How to Write a 12-Tone Composition

Twelve-tone is a 20th-century compositional technique created by Arnold Schoenberg.  Rather than setting a piece of music in a diatonic key, the goal of 12-tone music is to use all 12 chromatic pitches equally.  To create a 12-tone composition, follow these steps:

  1. Begin with a 12x12 grid.  Label your grid as in the example below:

      Inverted  
    Original Row
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
    Retrograde
      Inverted Retrograde  
  2. Next, arrange the 12 chromatic pitches in any order you like.  This is your 12-tone row, which will form the basis of your composition.  Fill in the first row of the grid with your 12-tone row.  Check to make sure that you have used each note exactly once.

    Example:

      Inverted  
    Original Row
    C A G D# E F D B A# G# C# F#
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
                           
    Retrograde
      Inverted Retrograde  
  3. Next, calculate the inversion of your row.  Do this by finding the inversion of each interval, and write the inverted row down the left column of your grid.

    Looking at the above example, the first interval is C to AA is a minor third down from C.  The inversion of this would be a minor third up from C, which is E-flat.  While not technically necessary, I like to keep all the accidentals the same; that is, using all sharps or all flats.  I find that this makes proofreading easier, and helps to avoid errors.  Since I've used sharps in my original row, I'll use D-sharp in my inverted row, which is enharmonically equivalent to E-flat.  (You can also write down both spellings of the chromatic pitches, such as D#/Eb.)

    The next interval in my original row is A to GG is a major second down from A, so in the inversion, you would calculate a major second up from D-sharp (E-flat), which is F.

    The resulting inverted row is as follows:

      Inverted  
    Original Row
    C A G D# E F D B A# G# C# F#
    D#                      
    F                      
    A                      
    G#                      
    G                      
    A#                      
    C#                      
    D                      
    E                      
    B                      
    F#                      
    Retrograde
      Inverted Retrograde  

    Double-check your inverted row to make sure you have used each note exactly once.
  4. Fill in your grid by transposing your 12-tone row into each key listed down the left column of the grid.

    Example:

      Inverted  
    Original Row
    C A G D# E F D B A# G# C# F#
    D# C A# F# G G# F D C# B E A
    F D C G# A A# G E D# C# F# B
    A F# E C C# D B G# G F A# D#
    G# F D# B C C# A# G F# E A D
    G E D A# B C A F# F D# G# C#
    A# G F C# D D# C A G# F# B E
    C# A# G# E F F# D# C B A D G
    D B A F F# G E C# C A# D# G#
    E C# B G G# A F# D# D C F A#
    B G# F# D D# E C# A# A G C F
    F# D# C# A A# B G# F E D G C
    Retrograde
      Inverted Retrograde  

    This is your palette.  Reading from left to right, you have your original row in all 12 keys.  Reading from top to bottom, you have the inversion of your row.  Reading from right to left gives you the retrograde of your row, and reading from bottom to top gives you the inverted retrograde.

    Again, you can spot-check your work by making sure any given row contains exactly 12 distinct pitches.
  5. Using the palette that you have created, write your composition.  The following rules apply:

    1. Select any row in your palette:  original, inverted, retrograde, or inverted retrograde.
    2. Once you begin a row, you must follow it to completion:  you must play all the pitches in order, you may not skip any pitches, and you may not repeat any pitches.
    3. Notes may occur in any octave any may last any duration.  You may begin two or more notes simultaneously, as long as they occur sequentially in the row.
    4. Any number of rows may be played concurrently.

    The following is a simple example of a composition based on the above palette.  Click the Play button at the top of the page to listen to the example.

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Comments
Beautiful composition! And great explication of this technique, thank you!
Posted by Gregory Tippett on Friday, March 17, 2017
Fantastic! Thank your for the post Carolyn, and the spreadsheet calculator Brent. It has helped me to understand the tonal matrix with greater clarity. Prior to reading here, the tonal matrix was just a mess of meaningless pitches. Now onto my uni assignment. Peter
Posted by Peter on Saturday, March 4, 2017
I have posted spreadsheet that calculates the matrix on my website, and it can be downloaded, used and distributed for free. See ... www.carolinaclarinet.org/trmc.xlsx
Posted by Brent Smith on Friday, January 6, 2017
I had a visitor ask, "Isn't your first interval going up a Major 6th?"

Yes, C to A can either be a minor 3rd down, or a Major 6th up.  It doesn't matter which one you use.  On the inversion row, you just use the same interval going the opposite direction.  So a minor 3rd up from C is E-flat, and a Major 6th down from C is also E-flat.  It works out the same.
Posted by Carolyn on Monday, November 7, 2016
Hi Michelle,
For your palette, you get to choose what order to put the notes in for your original row. That's part of your composition. Then, when you create your music, you can choose whatever row you want, in any direction and in whatever key you like, and start it at any time you like. And you can have as many rows playing simultaneously as you please.
Posted by Carolyn on Thursday, July 21, 2016

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