MUS 312 Form & Analysis

M. 12-Tone Matrix

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Introduction

Arnold Schönberg composed the first 12-tone 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 "method--he insisted it was not a system--of 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 12-tone composition is serialism.  The term "dodecaphonic" has also been used as an adjective to describe the 12-tone method; however, the term serialism is preferred.

12-Tone Matrix Systems

There are two ways of approaching the 12-tone 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 Mod-12-Integer Analysis, or more simply, Mod-12 (technically, Modulus-12).

Both systems rely on the same principle--intervallic relationships.  The Mod-12 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 both--and they are quite similar.

Basic Terminology

The core of the 12-tone 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 (12-tone) 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 0-11, which specifies the transposition of the row in half-steps.  For example: P-3, R-4, I-9, RI-6

Traditional Matrix Construction (using letters)

To construct a traditional matrix using note letter names, follow these steps:

1. Determine the original row and fill in P-0 along the top row.

2. Write the transposition numbers above the matrix by numbering the pitches chromatically, beginning with 0 for the first note and moving up by half-step.

• For example, if E = 0, then F = 1, F# = 2, G = 3, etc.  Write these same numbers beneath the matrix.

3. 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.

4. Find the horizontal row with a 1 in front of it, and transpose all of P-0 up one half-step into that row.

5. Find the horizontal row with a 2 in front of it, and transpose all of P-1 up one half-step into that row.

6. Continue this process until P-10 is transposed up one half-step into P-11, and the matrix is complete.

Mod-12 Matrix Construction

To construct a Mod-12 matrix using numbers, follow these steps:

1. 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 Mod-12, however, the numbers are placed inside the boxes of the top row (P-0).  (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.

2. In the upper triangle above each number, indicate by number and a plus (+) sign how each number differs in half-steps 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.

3. Now, to determine the Inversion of the row (first vertical column, I-0), first you must fill in the upper triangle to the left of each I-0 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 I-1 at the top is +1, then the first triangle would be -1. If I-3 is +3, then the top half of the second triangle is -3, etc.

4. 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.

5. Now, fill in the numbers for I-0 (first vertical column) by transferring the positive numbers from the lower triangles to the left of the matrix.

6. 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 I-0 (far-left 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.

7. 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:  I-0 I-1 I-3 I-9, etc., indicating Inversion.  Place the same numbers beneath the matrix, indicating Retrograde Inversion, for example: RI-0 RI-1 RI-3 RI-9, etc.  To the left of the matrix, indicate Prime form, for example: P-0, P-11 P-9, etc.  Place these same numbers to the right of the matrix to indicate Retrograde, for example: R-0 R-11 R-9, etc.

8. Go through the piece and write in the corresponding numerical equivalents of P-0 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.

9. 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 12-tone 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, 12-tone music follows the convention of omitting natural signs: any note without an accidental is natural.  Each note of the row is assigned a number (1-12) 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 P-3, R-9, etc., specify the transposition of the row form in half-steps.  The very first row form in a composition is numbered P-0.  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 P-0 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 all-interval 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 1-3					R of 1-3					I of 1-3

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 3-1 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 6-5 within the row is reversed later as 5-6, and it is also part of a larger pattern that reverses itself: 6-5-3-5-6.  Such a pattern may indicate that a segment of the row is its own retrograde or retrograde inversion.  Here, the patterns 6-5-3-5-6, 3-1-1 and 1-3-1 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.

Pitch-Class Set Types

STEP 3.

In set-theoretic integer notation, 0 is mapped to the first pitch class of a pitch-class 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 pitch-class 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 12-tone 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 [0-1-3-7-8].  "Set 2" may be those which contain pitch classes [0-3-4-5], 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 ...

 

 

 

Pitch-Class 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 six-sonority 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 six-sonority chords.

In an analysis, one would assign a number to each pitch-class 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 12-tone 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 figures--using 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 P-0 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 commonly-used 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 P-0 and nothing else.  Not many 12-tone 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 RI-5 form.  Notice that the last hexachord of P-0, when combined with RI-5, 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:

P-0

RI-5

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:

P-0:

Bb

F

C

B

A

F#

C#

D#

G

Ab

D

E

/\ | \/

I-5:

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 12-tone aesthetic.

Analyzing Serial Music

In analyzing the use of rows in a serial piece, it is often enough to label the row forms (P-0, 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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