MUS 312 Form & Analysis
M. 12Tone Matrix
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Introduction
Arnold Schönberg composed the first 12tone piece in the summer
of 1921. He carried to a conclusion the developments in chromaticism
that had begun many decades earlier. The assault of chromaticism on
the tonal system had led to the nonsystem of free atonality,
and now Schönberg had developed a "methodhe insisted it was
not a systemof composing with twelve tones that are related
only with one another."
Many composers seem to have been convinced that atonality could
best be achieved through some sort of regular recycling of the 12
pitch classes, but it was Schönberg who came up with the idea of
arranging the 12 pitch classes into a particular series, or row,
that would remain essentially constant throughout a composition.
Another term for 12tone composition is serialism. The
term "dodecaphonic" has also been used as an adjective to
describe the 12tone method; however, the term serialism is
preferred.
12Tone Matrix Systems
There are two ways of approaching the
12tone matrix: (1) The traditional approach is to construct the
matrix using the letter names of the tone row notes. (2) A
second way is to substitute numbers for the letter names.
This latter approach is referred to as Mod12Integer Analysis,
or more simply, Mod12 (technically, Modulus12).
Both systems rely on the same principleintervallic
relationships. The Mod12 approach may initially involve more work
in that the notes of a composition must be converted to numbers.
However, if these numbers are transferred to a clean sheet of paper
where they may be viewed purely as numbers without the distractions
of accidentals, rhythms, etc., "row chasing" tends to be
easier.
Although the system you prefer is a matter of personal taste, it
is advantageous to be familiar with bothand they are quite
similar.
Basic Terminology
The core of the 12tone system is the tone row, an ordered
arrangement of the twelve pitch classes, with each pitch occurring
only once before being repeated ("classic" version).
The
row itself has four basic forms:
 Prime (or Original): the original set
 Retrograde: the original set in reverse order
 Inversion: the mirror inversion of the original set
 Retrograde Inversion: the inversion in reverse order.
When analyzing a serial (12tone) composition, we label the row
forms using the following abbreviations:
 P = Prime (or O = Original)
 R = Retrograde
 I = Inversion
 RI = Retrograde Inversion
After the abbreviation comes a number, from 011, which specifies
the transposition of the row in halfsteps. For example: P3, R4,
I9, RI6
Traditional Matrix Construction (using letters)
To construct a traditional matrix using note letter names, follow
these steps:
 Determine the original row and fill in P0 along the top row.
 Write the transposition numbers above the matrix by
numbering the pitches chromatically, beginning with 0 for the first
note and moving up by halfstep.
 For example, if E = 0, then F = 1, F# = 2, G = 3, etc. Write
these same numbers beneath the matrix.
 Fill in the transposition numbers along the outside of
the left border by subtracting each of the numbers on the top
border from 12, with the exception of the first number, which is
always 0.
 Find the horizontal row with a 1 in front of it, and transpose
all of P0 up one halfstep into that row.

