Introduction to Set Theory  

Part 2. 12-Tone Pantonal Composition & Serial or Ordered Sets

copyright © 2002 by Larry Solomon 

Contents

Set Theory Glossary
Introduction
How to Construct a 12-tone Matrix
How to Select Organizational Properties of a Row: Derived Rows
Unity vs Variety in Row Structure: All-interval Rows
Combinatoriality
Invariants
Linear Set Presentations
Partitioning
Twelve-Tone Tonalities
Harmony and Meter
Guidelines for Serial (Pantonal) Composition
Bibliography

Introduction

The construction of the set of twelve tones derives from the intention to postpone the repetition of every tone as long as possible. [Arnold Schoenberg, "Composition with Twelve Tones", Style and Idea, 246.

Schoenberg's purpose in his new compositional method of twelve tones was to supplant tonality with a new means of order, of which he approved calling "pantonality", and disproved of the term "atonality". " Pantonality means an equality of tones, effectively a democracy of tones. Thus, the basic principle behind composition with twelve tones is the even distribution of pcs with none dominating the others. Repeating some tones makes them sound more important, and hence contradicts the goal.

One way to guarantee the "postponement" of tone (pc) repetitions is to place all twelve in an order where none is repeated. This is called a twelve-tone row, or series. These pcs may be represented as letter names (C, D, F#, etc.) or by pitch numbers. The latter enables mathematical operations on the set which facilitate various transformations and constructions. Thus, the number representation has definite advantages over the letter notations.

Tone Row. A group of pcs (usually the 12 chromatic pcs) placed in a particular order to be so used in a composition.

Pitch number (pin). Each pc can be represented by a number from 0 to 11 in the twelve-tone system. In pantonal music, since tonic is neutralized, enharmonics are equivalent, i.e., C# = Db, D# = Eb, F=E#, etc.

C C# D D# E F F# G G# A Bb B
0 1 2 3 4 5 6 7 8 9 10 11
0 1 2 3 4 5 6 7 8 9 A B

The first row of numbers in this table indicates the decimal notation for each pc. The last row shows the same pcs in hexadecimal (base 16) notation. Hexadecimal will be used here, because it gives a more compact notation of a set, not requiring separators between the numbers. The table shows what is known as absolute-do (abdo) notation, where C is always zero (0). In the relative-do (reldo) notation, any pc may be set to zero, usually the first. Thus, FAC is represented as 590 in abdo, but as 047 in reldo.)

Many thousands of different 12-tone rows many be constructed from the above pcs. In fact, there are 12! (12 factorial) or 479,001,600 possible permutations of the chromatic set. Each is an ordered set or series. Any one of these is really an interval series, because the set may be transposed to begin on any of 12 pcs. This interval series may also be reversed in order or inverted. Inversion may be derived by subtracting the pins from 12. These may also be transposed. Thus, for each set there are 12 possible transpositions of the prime (P) set, 12 of the inversion (I), 12 of the retrograde (R), and 12 of the retrograde-inversion (RI), making 48 possible conventional transformations all together. Some special, advanced transformations, called Quadrates, are discussed at http://www.solomonsmusic.net/diss8.htm.

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How to Construct a 12-Tone Matrix

A 12-tone matrix is a concise way of representing all 48 forms of a row on a grid (matrix). As an example, take the 12-tone row G A# B F E C# C A G# D D# F#. This may be represented in pins as 7AB541098236 (hexadecimal).

Since we will be using the relative-do (reldo) system here, set the first PC to zero by transposition (subtract 7): 034A9652178B. This sets the pc G to zero, and the set that begins with it is Po. (Note that matrix construction would be different in the absolute-do [abdo] system.) To construct a reldo matrix first write the row horizontally. Then write its inversion vertically down the left side, starting with 0. Subtract the horizontal numbers from 12 for each successive pc of the inversion down the left side. Each diagonal pair will be the complement of 12. Remember that A=10, and B=11.

0  3  4  A  9  6  5  2  1  7  8  B
9
8
2
3
6
7
A
B
5
4
1

Now fill in the rest of the matrix by writing the rest of the transposed prime forms horizontally starting with each number that appears in the first column. The second row will be P9, the third row P8, etc. For example, the second row starts with 9, which means that 9 must be added to each number in the first row to get the second row of numbers. Using mod12 arithmetic, imagine a clock (except that zero is in the 12 o'clock position). To get the second number of the second row add 9 hours to 3 (first row) to get 12. But 12 in mod12 is zero (0). Add 9 hours to 4 to get 1 (column 3). {4 o'clock plus 9 hours gets you 1 o'clock., or 13 in European time). At the top of column 4 is hexadecimal for 10. So, 10 + 9 = 7 (second row). Continue with the rest of  the numbers in row 1, adding 9 to each to get the numbers of the second row. (Instead of adding 9, one could subtract 3 instead.)

