Introduction
In the early 1920s, Arnold Schoenberg, Anton
Webern, Alban Berg and others were composing with serial (from series = ordered)
motives. The basic idea of using a motive in an "original"
or prime
form, the motive inverted, the motive backwards (or in retrograde), and the motive
backwards and inverted (or in retrograde
inversion), dates back to the 1400s in
the Franco-Flemish School and was still used by Bach in the Art of Fugue
and other works -- particularly in "puzzle canons." Webern was initially a musicologist and did his dissertation work on the composer Heinrich Isaac, where he likely encountered these types of contrapuntal devices. In addition to serial elements, twelve-tone music incorporates "complementation," or the use of all twelve notes. We can find examples in the fugue subject to Bach's WTC Book I, fugue 24, and the theme to the first movement of Liszt's "Faust Symphony." When serial motives containing all twelve pitch-classes
were used as the basis of compositions,
Basic Elements
Series: a specific ordering of any number of pitch-classes. There may be only different ("distinct") notes in the series, or there may be some doublings. In Stravinky's Dylan Thomas Pieces, a five-note series (E-Eb-C-C#-D) is used. A series is combinational, which means that different forms of the series introduce new notes. For instance, E-Eb-C-C#-D combined with D-Eb-Gb-F-E adds the notes F,Gb so that the notes combine into the C-Gb tritone.
In a series, we can find many of the elements described below.
12-tone row: a specific ordering of the
twelve distinct pitch-classes. Because all the notes are present, different forms of a row are permutational, which means that the notes are rearranged only, with no new notes added. Another name for the 12-note collection
is aggregate. A row may be labelled with pitch-class numbers in
a "fixed do" system, where C = 0, C#/Db = 1 etc. to
Bb = 10 (written as "A" or "t") and B = 11
(written as "B" or "e"). The ordered place
of notes in a row is indicated by order position numbers, from
0 to 11; order position numbers are underlined and
start from 0. (row from Alban Berg, Violin Concerto)
| Pitch-class nos. | 7 | A | 2 | 6 | 9 | 0 | 4 | 8 | B | 1 | 3 | 5 |
| Order positions | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
INT The INT is the ordered interval pattern in the row, expressed with + and - or just with + intervals. For instance, the Berg row INT, which is for the P row, is as follows (the four forms are included):
The Stravinsky series is
Notice that P - R(P) and I - R(I) are retrograde-related in terms of pitch, but that P - R(I) are retrogrades when intervals are considered, as are R and I.
Series set: When the series is not comprised of twelve notes, it can be labelled by it's set-class information. The Stravinsky series E-Eb-C-C#-D makes up [01234]. This set-classes has only 12 forms, since the inversion yields the same set-classes.
Harmonic Profile: A row is defined by it's consecutive intervals and interval groups
(set-classes). The harmonic profile lists the consecutive intervals
(ascending) and set-classes that result from the ordered intervals in the row. The harmonic profile for the row of Berg's Violin Concerto is the following, where each interval or set begins from 0, then 1, etc. Thus, trichord 012 ( <
G,Bb,D > ) is in set-class [037]; tetrachord 2345 (<
D,F#,A,C>) is in set-class [0258], etc. It is important to