Find the horizontal row with a 2 in front of it, and transpose
all of P1 up one halfstep into that row.
 Continue this process until P10 is transposed up one
halfstep into P11, and the matrix is complete.
Mod12 Matrix Construction
To construct a Mod12 matrix using numbers, follow these steps:
 Determine the original row, and convert the row to numbers.
The first note of the row is 0. For example, if E is the first note,
E = 0, F = 1, etc., as with the traditional matrix. In Mod12,
however, the numbers are placed inside the boxes of the top
row (P0). (Note: "0" is movable. If Ab is the
first note of the row, then Ab = 0, A = 1, etc.) To easily
determine the integer equivalent from any 0 note, use the
"clock" on the matrix sheet.
 In the upper triangle above each number, indicate by
number and a plus (+) sign how each number differs in halfsteps
from the number in the top left corner box (0). For example, if the
row is 0 1 3 9 2 11 4 10 7 8 5 6, in the upper triangle above
1, you would write +1; above 3 write +3, above 9 write +9, etc.,
because that's how each of the successive numbers differs from 0.
 Now, to determine the Inversion of the row (first vertical
column, I0), first you must fill in the upper triangle to
the left of each I0 box. Write the same numbers going down the
triangles that you wrote in going across earlier, except change all
the plus (+) signs to minuses(). For example, if I1 at the top is
+1, then the first triangle would be 1. If I3 is +3, then the top
half of the second triangle is 3, etc.
 Now, go back and fill in all the lower triangles. The
number in the lower triangle will have an opposite sign,
positive (+) or negative (), than that of the upper number.
To determine the lower number, simply subtract the number
from 12, temporarily ignoring any plus or minus signs. It may be
helpful to again think of the face of a clock in realizing this
positive/negative relationship. For example, +9 = 3, 4 = +8, etc.
 Now, fill in the numbers for I0 (first vertical
column) by transferring the positive numbers from the lower
triangles to the left of the matrix.
 You may now begin filling in the matrix with the help of the
numbers in the triangles across the top, in relation to
the numbers in I0 (farleft vertical column). Either add the
positive number from the upper triangle, or subtract the lower
number, whichever will result in a number of 11 or less.
 Write in above the top matrix squares the form of the
row for each column in order to be able to identify each form of the
row being used. Keeping our same row, the top would read:
I0 I1
I3 I9, etc., indicating Inversion. Place the same
numbers beneath the matrix, indicating Retrograde
Inversion, for example: RI0 RI1 RI3 RI9, etc. To the left
of the matrix, indicate Prime form, for example: P0, P11
P9, etc. Place these same numbers to the right of the
matrix to indicate Retrograde, for example: R0 R11 R9,
etc.
 Go through the piece and write in the corresponding numerical
equivalents of P0 beside each note of the piece. It is easier if
you do one number at a time. Be sure not to miss enharmonic
equivalents. For example, if F# = 2, Gb also = 2.
 Your analysis will be speedier if you then transfer all the
numbers, in proportional spacing to the rhythms, on a clean sheet of
paper.
Tip: If the matrix has been completed successfully, "0"
should appear in every square between the top left and bottom right
square, diagonally. (Likewise, if completing a traditional matrix,
the same letter name should appear diagonally in these squares.)
Basic Terminology Continued
As was mentioned earlier, the core of the 12tone system is the tone
row, sometimes referred to as the basic set or series,
which is an ordered arrangement of the 12 pitch classes in four
basic forms: Prime, Retrograde, Inversion and Retrograde
Inversion.
The notes may be written in any octave or with any enharmonic
spelling. Typically, 12tone music follows the convention of
omitting natural signs: any note without an accidental is
natural. Each note of the row is assigned a number (112) simply
to indicate each note's position in the row form. These are called order
numbers.
The numbers used in conjunction with the matrix, such as P3,
R9, etc., specify the transposition of the row form in
halfsteps. The very first row form in a composition is numbered
P0. Transpositions are always figured above the original,
regardless of the octave in which they occur.
A row does not always have to proceed strictly from the first
note to the last. And segments of various rows may appear
simultaneously.
Analyzing a Row
Since the row serves as the source of the pitch material of a
composition, one should analyze the row itself before beginning the
analysis of the piece.
STEP 1.
The first step should be to play the music several times. Listen
for sequences or familiar patterns. In general, composers avoid
using any combination of pitches that would recall tonal music, such
as triads, but there are exceptions. For example, look at P0 from
Berg's Lyric Suite.
Suggests F tonality 
Suggests B tonality 
<<< 
<<< 
<<< 






>>> 
>>> 
>>> 
F 
E 
C 
A 
G 
D 
G# 
C# 
D# 
F# 
A# 
B 

A
minor? 




D#
minor? 