Row 3 is constructed in the same way. Since the first number is 8, then 8 must be added to each number of row 1 to get row 3. All the rows, or transpositions, can be constructed in this manner to complete the matrix. For row 4, add 2 to row 1, etc.

Table 1. The Matrix in Reldo

0

3

4

A

9

6

5

2

1

7

8

B

9

0

1

7

6

3

2

B

A

4

5

8

8

B

0

6

5

2

1

A

9

3

4

7

2

5

6

0

B

8

7

4

3

9

A

1

3

6

7

1

0

9

8

5

4

A

B

2

6

9

A

4

3

0

B

8

7

1

2

5

7

A

B

5

4

1

0

9

8

2

3

6

A

1

2

8

7

4

3

0

B

5

6

9

B

2

3

9

8

5

4

1

0

6

7

A

5

8

9

3

2

B

A

7

6

0

1

4

4

7

8

2

1

A

9

6

5

B

0

3

1

4

5

B

A

7

6

3

2

8

9

0

As if by magic, all 48 forms (including the inversions, retrogrades and retrograde-inversions) are represented on this matrix with inversions read down from the top. Retrogrades are read from right to left, and RI forms are read from bottom to top. The first number becomes the index number of the transposition in reldo; e.g., the second row is P9, the second column is I3, the third row of the retrograde, reading from right to left, is R7, etc. One should get familiar enough with the number system that one can sing or play the pitches they represent quickly, without hesitation. It helps to remember landmarks, like in abdo, E=4, G=7, and Bb=10 or A. From these one can "fetch" the others quickly.

The following axioms apply to the Reldo system:
Axiom 1. The Prime (Po) form starts with zero (reldo) which sets the index numbers for all other forms.*
Axiom 2. Transpositions are determined by adding a transposition number to the pins (mod12).
Axiom 3. The Inversion (I) pins are determined by subtracting the prime pins from 12.
Axiom 4. The Retrograde (R) pins are the prime pins in reverse order.
Axiom 5. The Retrograde Inversion (RI) pins are the reverse of the inversion pins.
Axiom 6. The index or transposition number is equal to the first pin number of each set form.*

*It should be noted that the other system in use (abdo) calls the last pc of Po the zero transposition index of Ro.
To construct the matrix in Abdo, place the original prime in the first row as above. Let x=double the number in the upper left corner, in this case 7x2=14, (14=2 in mod12). After placing the original prime in the first row, construct the inversion by the complements of x in the leftmost column; e.g., A+4=2, B+3=2, 5+9=2, etc. Fill in the rest of the matrix using the same method as in reldo.

Table 1. The Matrix in Abdo

7
A
B
5

4

1

0

9

8

2

3

6

4
7
8
2
1
A
9
6
5
B
0
3
3
6
7
1
0
9
8
5
4
A
B
2
9
0
1
6
7
3
2
B
A
4
5
8
A
1
2
8
7
4
3
0
B
5
6
9
1
4
5
B
A
7
6
3
2
8
9
0
2
5
6
0
B
8
7
4
3
9
A
1
5
8
9
3
2
B
A
7
6
0
1
4
6
9
A
4
3
0
B
8
7
1
2
5
0
3
4
A
9
6
5
2
1
7
8
B
B
2
3
9
8
5
4
1
0
6
7
A
8
B
0
6
5
2
1
A
9
3
4
7

An economical way to represent the 48 forms is with their divs (directed-interval vectors). For example, the row for Schoenberg's Septet, Op. 29 is 376A2B098451. The div for this set is 4B44919B818. This will be the div for all transpositions of the prime and is a more elegant representation of a set. The inversion div is simply the pin complements of 12; i.e., 81883B314B4. RI is the reverse div of P: 818B91944B4, and R is the reverse div of I: 4B413B38818 . A complete matrix of the divs can be represented:

P: 4B44919B818
I : 81883B314B4
R: 4B413B38818
RI:818B91944B4

This is a much more compact way to express the matrix, i.e., more elegant.

