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Today we’re in a dodecaphonic state of mind…
How about a glance at what Arnold Schoenberg was thinking:
He wanted a “pantonal” system, where no one note had any more importance than any other note. The best way he found to achieve this was to use Dodecaphonic Tone Rows. The result would of course be “atonality” (though he preferred the term pantonality which means “all tonal” instead of “non tonal”).

Many composers have achieved this in many different ways and Schoenberg’s was just one way. However, his was one of the (if not the) earliest and he took the trouble to codify what he was doing. He also had students who learned from him and modified the system for their own purposes. As such, Schoenberg’s system is widely studied and emulated.

A Tone Row is a series of 12 notes (it’s also called a 12 Tone Series or 12 Tone Row, etc.). It has one, and only one of each of the 12 chromatic notes: A A# B C C# D D# E F F# G G#

Since part of what S wanted to do was avoid the “cliches of tonality”, he ordered the notes in the row so that traditional tonal elements were avoided – scalar segments, outlining of triads or diminished 7th chords, etc.

This led to a series of tones, or a Tone Row that would be the basis for an entire piece, or movement, or large section of a piece, etc.

His method of implementing the Tone Row was that each note in the series need to be sounded in order and be heard only once until the completion of the entire Row. He did allow immediate repetitions (reiterations) like C-C-C-A but after that, you wouldn’t hear the C again until the Row was completed. He also allowed for simultaneous occurrences like a chord could be made out of the first 4 notes of the series – and even though you can’t tell which of the 4 was first, the remaining 8 notes had to sound before any of those first 4 were heard again.

By doing this, S felt that the lack of repetition would prevent any one tone or harmonic combination from getting undue emphasis.

One issue is though that using a single row can get quite boring after a while. It’s a little more complex than simply “hearing the same notes in the same order over and over” but essentially, that’s what happens.

So, just like Tonal composers might write a piece in the key of C Major then modulate to G Major and back, S would used either different Tone Rows or “transformations” of a Tone Row.

Obviously, when you write a piece of music that throws out all of the conventions of traditional tonal music, what is left to tie it together? S felt that if he used Tone Rows that were “related”, there’d be much more internal cohesion in the piece. So just like C Major and A minor are related, he found “related” versions of his “primary” Tone Row to use.

So think about this – C Major is relative to A minor. In fact, they have the same notes, just starting on a different one. So even though the notes are the same, the “focus” is on a different note. Now S didn’t want to focus on any particular note, but the idea of two Tone Rows being related sort of gave each transformation a “reason for being in the piece”.

You can transform a Tone Row in four ways: Transpose it, Retrograde it, Invert it, and Invert AND Retrograde it (and of course you can transpose the inversions and retrogrades, etc.).

By putting the Tone Row (the original being called the Prime) in a 12×12 grid called a Matrix, one could calculate all of the related row forms.

Some Tone Rows will produce 48 distinct forms. All those fancy letter/numbers just refer to where it is in the grid.

P0 is the Prime Row. P2 would be the Prime Row transposed up 2 semitones. R0 would be the Retrograded Row, and R5 would be the Retrograded Row transposed up 5 semitones. Etc.

However, many composers became interested in Row forms the produced shared elements, or even a limited number of Row Forms. A simple example is that if your Row is a chromatic scale, then the Retrograde Inversion is also just a chromatic scale. So there are really only 24 forms of this scale (P with 12 transpositions, and I with 12 transpositions).

So sometimes a Row might be chosen for a piece because of a particular characteristic – they all end with the same 3 notes for example.

So it’s rare for a composer to use all 48 forms – more typically they use 3 or 4 (and to some degree transpositions are often treated like a modulation, rather than change of mode for example).

You can download a 12 Tone Matrix Generator at