realize that the same intervals and set-classes result from the same order positions in any form of this row (P, I, R(P), R(I), the four forms are given below). Series Forms: The forms of the series can be listed in prime and inversion forms
Row Chart (or Row Matrix): Twelve-tone rows come in four
different forms. The Prime (original) form is usually considered to be the row form
that appears first or most prominently near the beginning of the piece. The Inversion form
is created by inverting the intervals of the prime form. The Retrograde form
is created by taking the pitch-classes of the Prime form in retrograde
order (which is the same as the intervals of the inversion in retrograde). The Retrograde-Inversion form is created by taking the pitch-classes of the
inversion form in retrograde order (which is the same as the intervals of the prime in retrograde). Each of the four forms has 12
different transpositions, therefore the total number of row forms
is 48. (Some rows have symmetrical interval patterns and thereby
less than 48 distinct forms, described below.) The 48 row forms are shown on a 12 X 12 row chart:
Primes (P) are left-right, Inversions (I) top-bottom, Retrogrades
(R(P)) right-left, and Retrograde Inversions (R(I)) bottom-top:
P: 3 - 4 - 4 - 3 - 3 - 4 - 4 - 3 - 2 - 2 - 2 I: 9 - 8 - 8 - 9 - 9 - 8 - 8 - 9 - t - t - t R(P): t - t - t - 9 - 8 - 9 - 9 - 8 - 8 - 9 R(I): 2 - 2 - 2 - 3 - 4 - 4 - 3 - 3 - 4 - 4 - 3
P: -1 -3 +1 +1;
I: +1 +3 -1 -1;
R(P): +1 +1 -3 -1;
R(I): -1 -1 +3 +1;
starting op
0
1
2
3
4
5
6
7
8
9
10
dyads
3
4
4
3
3
4
4
3
2
2
2
trichords
[037]
[048]
[037]
[036]
[037]
[048]
[037]
[025]
[024]
[024]
tetrachords
[0148]
[0148]
[0258]
[0258]
[0158]
[0148]
[0358]
[0247]
[0246]
pentachords
[01348]
[02458]
[02469]
[02458]
[01348]
[03458]
[01358]
[02469]
hexachords
[013468]
[013579]
[013579]
[013468]
[013458]
[013458]
[012469]
P4: E-Eb-C-C#-D I0: C-C#-E-Eb-D
P5: F-E-C#_D_D# I1: C#-D-F-E-D#
P6: F#-F-D-D#-E I2: D-D#-F#-F-E
. . . . . .
P2: D-Db-Bb-B-C It: Bb-B-D-Db-C
P3: Eb-D-B-C-C# Ie: B-C-D#-D-C#
G
Bb
D
F#
A
C
E
G#
B
C#
D#
F
E
G
B
D#
F#
A
C#
F
G#
A#
B#
D
C
Eb
G
B
D
F
A
C#
E
F#
G#
Bb
Ab
Cb
Eb
G
Bb
Db
F
A
C
D
E
Gb
F
Ab
C
E
G
Bb
D
F#
A
B
C#
Eb
D
F
A
C#
E
G
B
D#
F#
G#
A#
C
Bb
Db
F
A
C
Eb
G
B
D
E
F#
Ab
F#
A
C#
E#
G#
B
D#
G
A#
B#
D
E
D#
F#
A#
D
F
Ab
C
E
G
A
B
Db
Db
E
Ab
C
Eb
G
Bb
D
F
G
A
B
B
D
F#
A#
C#
E
G#
B#
D#
E#
G
A
A
C
E
G#
B
D
F#
A#
C#
D#
E#
G
I7
IA
I2
I6
I9
I0
I4
I8
IB
I1
I3
I5
P7
7
A
2
6
9
0
4
8
B
1
3
5
R(P7)
P4
4
7
B
3
6
9
1
5
8
A
0
2
R(P4)
P0
0
3
7
B
2
5
9
1
4
6
8
A
R(P0)
P8
8
B
3
7
A
1
5
9
0
2
4
6
R(P8)
P5
5
8
0
4
7
A
2
6
9
B
1
3
R(P5)
P2
2
5
9
1
4
7
B
3
6
8
A
0
R(P2)
PA
A
1
5
9
0
3
7
B
2
4
6
8
R(PA)
P6
6
9
1
5
8
B
3
7
A
0
2
4
R(P6)
P3
3
6
A
2
5
8
0
4
7
9
B
1
R(P5)
P1
1
4
8
0
3
6
A
2
5
7
9
B
R(P1)
PB
B
2
6
A
1
4
8
0
3
5
7
9
R(PB)
P9
9
0
4
8
B
2
6
A
1
3
5
7
R(P9)
R(I7)
R(IA)
R(I2)
R(I6)
R(I9)
R(I0)
R(I4)
R(I8)
R(IB)
R(I1)
R(I3)
R(I5)
Labelling Rows: Rows are are labelled by the number corresponding
to the first pitch-class. This is the "fixed do" system.
For instance, a prime row beginning on F or 5 is called P5; an
inversion row beginning on F or 5 is I5. For retrogrades, the
row is named as the retrograde of a row. For instance, the R(P5)
is the retrograde of P5, and so it ends on F. The R(I5) is the
retrograde of I5, and so it ends on F. R and RI rows are labelled
with order position numbers backwards, as 11 10 9 8
7 6 5 4 3 2 1 0.
