What Is A 12-Tone Matrix?

Who Invented It?

Invented by 20th century comopser Arnold Schoenberg, the 12-tone matrix is a compositional tool used by composers to produce 12-tone music. ("Twelve-tone technique") The technique's purpose is to ensure that the consequent music is atonal, meaning that no particular pitch occurs more frequently than another and there is no perceptible key. ("Twelve-tone technique") It serves to emphasize the intervallic relationships created by the composer's matrix rather than relying on the intervallic relationships present in tranditional tonal music. ("Twelve-tone technique") 12-tone music is considered a form of serialization and therefore relies on the very particular ordering of the 12 pitches in the octave. ("Twelve-tone technique") It also relies on the notion of octave equivalence, meaning that the pitches within different octaves are considered equivalent. ("Twelve-tone technique")

black and white portrait of Arnold Schoenberg

(Homolka, 1948)

The 12-Tone Row:

The 12-tone matrix is based upon a single 12-tone row. ("Twelve-tone technique") This is an ordered sequence of the 12 pitches present in an octave. ("Twelve-tone technique") Twelve tone rows are typically created with very particular intervallic relationships in mind. ("Twelve-tone technique") The 12-tone matrix shows all of the possible transformations of a particular row. These transformations are then used to create 12-tone music. ("Twelve-tone technique") There are four possible transformations that can be performed on a row:

  1. Transposition: Each note in the row is moved up or down by a consistent interval. The intervallic relationships between the notes of the row are preserved. ("Twelve-tone technique")
  2. Inversion: The intervals between each note in the row are reversed. ("Twelve-tone technique")
  3. Retrograde: The notes of the row are reversed. ("Twelve-tone technique")
  4. Retrograde Inversion: The row is inverted and then reversed. ("Twelve-tone technique")

Possible Row Transformations:

Prime Row:

An example of a 12-tone row.
("Twelve-tone technique")

Retrograde: the prime row in reverse order:

The previous example with the notes in reverse order.
("Twelve-tone technique")

Inversion: the prime row with the intervals inverted:

The previous example with the intervals inverted.
("Twelve-tone technique")

Retrograde Inversion: the inverted row in retrograde:

The inverted version of the row in retrograde.
("Twelve-tone technique")

Rules of the 12-tone matrix:

The adjacent diagram shows the 12-tone matrix along with all of the possible row transormations. The procedure for making a 12-tone matrix by hand is as follows:

  1. Make your prime row the first row of the matrix. ("Create a Twelve-Tone Melody With a Twelve-Tone Matrix")
  2. Populate the first column of the matrix by calculating the inversion of the prime row. ("Create a Twelve-Tone Melody With a Twelve-Tone Matrix")
  3. Populate the remaining rows of the matrix by transposing the row according to the first note populated in step two. ("Create a Twelve-Tone Melody With a Twelve-Tone Matrix")
A 12-tone matrix with labels on the rows and columns.

("Music Of The 20th Century")

Additional Information:

Sources:

Wikimedia Foundation. (2021, October 9). Twelve-tone technique. Wikipedia. Retrieved November 13, 2021, from https://en.wikipedia.org/wiki/Twelve-tone_technique.


Richardson. (n.d.). Lessons ten, eleven & twelve - serialism. Music Class Hub. Retrieved November 13, 2021, from https://musicclasshub.weebly.com/lessons-ten-eleven--twelve---serialism.html.


Hutchinson, R. (n.d.). Music theory for the 21st-century classroom. Twelve-Tone Technique. Retrieved November 13, 2021, from https://musictheory.pugetsound.edu/mt21c/TwelveToneTechnique.html.


Oglesby, & Instructables. (2017, November 6). Create a twelve-tone melody with a twelve-tone matrix. Instructables. Retrieved November 13, 2021, from https://www.instructables.com/Create-a-Twelve-Tone-melody-with-a-Twelve-Tone-Mat/.