Theorum 1. The div of a row inversion (I) is determined by subtracting the prime (P) div from 12.
Theorum 2. The div of RI is the reverse of P.
Theorum 3. The div of R is the reverse of I.
Theorum 4. If the div numbers of P are complements of 12 about the central axis of symmetry, P=R and I=RI.
Theorum 5. If the div numbers of P are equal about the central axis of symmetry, P=RI and I=R.

Yet another way to represent the set in a matrix is by using the dcv (directed-class vector). The numbers over 6 are converted to their complements as negatives. The negatives are here represented as smaller numbers on the matrix. They may be represented in long hand by placing a small minus sign before each negative number as in example 1 above.

P: 41443131414
I : 41443131414
R: 41413134414
RI:41413134414

With this last representation, there are no numbers larger than 6, and it becomes easier to memorize the set, especially if it is partitioned into segments and visualized or "auralized": 4144, 3131, 414. Thus, one could start with any note and generate a transformation with this dcv; e.g., Po = CED#GB, G#AF#F, C#DA#. Inversional forms simply switch positive and negative numbers:
4144, 3131, 414. Thus, I2 = DA#BGD#, F#FG#A, C#CE. Retrograde-Inversional forms (P/RI and I/R) reverse the order of`div.

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How to Select Organizational Properties of a Row: Derived Rows

If greatest unity is desired, a row may be constructed to have recurring subsets within the series, e.g., in four 3-note groups. Such a constructed row would be used to explore the motivic and/or harmonic properties of the 3-note figures.

Such a row is the one in Table 1. This row has very special properties: 034A9652178B. Its div is 316B9B9B613. Notice that the numbers are symmetrical about the central B. Since RI has the reverse div of P and the div is symmetric, then P=RI. One can prove this by comparing Po with the RI that starts with 0 (reading up)(RIo). The same relationship holds between I and R; so I=R. Comparing I4 with R4 on the matrix, they are identical.

But there are even further principles organizing this row. If the first 3-note figure is taken as a little prime (p0), the second trichord is rA, the third trichord is i5 and the last is ri7! Thus, the row itself consists of an initial trichord with transformations derived from it to create the rest of the row. Additionally, each hexachord is inversionally symmetric; the second is the inversion of the first. The row is also combinatorial (Po+I5 and Po+IB). This is called a derived row and has profound implications for the composition that uses this row. Additionally, each trichord, if its unordered pc content is considered, consists of the Forte prime 014; i.e., they are identical in ic (unordered) content.

This row also has a most remarkable property if considered cyclic. All the trichords of all transformations maintain a constant ordering. Therefore, effectively, P=I=R=RI, often considered to be an impossible condition. Also, each trichord is followed by its own retrograde. Po, RI

This row demonstrates an extreme condition of maximum internal organization and therefore contains maximum repetition. Repetition eases comprehension. Anton Webern was fond of using derived rows in his compositions, which is a probable reason that his music is more accessible than that of Schoenberg's. (Another is that Webern's music is generally thinner in texture, hence more transparent, and shorter in length than Schoenberg's.)

Theorum 6. There is no transposition of P that is congruent with I (nor R with RI) unless the row is cyclic.
Theorum 7. Interval classes are invariant after Inversion (TnI).

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Unity vs Variety in Row Structure: All-interval Rows

"Repetition is the means by which this unity, and comprehensibility, is achieved.... All formal construction is built upon it." [Anton Webern, The Path to the New Music, 17]
"Composition with twelve tones has no other aim than comprehensibility. [Arnold Schonberg, Style and Idea, 103]

Derived rows are definitely an advantage if one is striving for the greatest amount of unity in a composition. Webern's chief concern was unity; thus, his rows are mostly of the derived type. For Schoenberg, however, variation was at least as important a compositional concern as unity. And, sometimes a composer prefers variation over repetition. In this case, a row might be chosen that has a minimum amount of internal repetition and maximum variation. Such rows are called 11-interval 12-tone rows, or "all-interval rows". In these rows all eleven intervals (di) are used without any repetition, assuring maximum variety. Such a row is found in Alban Berg's Lyric Suite: 5409728136AB. However, not all these rows have the same degree of variation, For example, Berg's row has a symmetrical dcv which makes P=R and I=RI. Here is a list of some all-interval rows expressed as div:

(1) B89A7652341 (Berg)
(2) 4B295681A37
(3) 529A16473B8
(4) 453126AB978
(5) 1432567A98B
(6) 523146B897A
(7) A195268B374
(8) B29476583A1
(9) 1A98567432B

For a complete list of rows of this type, see: Morris, Robert, and Daniel Starr, "All-Interval Series", Journal of Music Theory, 18 (1974), 364-398

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Combinatoriality

If only one set is used the order of notes can be easily maintained without ambiguity. But, if two row forms are presented at once, say in counterpoint, pcs will be repeated unless special procedures are followed. An aggregate is a collection that includes all twelve pcs. One of the aims of twelve tone composition is to complete an aggregate before repeating pcs, even when row forms are combined.