Thus, 012 of I5 is <F,D,Bb> and 210 of R(I5)
is <Bb,D,F>. This labelling system allows us to relate P/I and R(P)/R(I) rows directly. (You may see labels like "R3" for the row we have labelled R(P5), indicating the retrograde row starting on Eb or 3; this label does not reveal that the row is the R of P5).
| P5: | F,Ab,C,E,G,Bb,D,F#,A,B,C#,Eb | 0123456789AB |
| I5: | F,D,Bb,Gb,Eb,C,Ab,E,Db,B,A,G | 0123456789AB |
| R(P5): | Eb,C#,B,A,F#,D,Bb,G,E,C,Ab,F | BA9876543210 |
| R(I5): | G,A,B,Db,E,Ab,C,Eb,Gb,Bb,D,F | BA9876543210 |
Series and Row Characteristics
Segment: An ordered adjacent section of a series or row, labelled with consecutive order position numbers, indicated using < >'s: in this famous passage from the Berg Violin Concerto, overlapping segments 012, 234, 456, 678, 89AB are harmonies with tonal implications: < G,Bb,D>, <D,F#,A>, <A,C,E>, <E,G#,B>, and <B,C#,D#,F>
Row Partition: A group of mostly non-adjacent notes extracted from
a series or row, also labelled with order position numbers; now the numbers
are not consecutive, but still relatively ordered. In this example from the Berg
Violin Concerto, the melody is created from three partitions:
<15158BAB > <F,G,F,G,Gb,C,Bb,C> from P2, then
< 04848 > <,A,G,F,G,F> from I9, then 1
<E> from P1. The bass line is created from partitions <
024024 > <D,A,E,D,A,E> from P2, then < 2367
> <D,Bb,C,Ab> from I9, then 0 <Db> from
P1.
Invariance
INVARIANCE:
If the set of a series has inversional or transpositional symmetry, it may be possible to arrive at the same notes in different ordered forms. In the Stravinsky Dylan Thomas series, E-Eb-C-C#-D, the set [01234] has inversional symmetry, and so P4 and I0 (C-C#-E-Eb-D) have the same notes.
Since rows have all twelve notes, composers tend to use smaller note groups as motives to relate different voices and their respective rows. Invariance results when
the same notes or set-classes are held in common between two row
forms, either in segments, partitions, or both. If the same set-class appears in two different order position placements in the row (as indicated in the harmonic profile), or if a set-class has symmetrical properties, the potential exists for invariance of pitch-classes in segments. Invariance may also include partitions.
Invariance between segments: In the row from Webern's Symphonie Opus 21, the symmetry of trichord 123 [012] allows for invariant notes < F#,G,Ab > and < Ab,G,F# > between P9 0123 <A,F#,G,Ab> and I5 0123 <F,Ab,G,F#>. The sum of inversion here is 2 (<F#,G,Ab> as <678> maps into <Ab,F,F#> as <876>).
Invariance between a Segment and a Partition: Invariance between segments and partitions means having the same pitch-classes (or set-classes) in a segment and a partition. The example below is from Schoenberg's Variations Opus 31, mm. 24-25 of the Introduction. Here there is segmental and partitional invariance between rows PA and R(IB). In the upper part PA 6789AB <D-C#-G-Ab-B-C> and R(IB) BA9876 <A-Bb-C#-D-G-G#>, there is segmental invariance with notes <D-C#-G-Ab> and <C#-D-G-G#>. The lowest voice in this example is the famous BACH motive, < Bb,A,C,H >. The notes of this motive are invariances: pitch-classes < Bb-A> in partition < 05 > of PA and segment < BA > of R(IB), and pitch-classes < C-B > in partition < 5,0 > of R(IB) and segment < AB > of PA.
