |
12
|
| 10 PCs (Decachords) |
6
|
| TOTAL |
220
|
Hints for finding prime forms
With a little practice, the task of analyzing a set becomes much less time-consuming.
Also, working at the piano not only makes finding the best normal order easier, but
it also helps you associate the sounds with the set types.There are also a few shortcuts
that may make the process easier. Following are some examples of possible shortcuts:
- It is usually not necessary to notate the inversion
and go through the process of finding the best normal order. For instance, the set
[E, F-sharp, G-sharp, B] is in normal order, but not necesarily in best normal order.
The normal order of the inversion would simply reverse the order of the intervals
(M2-M2-m3 becomes m3-M2-M2). Without bothering to notate the inversion, we can see
that the set is already in best normal order because its first interval (M2) is smaller
than the first interval of its inversion (m3). The best normal order is, thus, [0247].
- The set [F-sharp, A, B-flat, C], also in normal
order, has the interval succession m3-m2-M2, while the normal order of its inversion
will have M2-m2-m3. Since M2 is smaller than m3, the inversion will be the best normal
order. However it is not still not necessary to write out the inversion; instead,
designate the top note as 0, and number the others in terms of half-steps
below the top note (6,3,2,0), reverse the order of the numbers, and the best
normal order becomes [0236].
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