Diatonic
to C or F
(hexachords) 
Diatonic
to F# or B
(hexachords) 
STEP 2.
The next step in the analysis may be to label the interval
classes (ICs):
Interval Class: 
Traditional Interval: 
1 
m2, M7 
2 
M2, m7 
3 
m3, M6 
4 
M3, m6 
5 
P4, P5 
6 
A4, d5 
For example, in Schönberg's Op. 25 we find:
IC: 
1 
2 
6 
5 
3 
5 
6 
3 
1 
3 
1 
Note: 
E 
F 
G 
C# 
F# 
D# 
G# 
D 
B 
C 
A 
Bb 

IC 1 
IC 2 
IC 3 
IC 4 
IC 5 
IC 6 
Totals: 
3 
1 
3 
0 
2 
2 
Interval classes 1 and 3 predominate. Some rows are composed so
as to emphasize particular intervals, as is the case above, while
others are not.
The allinterval row, when spelled in an ascending
fashion, contains one appearance of each interval, such as the one
from Berg's Lyric Suite:

M7 

m6 

M6 

m7 

P5 

TT 

P4 

M2 

m3 

M3 

m2 

F 

E 

C 

A 

G 

D 

G# 

C# 

D# 

F# 

A# 

B 
The interval construction of a row has a distinct bearing on the
resultant sound quality.
Some rows use the first three, four, or six notes as a pattern
from which the rest of the row is derived. Such a row is called a derived
set. In such a set the pattern is transposed, inverted,
retrograded, or inverted and retrograded to "generate" the
remainder of the set. For example, the first hexachord (scale of six
notes) of Berg's row from the Lyric Suite in retrograde and
transposed by a tritone would produce the second hexachord, so the
row is a derived set. The row of Webern's Concerto, Op. 24,
is generated by applying the RI, R and I operations to the first
trichord:
IC: 

1 

4 


4 

1 


4 

1 


1 

4 


B 

Bb 

D 
Eb 

G 

F# 
G# 

E 

F 
C 

C# 

A 

1 

2 

3 
4 

5 

6 
7 

8 

9 
10 

11 

12 

pattern 
RI of
13 
R of
13 
I of
13 
Even if the row is not a derived set, it may well contain pitch
patterns that are transposed, inverted, or retrograded. Patterns of
ICs that are repeated or reversed may help in finding these pitch
patterns. In Schönberg's Suite, the repeated 31 at the end
of the row is caused by two overlapping statements of a trichordal
pattern:






>>>>>> 

<<<<<< 





IC: 

1 

2 

6 

5 

3 

5 

6 

3 

1 

3 

1 

Note: 
E 

F 

G 

C# 

F# 

D# 

G# 

D 

B 

C 

A 

Bb 















D 

B 

C 























C 

A 

Bb 
The pattern 65 within the row is reversed later as 56, and it
is also part of a larger pattern that reverses itself: 65356.
Such a pattern may indicate that a segment of the row is its own
retrograde or retrograde inversion. Here, the patterns 65356,
311 and 131 are all reversible. With these particular row
segments, the retrograde inversion of the segment reproduces the
segment (but transposed):
Interval Classes
(IC): 