rows:
PA: Bb,Fb,Gb,Eb,F,A,D,C#,G,Ab,B,C
R(IB): A,Bb,C#,D,G,G#,C,Fb,Gb,Eb,F,B
PA / R( IB ): 1 3 / 4 2 = <F,Eb,Fb,Eb>
PA / R( IB ): 2 4 / 3 1 = <Gb,F,Gb,F>
PA / R( IB ): 0 5 / 5 0 = < Bb, A, C, H >
Row Order Invariance: Row order invariance occurs when a prime row form is equivalent to either a retrograde or retrograde inversion form. With the row of Webern's Symphony Opus 21, the row is equivalent to its retrograde transposed by 6, below P3 = R(P9) and IB = R(I5). Thus there are only 24 distinct forms of the row (as opposed to the usual 48 forms shown in the row chart), but the labelling depends on the row use.
| P9: | A,F#,G,Ab,E,F,B,Bb,D,C#,C,Eb |
| R(P9) = P3 | Eb,C,C#,D,Bb,B,F,E,Ab,G,F#,A |
| I5: | F,Ab,G,F#,Bb,A,Eb,E,C,C#,D,B |
| R(I5) = IB | B,D,C#,C,E,Eb,A,Bb,F#,G,Ab,F |
Row Chaining: Row chaining occurs when two rows can overlap, such as the overlap on C,Eb between the end of P9 and the beginning of I0.
| P9: | A,F#,G,Ab,E,F,B,Bb,D,C#, | C,Eb = C,Eb, | D,C#,F,E,Bb,B,G,G#,A,F# | I0 |
| I5: | F,Ab,G,F#,Bb,A,Eb,E,C,C#, | D,B = D,B, | C,Db,A,Bb,E,Eb,G,F#,F,Ab | P2 |
Isomorphic Partitions: (Isomorphic means same structure). When the same
partition structure is used for different rows. The result is
invariant set-classes and possibly pitch-classes.
In the passage from Schoenberg's Variations Opus 31, Variation II,
below, row PA is in the upper stave and row I7 is in the lower stave. In each row, the instrumental parts criss-cross, so that there are registral voices and timbral voices. In the upper row PA, the timbral voices are 0567AB Bb-A-D-C#-B-C and 123489> E-F#-Eb-F-G-Ab, and the registral voices are 0589 Bb-A-G-Ab and 123467AB E-F#-Eb-F-D-C#-B-C. In the lower row I7, the timbral voices are 123489 C#-B-D-C-Bb-A and 0567AB G-Ab-D#-E-F#-F, and the registral voices are 05AB G-Ab-Bb-A and 1234ABC#-B-D-C,D#-E-F#-F.
Invariances:
< G,Ab > from PA <
89 > and I7 < 0,5 >,
< Bb, A >
from I7 < 89 > and PA < 05 >,
{E,F#,Eb,F} and {D#,E,F#,F} (the same unordered pitch-class
set) from PA <1234 > and I7 < 67AB >,
{D,C#,B,C} and {C#,B,D,C} (also the same unordered pitch-class set) from I7 < 1234 > and
PA < 67AB >.
I7: 0 5 8 9 <G,Ab,Bb,A>, 1 2 3 4 6 7 A B <C#,B,D,C,D#,E,F#,F>
Derived Row: a row created from the same set-class in consecutive segments, usually either trichords or tetrachords. In a derived row, there are lots of possibilities for invariance because of the repeated set-class. The following derived row is created from repeated [037] trichords. The inverted row form given has invariant trichords with the derived row, {G,Bb,D}, {C,A,F}, {G#,C#,E}, and {D#,F#,B}, but they occur in a different order overall, and the trichords are ordered differently internally.
Axis of Symmetry / Sum of Inversion: In this arrangement of rows associated with Webern's music, Prime and Inversion rows are associated either with fixed middle points, or axes of symmetry, or more generally at a constant sum (mod 12), called the sum of inversion. Webern tends to use even sums, where two notes map into each other and the other 10 notes pair into 5 dyads, such as the note pairs at sum 6: A-A, Bb-G#, B-G, C-F#, C#-F, D-E, Eb-Eb. For an example, in Webern's Symphony Opus 21, movement I, there are three sections. The texture consists of four rows, which pair into inversional canons. One of the canonic row pairs is shown below, with consistent axes and sums: axis A3, sum 6 in section I, axis E5, sum 8 in section II, and axis Eb5, sum 6 in section III. The third section has the same sum as in section I, but with a tritone-related axis.