Hexachordal combinatoriality means that hexachords of the same order can be combined without pc duplication, or, in other words, when they complete an aggregate. Hexachordal combinations are the only ones that we will consider here, because they are the most frequently used. Other types are possible but are beyond the scope of this exposition.

All series are combinatorial with their retrograde at a transposition that corresponds with the last pin of Po (abdo). This is referred to here as Ro combinatoriality. Since this is automatic, there is nothing special about it. Webern uses retrograde combinatoriality in his Piano Variations, Op. 27. At the beginning he uses Po+Ro where he aligns the first hexachord of each followed by their second hexachords. See analysis. This results in no pc duplication before all 12 pcs sound. The same result will hold when I is combined with RI at the proper transpositions. Since Ro combinatoriality is a property of all 12-tone rows, this form is trivial and will not be studied here. It is possible, however, for another transposition of R to be combinatorial, and this will be referred to here as Retrograde combinatoriality. This form is not trivial and must be dealt with. R combinatoriality is possessed by any hexachord that has a 6 as one of its iv (interval vector) entries, or a 3 in the tritone position.

The prime form of a set may also be combinatorial with itself at some transposition. This may be called transpositional or Prime combinatoriality, or TH. And, a set may be combinatorial with its inversion. The last type, called hexachordal-inversional combinatoriality, or IH, has received considerable attention among 12-tone composers. A row may also have RI combinatoriality.

Composition of a row with TH is fairly easy. If for instance, one constructs the initial hexachord from some ordering of the first six notes of a chromatic scale, the second hexachord can be made from an ordering of the other six. When the whole row is transposed by a tritone, i.e., P6 , it will be combinatorial with Po. Transpositional combinatoriality is possessed by any hexachord whose iv contains a zero.

Over half of all twelve-tone rows are IH combinatorial. The combinatorial properties can be determined from examination of the first unordered hexachord of the row. Combinatoriality does not depend upon how the notes are ordered but simply by their content. Thereby, these hexachords can be identified in the complete Table of (unordered) Pc Sets. On this list "comb" means combinatorial. The forms of combinatoriality are listed following "comb". The transposition numbers apply only to the hexachord in prime form. The table shows all forms of hexachordal combinatoriality except for the Ro type, which all hexachords possess.

Theorum 8. A necessary condition for IH is that the sum of pins of the same order number must be odd.

IH combinatorial row forms by themselves do not guarantee the avoidance of pitch class duplications. This also depends upon the rhythmic alignment of the rows in a composition. Schoenberg avoided pc duplication in his late works by rhythmically aligning the respective IH hexachords. To extend this principle even further he created secondary sets that formed aggregates when two row forms were used in succession.

Invariants

Through the use of operations of transposition, I, R, and RI, certain properties of the original set may be preserved. These are called invariants and take the form of pcs, subsets, and intervals (di) or interval classes (ic). Additionally, combinations of row forms may result in invariant sets of various sizes.

Theorum 9. Pc dyads are held invariant between inversionally related forms (TnI) if the transposition number is the sum of pins of the same order. (e.g., Webern Op. 27/2)

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Linear Set Presentation

Although Schoenberg used the row as a linear/melodic line in the earlier twelve-tone works his later use became more sophisticated. A strict linear statement is relatively rare in his mature compositions and is usually reserved for marking important structural or dramatic points in the composition, e.g., Moses and Aron. However, a linear statement is often given at the beginning in order to present the row in its clearest form, i.e., as an aid to perception.

Contrary to novice beliefs, pitches can be repeated in a row composition. A repetition is not considered different from holding a note, as in voice leading. Notes can be held and repeated but these repetitions should be at the same pitch level (rather than octave displaced). Further, a series subset may repeat, e.g., 1234 1234 567 456 78 78 9AB, again keeping the same pitch level. Although backtracking is permitted, it is the order that is maintained.

The series can be used to form chords where ordering is indeterminant. Thus, the pcs need not occur in any special arrangement (bottom to top, or vice versa). A horizontal presentation may be combined with a vertical one using various row segments.