6 

5 

3 

5 

6 


3 

1 

3 


1 

3 

1 

Prime: 
G 

C# 

F# 

D# 

G# 

D 
D 

B 

C 

A 
B 

C 

A 

Bb 
Retrograde Inversion: 
D 

G# 

C# 

A# 

D# 

A 
A 

F# 

G 

E 
Bb 

B 

G# 

A 
These various patterns may have implications in the piece,
motivically.
PitchClass Set Types
STEP 3.
In settheoretic integer notation, 0 is mapped to the first pitch
class of a pitchclass set arranged in normal order. Normal
order is a way of arranging pitch classes so that the sum of the
integers, read either from bottom to top or top to bottom, is the smallest
possible sum. In instances where two possible arrangements have
the same integer sum, then the normal order is that in which the
smaller intervals come first.
Newton Hoffman (heh, heh, my old theory coach) has suggested that
the simplest way to accomplish this is by first indicating the
pitches on a circle and then inspecting them to see which reading
produces the normal order.
The convention of a normal order makes it possible to search for
similarities and differences among pitch sets used in music. For
example, Chord C below is similar to Chord B, a fact that becomes
obvious only when both are reduced to their normal ordering.
The two chords are based on the same set type, which means
that the pitchclass set of one is the same as that of the other
under transposition or inversion or both. Notice that
"inversion" here has no relation to chord inversion in
traditional theory.
This type of analysis is helpful in analyzing simultaneous
sonorities encountered in 12tone music, and other music, when
conventional harmonic analysis is difficult or impossible.
When harmonic patterns are discovered, one simply assigns numbers
to various set types in analyzing a particular composition.
For example, a "Set 1" may be sonorities which contain
pitch classes [01378]. "Set 2" may be those which
contain pitch classes [0345], etc.
 Start "0" where the majority of notes will be
closest together, resulting in the smallest possible sum.
Go clockwise or counterclockwise, whichever direction
will accomplish the preceding criteria.
 Below, C = "0," but zero is movable ...
PitchClass Set Types ... Example
Schönberg Suite, Op. 25: (Prime)
E 
F 
G 
C# 
F# 
D# 
G# 
D 
B 
C 
A 
Bb 
Following are examples of three, four, five and sixsonority
chord structures which would be possible in completing a more
thorough analysis of a serial work. Of course, in practice it is
usually enough to analyze only the trichords and tetrachords, unless
there are a large number of five or sixsonority chords.
In an analysis, one would assign a number to each pitchclass set
for easy identification. For example, for trichords, the most
frequently occurring pitch set is [0,1,3] (normal order), so
this set could be labeled Set No. 1. The next most frequently
occurring pitch set is [0,2,5], which could be labeled Set No. 2,
etc.
The chord sets are determined in the following manner:
 The first
trichord would be comprised of the first three notes in the row:
notes 1, 2, 3.
 The second trichord would be comprised of notes 2, 3, 4.
 The third trichord would be comprised of notes 3, 4, 5, etc.

Then, the normal order of the three pitches is determined in
order to determine the set description.
(The numbers below are "normal order" numbers, not
matrix numbers.)
etc. 


First Note 
Trichords 
Tetrachords 
Pentachords 
Hexachords 


E 
[0,1,3] 
[0,2,3,6] 
[0,1,2,3,6] 
[0,1,2,3,4,6] 

F 
[0,2,6] 
[0,1,2,6] 
[0,1,2,4,6] 
[0,1,2,3,5,7] 
G 
[0,1,6] 
[0,1,4,6] 
[0,1,2,5,7] 
[0,1,2,5,6,7] 

C# 
[0,2,5] 
[0,2,5,7] 
[0,1,2,5,7] 
[0,1,2,4,7,9] 


F# 
[0,2,5] 
[0,1,4,6] 
[0,1,4,6,9] 
[0,1,3,4,7,9] 


D# 
[0,1,6] 
[0,1,4,7] 
[0,1,3,4,7] 
[0,1,3,4,6,7] 


G# 
[0,3,6] 
[0,2,3,6] 
[0,1,3,4,6] 
[0,1,2,3,4,6] 


D 
[0,1,3] 
[0,2,3,5] 
[0,1,2,3,5] 



B 
[0,1,3] 
[0,1,2,3] 




C 
[0,1,3] 