| Section I: axis A3, sum 6 | Section II: axis E5, sum 8 | Section III: axis Eb5, sum 6 | ||
| P9 (m. 1), P6 (m. 13) | P4 (m. 27), R(P4) (m. 36) | P9 (m. 42), I0 (m. 50) | ||
| I9 (m. 3), I0 (m. 11) | I4 (m. 25), R(I4) (m. 38 | I9 (m. 44), P6 (m. 53) |
Klangfarbenmelodie: Literally "sound color melody." In one aspect of this concept, a voice continually changes timbre. For instance, in Webern's Symphony Opus 21, movement I, the first row, P9 starting on A3, passes from the horn (A,F#,G,Ab) to the clarinet (E,A,B,Bb) to the cello (D,C#,C).
Combinatoriality
Combinatoriality: Rows combined to create vertical as well as horizontal aggregates. The practice is associated with Schoenberg's music, and is developed in Babbitt's music. The prime / inversion pairings in combinatoriality are all at odd sums, where the twelve pcs group into six dyads, such as the dyads at sum 9: E/F, Eb-F#, D-G, Db-G#, C-A, B-Bb.
Hexachordal Combinatoriality: The hexachords of two aligned rows yield vertical
aggregates.
In any row, the first hexachord plus the second hexachord yields
an aggregate; thus any row has the possibility of hexachordal
combinatoriality (HC) with its own retrograde. The two hexachords
are labelled H1 and H2.
(Schoenberg, Variations for Orchestra Opus 31)
H2 ||||||||||||||||||||||||||||||||||||||||| H1
More intriguing is the case in which different row forms combine into vertical aggregates with Hexachordal Combinatoriality.
I7: H2 ||||||||||||||||||||||||||||||||||||||||| H1
Here the first hexachord of PA plus the first hexachord of I7 yield an aggregate. Similarly, the second hexachords of P2 and of I7 yield an aggregate. The same relationship holds between R(P2) and R(I7). Note that the hexachords are not ordered the same way internally in P2 and I7; with Hexachordal Combinatoriality the hexachords are (usually) ordered differently, it is the content that matters.
For a row to have combinatorial hexachords, the two hexachords must be in the same set-class. In the row above, the two hexachords are in set-class [012367]. Of the fifty hexachordal set-classes, there are twenty in which the hexachord and its complement are in the same set-class. Thus, two complementary hexachords are either Z-related (30 = 15 pairs) or they are in the same set-class.
Once you have established that the row hexachords are in the same set-class, to find which row forms are hexachordally combinatorial with a given row form, look on the row chart, comparing inversionally-related rows that begin on notes from the second hexachord of the given row form. (Most of the time you are looking for an inverted row form.) With row PA, look for an inverted form starting on D, then C#, then G -- with G you've found the combinatorial row with I7 starting on G.
Hexachordal Areas: The pitch-class content of hexachords H1 and H2 of the row is called a hexachordal area. The four rows, PA,I7,R(PA), and R(I7) form a combinatorial group with the same hexachordal area, H1/H2. Within the 48 total row forms, there are 12 such areas.
PA, I7, R(PA), R(I7) Area 1
PB, I8, R(PB), R(I8) Area 2
P0, I9, R(P0), R(I9) Area 3
etc.
P9, I6, R(P9), R(I6) Area 12
These hexachordal areas are used to delineate formal areas, with a composition beginning in one area ("home"), leaving that area for others ("contrasting"), then returning ("home").
Of the 20 hexachordal set-classes that allow for HC, the following hexachordal set-classes all combine with themselves in 12-tone rows to yield 12 areas of 4 rows each:
[012346] [012357] [012367] [012458] [012468] [012578]
Hexachord [013679] combines with itself in 12-tone rows to yield
6 areas of 8 rows each: Pn, Pn+6, In, In+6, etc. and their retrogrades.
[013458] [013469] [013579] [013589]
[014568]
[023468]
[023579]
For instance, Schoenberg's Op. 31 row with [012367] hexachords has a combinatorial group of Pn, In+9 and their retrogrades.
All-combinatorial hexachords: Six hexachords have multiple possibilities for hexachordal combinatoriality, including P, I, R(P), and R(I) row forms. The fewer the number of distinct forms of the hexachord, the smaller the number of hexachordal areas, but the greater the number of rows in those areas.
