Composing Using Systems

A closeup of two wooden gears interlocked with one another. The light wood gears are held in place with black screws. Gears in the Astronomical Clock in the Copenhagen City Hall. Older clocks use a complicated system of gears and springs to turn the hands at the appropriate speed.
(Stefan Weihs | CC BY-2.0)

The process of writing music is usually a deeply human act, involving emotion, lived experience and aesthetics. Some composers have experimented with removing the human component of composition to various degrees, by devising systems in which music is created based on non-human systems.

Serialism

Serialism is a system of musical composition in which a series of musical elements — a set of different pitches, for example — is used as a seed for a piece of music. A serialist system will usually have specific rules for how music is generated from the seed, specifically which aspects of the composition are prescribed by the system and which are left to composer discretion. In most cases, serialist systems are designed to propogate characteristics of the seed through the entire piece.

Figure 1: Five Bagatelles, Op. 49, a 1962 set of piano works by English composer Elisabeth Lutyens. Lutyens used a serial system of her own design to create a colorful atonal texture.

Twelve-Tone Row

Perhaps the most well-known serialist system was devised in 1923 by Austrian composer Arnold Schoenberg. Though the composer labeled the system a "method of composing with twelve tones which are related only with one another," theorists commonly refer to this system as twelve-tone row composition.

Twelve-tone row composition serializes pitch, and is a means of creating a purely atonal composition by ensuring that all twelve notes of the chromatic scale are used equally. The process is built upon a single series which contains all twelve tones of the chromatic scale only once. The order of this series is determined by the composer, but should avoid any tonal structures like major or minor triads.

A sequence of twelve notes: C sharp, F sharp, B, C, G, A, D, A sharp, D sharp, G sharp, E and F, written in musical notation and in a single-row, twelve-cell table.
Figure 3: A twelve-tone row. In the twelve-tone row system, enharmonics are equal, and calculations are sometimes easier if only one type of accidental is used.

The composer then constructs a composition using variations of the original or prime row: an inversion of the row, in which the component intervals are inverted to create a new, similarly atonal, row; a retrograde of the row, consisting of the notes in the original played in reverse order, a retrograde inversion, a reversal of the inverted form, and all possible transpositions of each of these four forms.

Four sequences of twelve notes. Prime for is C sharp, F sharp, B, C, G, A, D, A sharp, D sharp, G sharp, E and F. Inversion is C sharp, G sharp, D, sharp, D, G, F, C, E, B, F sharp, B flat and A. Retrograde is F, E, A flat, E flat, B flat, D, A, G, C, B, F sharp and C sharp. Retrograde inversion is A, B flat, F sharp, B, E, C, F, G, D, D sharp, G sharp and C sharp.
Figure 5: Four versions of the same row: prime, inversion, retrograde, and retrograde inversion.

Theorists find it useful to determine and label these 48 variations of the row using a matrix, where the prime row appears along the top, the inversion appears down the left side, and transpositions of each are used to complete the diagram. Rows are labeled with the letters P, I, R, and RI, along with a number representing the transposition from the original.

A twelve-by-twelve matrix of note names, with the row from Figure 2 across the top, and the inversion from Figure 3 down the left side; each row shows a transposition of the top line, starting on the appropriate note from the inversion. The completed row shows C sharp in a diagonal line from top left to bottom right.
Figure 6: The completed matrix. Rows are numbered based on the number of half-steps transposed from the original row or inversion; thus, P-1 is the fourth row from the top, starting on D; RI-1 is the eighth column from the left, read upward from the F# at the bottom.

The twelve-tone row system uses the following guidelines to ensure atonality, preserve the character of the original row and to facilitate future analysis:

  • The composition must begin with a statement of P-0, but thereafter may use any rows in any order. It is not required to use all the rows in the matrix.
  • The notes within a row must be played in the prescribed order. Chords may be created from consecutive notes in a row.
  • Notes may be repeated before moving on to the next note in the row.
  • Rows may overlap one another, or pass between different instruments or ranges.
  • Only complete rows may be used.

While the use of pitches are subject to these rules, other elements of music, such as rhythm and dynamics, are not; in fact, in Schoenberg's own works, musical elements other than pitch are similar to the works of Classical or Romantic composers.