A 






Bb 




(E.g., the first trichord is EFG=013; the second trichord is FGC#=026. The first tetrachord is EFGC#=0236; the
second tetrachord is FGC#F#=0126, etc.)
The above three steps should provide a foundation for analyzing a
serial composition.
Compositional Uses of the Row
There are a number of ways in which rows are actually used in
compositions. Generally, a 12tone work consists of the presentation
of various row forms at a number of transpositions, the forms being
used sometimes in succession and sometimes simultaneously.
The notes may appear in any octave, and the order of the notes of
each row is usually preserved, but there are exceptions.
Notes can be sounded simultaneously, as in a chord, and there is no
"rule" as to how the notes in this case must be arranged.
Repeated notes are not considered to alter the order of the row, and
neither are tremolo figuresusing two of the notes repeatedly in
alternation.
Since most music involves more than a single line, the composer
must either present two or more row forms simultaneously or
distribute a single row form among the various voices. Both of these
approaches are widely used, which complicates the task of
determining P0 at the beginning of an analysis.
Assuming a serial piano composition in which the row is
distributed between the two clefs, following are two possible
distributions of the row:
1 
2 
3 

6 
7 
9 

11 
12 

or 

1 
2 
3 

4 
5 
6 

7 
8 


4 
5 

8 

10 






9 
10 

11 

12 


The diagram to the left is the more commonlyused method of
distributing a row, but the second possibility cannot be ruled out.
With either method, we cannot be sure of the exact order in that
some of the pitches are displayed simultaneously. We would have to
analyze more of the piece to find out. In art songs, sometimes the
row first appears in the vocal part.
Set Succession
Although a composer has 48 row forms available, few compositions
make use of them all. In fact, some compositions use P0 and nothing
else. Not many 12tone works are so restricted, however.
Most employ
all four basic forms and several transpositions.
One of the more difficult tasks of the analyst is attempting to
determine why a particular row form and transposition have
been chosen. Sometimes there may be no obvious explanation.
Often,
however, it is not uncommon for one or more pitch classes to serve
as common tones between two row forms in this manner.
Combinatoriality
Sometimes the choice of row forms or transpositions is governed
by a desire to form aggregates between portions of row forms.
For example, in the following diagram, the row that Schönberg used
for his Piano Piece is followed by its RI5 form. Notice that
the last hexachord of P0, when combined with RI5, forms an
aggregate. In effect, a new row has been created, called a secondary
set, by combining two hexachords from two different row forms.
HORIZONTAL:
P0 

RI5 
Bb 
F 
C 
B 
A 
F# 
C# 
D# 
G 
Ab 
D 
E 

A 
B 
F 
Gb 
Bb 
C 
G 
E 
D 
C# 
G# 
D# 






aggregate
(secondary set) 






This combining of row forms to form aggregates is called combinatoriality,
and it is an important aspect of some serial compositions. Most
often the combining is done vertically:
VERTICAL:
P0: 
Bb 
F 
C 
B 
A 
F# 

C# 
D# 
G 
Ab 
D 
E 
/\

\/ 
I5: 
D# 
G# 
C# 
D 
E 
G 

C 
Bb 
Gb 
F 
B 
A 

aggregate 

aggregate 

Other aggregate combinations are also possible, using not only
hexachord combinations, but trichord or tetrachord combinations.
Combinatoriality guarantees a more controlled recycling of the 12
pitch classes, and to some it seems a necessary extension of the
12tone aesthetic.
Analyzing Serial Music
In analyzing the use of rows in a serial piece, it is often
enough to label the row forms (P0, etc.) without writing the order
numbers on the music. If the texture is complex, or if some unusual
row technique is being employed, it may be necessary to write the
order numbers near the noteheads and even to join them with lines.
Always work from a matrix. If you get lost, try to find several
notes that you suspect occur in the same order in some row form, and
scan the matrix for those notes, remembering to read it in all four
directions.
The labeling of row forms and the consideration of the details of
their use is only a part of the analysis of a serial composition,
somewhat analogous to identifying the various tonalities of a tonal
work. Questions regarding form, thematic relationships, texture,
rhythm, and other matters are just as relevant here as in the
analysis of more traditional music. The music of classical serialism
is not especially "mathematical," and it is not composed
mechanically and without regard to the resulting sound or the effect
on the listener. Probably the best way to appreciate the processes
and choices involved in serial composition is to try to compose a
good serial piece yourself.
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