[012345] 6 hexachordal areas of 8 rows each
[023457] 6 hexachordal areas of 8 rows each
[024579] 6 hexachordal areas of 8 rows each
[012678] 3 hexachordal areas of 16 rows each
[014589] 2 hexachordal areas of 24 rows each
[02468A] 1 hexachordal area of 48 rows
Example: [012678] hexachords: 3 areas of 16 rows each:
1) H1/H2 (H2/H1): P0,P3,P6,P9,I2,I5,I8,IB
and retrogrades
2) H3/H4 (H4/H3): P1,P4,P7,PA,I3,I6,I9,I0 and retrogrades
3) H5/H6 (H6/H5): P2,P5,P8,PB,I4,I7,IA,I1 and retrogrades
1) T0/T3 {012678}/{3459AB}
| P0: | 0 2 1 7 8 6 | | | 5 3 4 B A 9 | (and retrograde) |
| P6: | 6 8 7 1 2 0 | | | B 9 A 5 4 3 | (and retrograde) |
| I2: | 2 0 1 7 6 8 | | | 9 B A 3 4 5 | (and retrograde) |
| I8: | 8 6 7 1 0 2 | | | 3 5 4 9 A B | (and retrograde) |
T3/T0 {3459AB} / {012678}
| P3: | 3 5 4 A B 9 | | | 8 6 7 2 1 0 | (and retrograde) |
| P9: | 9 B A 4 5 3 | | | 2 0 1 8 7 6 | (and retrograde) |
| I5: | 5 3 4 A 9 B | | | 0 2 1 6 7 8 | (and retrograde) |
| IB: | B 9 A 4 3 5 | | | 6 8 7 0 1 2 | (and retrograde) |
To create hexachordal combinatorial combinations, take P0 / P3, P0 / P9, P6 / IB, etc.
Stravinsky's Verticals: Stravinsky began composing with serial, then 12-tone techniques in the 1950s, and developed his own technique of creating vertical chords. This example is from the row of "Requium Canticles." The row divides into hexachords, each hexachord is rotated successively, and transposed so that it starts on the first note of the initial hexachord. The verticalities thus consist of a single note, then chords, in each hexachord. The transposition levels are the inverted intervals between the first note and each successive note in each hexachord. Stravinsky does the same operations with an inverted form of the row.
The resulting vertical chords are [0], then a symmetrical pattern of five chords, with the middle chord itself symmetrical. Note that the vertical chords have note duplications.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | |
| 1 | 2 | 3 | 4 | 5 | 0 | 7 | 8 | 9 | A | B | 6 | |
| 2 | 3 | 4 | 5 | 0 | 1 | 8 | 9 | A | B | 6 | 7 | |
| 3 | 4 | 5 | 0 | 1 | 2 | 9 | A | B | 6 | 7 | 8 | |
| 4 | 5 | 0 | 1 | 2 | 3 | A | B | 6 | 7 | 8 | 9 | |
| 5 | 0 | 1 | 2 | 3 | 4 | B | 6 | 7 | 8 | 9 | A |
| t=0 | 0 | 7 | 6 | 4 | 5 | 9 | 8 | A | 3 | 1 | B | 2 | t=0 | |||
| t=5 | 0 | B | 9 | A | 2 | 5 | 8 | 1 | B | 9 | 0 | 6 | t=A | |||
| t=6 | 0 | A | B | 3 | 6 | 1 | 8 | 6 | 4 | 7 | 1 | 3 | t=5 | |||
| t=8 | 0 | 1 | 5 | 8 | 3 | 2 | 8 | 6 | 9 | 3 | 5 | A | t=7 | |||
| t=7 | 0 | 4 | 7 | 2 | 1 | B | 8 | B | 5 | 7 | 0 | A | t=9 | |||
| t=3 | 0 | 3 | A | 9 | 7 | 8 | 8 | 2 | 4 | 9 | 7 | 5 | t=6 |
verticals:
first hexachord: [0], [013569], [012456], [012678], [012456], [013569]
second hexachord: [0], [01348], [01268], [0268], [01268], [01348]
