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Twelve-tone series in transposed, inverted and retrograde form

Tone Series Forms - Version 1.2

See table at centre for transposed (rows), inverted (columns) and retrograde forms of a tone series. Click on the [Update] Control Button below to initialize the table for the default Tone Series. [More user guidelines >>]

Tone Series Input
Pitch 1 to 6:  
Pitch 7 to 12:  
T\I I0 I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11
Interval-class content
01,112,103,9 4,85,76
Control Buttons
Message Window

Guidelines for the user

This JavaScript GUI lists all the forms of a tone series. It will handle any series between 2 and 12 pitches (twelve-tone series), and determine the transposed, inverted and retrograde forms of this series. It will also analyse the melodic interval-class content. This GUI is divided into a number of sections:

  • The Tone Series Input section at the top consists of a set of selection lists, grouped into two hexachords on two lines. Each selection list contains note names from the chromatic scale (the collection of 12 pitches at equally-tempered semitone tuning). Note names represent their enharmonic equivalent, C#=Db, F#=Gb, etc. When loading the page there is a default 12-tone series, that may be edited; choose any pitch series between 2 and 12 tones. Doubled and repeated notes are allowed. Close the series with the terminator symbol (-), that is present in the selection list from the 3rd pitch onward. A series may contain more than one terminator sign. The program will look for the leftmost terminator symbol.

  • The table at the centre will show all possible forms of a tone series, either as numbers or as note names (depending on the control settings). Reading from left to right each row represents a transposed (T) version of the input series; the label in the first column indicates the transposition, T0 is the original series (also called the prime form), T7 is the transposition at 7 semitones higher (a perfect fifth), etc. Reading the rows from right to left yields the retrograde form of the transposed series. The 2nd column, with the label I0, contains the inversion (I) of the original series. Columns 3 to 13 contain the transposed, inverted forms of the tone series. And finally, reading a column from bottom to top yields the retrograde inversion (RI) form of the series. For a twelve-tone series the table will display all the variants of the original series, for less than 12 pitches, there will be empty spaces in the table. Note the symmetry along the diagonal of the table (grey table cells). The bottom row (labeled Int, with coloured background) of the table shows the interval series, indicating the melodic step or leap (expressed in the number of semitones) to go from one pitch to the next in the tone series.

  • The Interval-Class Content table will list the number of melodic intervals (steps and leaps) in the tone series. Note, that for an N-length tone series, there will be a total of N-1 counts in the table (see the Pitch-Class Set GUI on this Website for a list of the total interval content, i.e., for all pitch combinations, for a given set). The column labels indicate the interval type, i.e., 0 = repeated note; 1,11 = 1 semitone up (minor second) or down (equivalent to 11 semitones higher, the major seventh); 2,10 = 2 semitones up (major second) or 10 semitones up (minor seventh); 3,9 = 3 semitones up (minor third) or 9 semitones up (major sixth); 4,8 = 4 semitones up (major third) or 8 semitones up (minor sixth); 5,7 = a perfect fourth or fifth up/down; 6 = a leap of an augmented fourth / diminished fifth.

  • The Message Window will report the status of the GUI. It will advice the user about what to do next.

  • The Control Buttons section has 4 buttons. After editing a tone series in the input selection lists, pressing the Update the table button will process this series and fill the table with the appropriate elements. The table will display the forms of the series either as note names (C,C#,...,B) or as numbers (0,1,...,11). Switching between these representations is done by pressing the Use note names or Use note numbers buttons. Finally, pressing the Reset the table button will reload the default twelve-tone series in the selection list at the top.

  • Depending on the status of the GUI, some buttons may be inactive.

Here's a few suggestions for tone series that you might want to consider:

  • The default twelve-tone series on this page, D-C#-A-Bb-F-Eb-E-C-Ab-G-F#-B, is the prime form from the 4th String Quartet, Mvt. 1 by Schoenberg. The first hexachord, P0 = D#-C#-A-Bb-F-Eb, through inversion and transposition I5 yields the pitches from the 2nd hexachord (differently ordered, though).

  • The Alban Berg Violin Concerto is based on the twelve-tone series G-Bb-D-F#-A-C-E-G-B-C#-Eb-F. Study the interesting character of this interval series.

  • Another composition by Berg, the Lyrische Suite, Mvt. 1, uses an all-interval series, based on the sequence of two all-interval hexachords, F-E-C-A-G-D and Ab-Db-Eb-Gb-Bb-B.

  • Anton Webern's Quartet Opus 22, Mvt. 1 uses the prime form C#-Bb-A-C-B-Eb-E-F-F#-Ab-D-G.

  • In George Perle's book Twelve-Tone Tonality, 2nd Edition, (University of California Press, 1996, ISBN 0-520-20142-6), many aspects are discussed around the series C-G-F-D-Bb-A-D#-E-G#-B-C#-F#.