Rhythm, dynamics, articulation, rests, texture, etc., are normally free in twelve-tone composition. But, to avoid confusion with a tonal conception and to maintain pc equality, there are compositional guidelines. In "total serialization" found in works by Milton Babbitt and Pierre Boulez, all elements of the composition are serialized, including dynamics and articulation.

Although the row was once believed to be rigid in its pc order, it is now used more flexibly. Often Schoenberg would partition his row into segments of three or four notes apiece, and although the order from one segment to another was maintained, the ordering within each segment was juggled. This principle was eventually extended to hexachordal segments. Thus, the linear rigidity of the row became much more flexible and began to function more as a governor of harmony.

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Partitioning

As strict linear ordering became less important in row composition, partitioning the row into sets composed of non-consecutive elements became increasingly important. These create what are known as secondary sets., i.e., sets formed from what may be non-consecutive pcs of the original row. Often these secondary sets are discrete equal units, but Schoenberg also began to use unequal units in his late compositions. By using features of invariance, these secondary sets can be organized and related to each other, overlapping and sharing subsets in a polyphonic texture.

These units may be composed so that their unordered contents are transpositions or inversions of one another, or have the same interval content (IV). This is called isomorphic partitioning.

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Twelve-Tone Tonalities

Schoenberg also sought a way to create an analogue to the key modulations that tonal music contains. This was to provide for temporal unity and variation beyond the surface level. Thus, the first hexachord was treated as an analogue of tonic and was presented at the outset of a composition. Subsequent divergences from and return to this metaphorical "tonic" could then signal important formal divisions. Again, the hexachord becomes an important harmonic unit in this scheme.

Harmony and Meter

Hramony in the twelve-tone music of Schoenberg is not arbitrary as is often claimed. In fact, one could argue that harmony was always the primary concern. The hexachord is used as the primary unit of harmony, but secondary sets are also very important. Non-adjacent elements were often combined to create local harmonies that repeat throughout a composition. Schoenberg used almost exclusively IH combinatoriality in his late works, which and regulates the possible harmonic combinations, effectively restricting them to one harmonic profile. Additionally, meter is often affected by the periodicity of this harmonic profile.

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Guidelines for Serial (Pantonal) Composition 

Bibliography

Babbitt, Milton. "Some Aspects of Twelve-Tone Composition", Score, 12 (1955), 53-61

_____ "Twelve-Tone Invariants as Compositional Determinants", Musical Quarterly, 46 (1960), 246-59. Reprinted in Paul Henry Lang, ed. Problems of Modern Music (1962), 108-21

_____ "Set Structure as a Compositional Determinant", Journal of Music Theory, 5 (1961), 72-94. Reprinted in Boretz and Cone, eds., Perspectives on Contemporary Music Theory (1972), 129-47

Beach, David. "Segmental Invariance and the Twelve-Tone System", Journal of Music Theory, 20 (1976), 157-84

Howe, Hubert S. "Some Combinational Properties of Pitch Structures", Perspectives of New Music, 4 (1965), 45-61

Hyde, Martha. "The Telltale Sketches: Harmonic Structure in Schoenberg's Twelve-Tone Method", Musical Quarterly, 66 (1980), 560-80

_____ Schoenberg's Twelve-Tone Harmony, Ann Arbor, 1982

Lewin, David. "A Theory of Segmental Association", Perspectives of New Music, 1 (1962), 89-116. Reprinted in Boretz and Cone, eds., Perspectives on Contemporary Music Theory (1972), 180-207

_____ "On Certain Techniques of Re-Ordering in Serial Music", Journal of Music Theory, 10 (1966), 276-287

Martino, Donald. "The Source Set and Its Aggregate Formations", Journal of Music Theory, 5 (1961), 224-73

Morris, Robert and Daniel Starr. "A General Theory of Combinatoriality and the Aggregate", Perspectives of New Music, 5 (1977), 50-84

_____"All-Interval Series", Journal of Music Theory, 18 (1974), 364-398

Perle, George. Serial Composition and Atonality, (1981)

Rahn, John. Basic Atonal Theory (1979)

Westergaard, Peter. "Toward a Twelve-Tone Polyphony", Perspectives of New Music, 4 (1966), 90-112. Reprinted in Boretz and Cone, eds., Perspectives on Contemporary Music Theory (1972), 238-60

Wuorinen, Charles. Simple Composition (1979)

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