I am completely new to the concept of Serialism. A music theory workbook I have only mentions about the concept in short, with an example of Schoenberg's Variations for Orchestra. It also mentions about the 4 types of tone rows: Original, Retrograde, Inverted and Inverted Retrograde, briefly stating the definitions only.

This has not helped me understand the concept in detail at all. At the end, I want to be able to write a 12 or 16 bar serial melody using Schoenberg's tone row. From where should I start?

  • 1
    Btw, if your library has "Analytic Approaches to Twentieth-Century Music" by Joel Lester, that will bring you up to speed with the practicalities of 12-tone music in about 60 pages. And it has a list of especially interesting 12-tone rows to use in your exam and dazzle your teachers with :-) Jun 7, 2019 at 5:15

3 Answers 3


Serialism means that musical material is derived from a series (hence "serial") of said musical material; this series is also commonly called a row. In early serialism, composers used the serial concept to determine the pitch content of the work. Most famously, the composers of the "Second Viennese School" (Schoenberg, Berg, and Webern) were proponents of twelve-tone serialism, where the given series were twelve pitches long.

The idea was that tonal music gave ultimate prominence to one pitch: the tonic. But these new composers famously sought "a democracy of tones" where all pitches were equal, with no one pitch hierarchically more important than the others. One way that they realized this democracy was by forcing themselves to use all twelve pitches before repeating any; this is what a twelve-tone row helps one to achieve.

In order to construct a twelve-tone row, just make a listing of all twelve pitches, using each exactly once and never repeating any. Here's the famous row from Berg's Violin Concerto; since it's the original row, we also call this the prime form of the row:

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Remember that this row is only a list of pitches; so we could realize this row by writing something like this:

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But we could also realize it with something like this:

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The example above shows the vital concept of octave equivalence. In short, octave equivalence states that a pitch need not be played in a given octave; as long as the opening G is some G somewhere in musical space, it is a G. (This gets into the distinction of pitch and pitch class, but that's not vital here.)

From that prime form of the row, there are four cardinal transformations used to change the original row into something else. First, we can transpose it to begin on any other pitch:

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We can also retrograde it, which just means we write it backwards:

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We can invert it, which means we use the intervals of the prime form of the row, but we go in the opposite direction (notice that the prime form of the row goes up by thirds, whereas this one goes down by thirds):

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We can also retrograde invert it, which is a combination of the two above transformations; the following row is just the inverted row form retrograded.

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Lastly, we can combine any of these transformations; we can transpose an inverted row form, for instance. Composers and theorists often use a tool called a twelve-tone matrix to determine all of these row forms, but you can discover that on your own.

Once a composer uses a row form, they're free to either continue using that row form or moving to a transformation of that row form. But typically, composers will stick with one prime form for an entire work (or movement), using only transformations of that one row; they won't, for instance, write a completely new prime form every time they exhaust all twelve pitches.

This was twelve-tone serialism, but you can also use serialism for fewer than twelve pitches, as Stravinsky famously did in his In Memoriam Dylan Thomas. You can also serialize aspects of rhythm. Other composers sought "complete serialism" (also called "total" or "integrated" serialism) where the serial process governed everything in the music: timbre, duration, dynamics, articulation, instrumentation, you name it. These composers belonged to what we call the "Darmstadt school" and include figures like Boulez and Stockhausen.

As for Schoenberg's Variations for Orchestra, here are the four row forms; try to create lines using these (or transpositions of these) and see what you come up with!

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    I wish I could upvote twice. Btw, for total serialism, check out Messiaen's "Mode de Valeurs et d'Intensitées" (1949) and Boulez's "Structures I" (1952) Jun 6, 2019 at 13:07

As a follow-up to Richard's answer, I'd like to give an example of the way a row or series is used in a composition for several instruments, and explain how the series to be used for a composition was chosen.

Here is a simplified version (omitting dynamics and articulation) of the opening measures of Schoenberg's String Quartet No.4 op.37:

%%score V1 V2 VA CL
V:V1 clef=treble name="Vn.I"
V:V2 clef=treble name="Vn.II"
V:VA clef=alto   name="Viola"
V:CL clef=bass   name="Cello"
% 1
[V:V1] "P0""_0"=D4 "_1"^C4 | "_2"=A,A,A,"_3"_B, "_4"=F2 "_5"_E2  | "_6"=EEE"_7"=c "_8"_B4   | ("_9"=G4G)"_10"^FF"_11"=B |(=B8 | B3) z "I5"z4 | "_0"=gz/2g/2 g z z2 "_1"_a z | "_4"=e z z2 z2 "_6"=fz/2f/2 | f z z2 z4 |
[V:V2] z2 "_3"_B, z "_8"_A, z "_10"^F z | z2 "_6"=E z "_11"=B, z "_1"^C z | z2 "_9"=G z "_0"=D z "_5"_E z | z4 "_1"^C2 C2 |^C2 "_5"^D2 (D4 | D) z "_6"=E z z "_0"=G"_1"_A"_2"=c | ("_3"=B4 B)("_4"=e"_5"^F2) | z "_6"=F"_7"=A"_8"^c ("_9"=D"_10"_e') (e'2 | (e'2) "_11"_B6) |
[V:VA] z2 "_5"^D, z "_6"=E, z "_9"=G, z | z2 "_7"=C, z "_10"^F, z "_2"=A, z | z2 "_11"=B, z "_2"=A, z "_4"=F, z | z4 "_0"=D,2 D,2 |=D,2 "_4"=F,2 (F,4 | F,) z "_8"_A, z z4 | "_2"=C,z/2C,/2 C, z z2 =c z | "_3"=B z z2 z2 "_8"^C,z/2C,/2 | C, z z2 "_9"=Cz/2C/2 C z |
[V:CL] z2 "_4"=F,, z "_7"=C, z "_11"=B,, z | z2 "_8"_A,, z "_9"=G,, z "_0"=D,, z | z2 "_10"^F, z "_1"^C, z "_3"_B,, z | z4 "_2"=A,,2 A,,2 |=A,,2 "_3"_B,,2 (B,,4 | B,,) z "_7"=C, z z4 | [K:C alto 4] "_1"_Az/2A/2 A z z2 "_0"=G, z | "_5"^F z z2 z2 "_7"=Az/2A/2 | A z z2 [K:C bass] "_10"_E,,z/2E,,/2 E,, z |

From the start up to the middle of measure 6, the first violin plays the non-transposed prime version of the row (often indicated as P0). After that, the second violin plays the inversion of the row transposed up by 5 steps (often indicated as I5).

%%score V1 V2
V:V1 clef=treble name="P0"
V:V2 clef=treble name="I5"
% 1
[V:V1] "_0"d "_1"^c "_2"A "_3"_B "_4"F "_5"_E "_6"E "_7"c "_8"_A "_9"G "_10"^F "_11"B
[V:V2] G _A c B E ^F F A ^c d _e _B

While P0 is being played by the first violin, the other three instruments do not each play a version of the row. Instead, they work together so that every measure (or group of measures 4, 5 and 6) contains all twelve pitch classes. If we number the pitch classes in the rows from 0 to 11 (as is conventional), these are the notes played by each instrument:

      P0      |         |          |        
V1    0  1  2   3  4  5    6  7  8   9 10 11
V2    3  8 10   6 11  1    9  0  5   1  5  6
VA    5  6  9   7 10  2   11  2  4   0  4  8
CL    4  7 11   8  9  0   10  1  3   2  3  7

You'll see that these zones don't strictly correspond to measures in the score; e.g. the 2 is not played in measure 1 but at the beginning of measure 2.

Starting from the middle of measure 5, the second violin plays I5, but now the other instruments work in a different way; they combine to form chords of adjacent notes 0-1-2, 3-4-5, 6-7-8 and 9-10-11. So the I5 series is played twice, once by the second violin, and once by the other three voices combined.

V1                   1  4           6
V2    0  1  2  3  4  5  6  7  8  9 10 11
VA                   2  3           8  9
CL                   0  5           7 10

As you can see, the idea of using the row or series isn't limited to the horizontal dimension, but is combined with the vertical in different ways.

This brings us to an important point: the rows used by Schoenberg are not chosen randomly. He uses rows that have interesting properties when combining different versions, and one of those properties, "hexachordal combinatoriality", can be seen in the example above. If you look at P0 and I5, you'll notice that the first 6 notes of P0 are the same as the last 6 notes of I5, and vice versa. This means e.g. that if two instruments play these two versions of the series, each group of six notes also contains all twelve notes.

Webern often used rows that were themselves inversions and transpositions of a shorter row of 3 or 4 notes. Again, this is done so that the combinations of different versions of the row would have interesting properties, e.g. that they contain the same adjacent notes (like the adjacent D-C#-A in P0 and A-C#-D in I5 in the example above).


Richard has given an excellent explanation. I just wanted to point out that the basic serial techniques can be applied not only to tones in your row, but to portions of your row.

For example, given the row:

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

you can segment the row, and apply the techniques to parts of it. Split it into thirds and retrograde the middle four tones and you get:

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

Split it into four segments ABCD and reorder them BADC and you get:

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

There are countless ways you can use a prime row to generate variations.

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