Measures 1 through 5 of Arnold Schoenberg's Klavierstuck, Op. 33a, showing row analysis. Measure one comprises row P-0 in three four-note chords, and measure two does the same for row RI-5. In measures 3 through 5, the right hand plays row RI-5 simultaneously with the left hand playing row R-0.
Figure 8: Austrian composer Arnold Schoenberg's 1929 Klavierstück, Op. 33a, showing twelve-tone row analysis.

Integral Serialism

The expansion of serialism to include elements other than pitch is commonly referred to as integral serialism. Pieces in this genre might serialize rhythms or note lengths, dynamics, or articulations. Systems for these elements may be simple, such as rotating through a certain set of items, or may involve the creation of matricies and other complex calculations.

Measures 1 through 5 of Pierre Boulez' Le Marteau sans maître, for flute, vibraphone, guitar, and viola. Notes are scattered rhythmically and across each instrument's range with no apparent connection between parts, and with widely varying dynamics and articulation.
Figure 9: Measures 1–5 of French composer Pierre Boulez' 1955 piece Le Marteau sans maître. This piece, a setting of the surrealist poetry of René Char, utilizes matrix-based serial systems for not only pitch but also rhythm, dynamics and articulations.

Generative Music

In serial composition, the system generates a catalog of elements, but the composer still decides which elements to include and how to include them. In generative music, the composer designs a starting configuration and system by which that configuration changes, so that the composition proceeds only according to the system.

A chart describing the construction of Arvo Part's Fratres, reproduced in the associated BRF file.
Figure 10: A chart illustrating Swedish theorist Linus Åkesson's analysis of the generative process of Estonian composer Arvo Pärt's 1977 work Fratres. The piece is written in nine sections, with each section playing a set of chords in the order described in the upper left. The chords themselves are determined from two cycles of notes: the outer voices use a descending harmonic minor scale, and the inner voice uses notes from an A minor triad. Each chord pattern starts on a note and proceeds counter-clockwise around the cycle, with the starting pitches moving two steps counterclockwise for each cycle.

Stochastic Music

When a composer combines a generative process with an element of randomness, the result is called stochastic music. Because the random values are subject to a specific generative process, this type of system effectively deals with probabilities, and often incorporates Markov chains — systems where the probabilities of one event are based upon the result of the previous event. For example, a piece where a random pitch is independently selected for each note is aleatoric, but a piece where each note moves by a random interval from the previous note is stochastic.

One of the most notable composers of stochastic music — and, in fact, the person who coined the term itself — was Greek composer Iannis Xenakis, who wrote several pieces based upon the statistical probabilities of natural processes: his work Pithoprakta, for example, is based on Brownian motion, the statistical calculations which govern the movement of molecules in a gas.

Figure 11: Pithoprakta, a 1956 piece by Greek composer Iannis Xenakis, with individual parts illustrated in a graphical format.

Composing with Systems: Summary

  • Serialism is the use of a series of musical elements to generate a composition.
  • Twelve-tone Row is a system for serializing pitch created by Arnold Schoenberg for the purpose of ensuring complete atonality.
    • The first step in the system is to create a prime row of all twelve pitches which is free of tonal implications.
    • Other versions of the row can be created by inverting each interval (inversion), reversing the order of the notes (retrograde), or combining these two techniques (retrograde inversion).
    • Further versions of the row can be created by transposing each of the four versions to other pitches.
    • All versions of the row can be shown in a matrix, with the original row at the top and the inversion down the left hand side.
    • Compositions are created by combining complete rows in any order. Chords can be created from consecutive notes, rows can overlap or pass between parts, and notes can be repeated before moving on to the next pitch in the row.
  • Integral serialism involves the serialism of multiple musical elements, such as dynamics, rhythm, or articulation.
  • Generative music is the composition of a piece by creating an initial state and a system by which parts follow set rules.
  • Stochastic music is a type of generative music that incorporates randomness, and often includes the use of Markov chains.

Exercises

Exercise 1: Analyzing Arnold Schoenberg's String Quartet No. 4