Basics of Dodecaphonic Composition

For more information on composing with twelve notes,
it is best to search out information on the credited founder
of the concept, Arnold Schoenberg.

Here is a very brief (but adequate for the Fairy Lights exercise )
description of the concept:

Basically, Schoenberg wanted to maintain maximum chromatic content
and thus avoid altogether any references to tonality.

Eventually he settled on the idea that one should pick pitches
one at a time, and not repeat one until all twelve of the chromatic
scale have been used.
This creates a virtual object known as a row.

An example of a row is the one from 'Fairy Lights':
E G# A C# D F G C B F# D# A#

This contains every chromatic pitch-class, but only once.
Its inversion (that is, each interval inverted in turn -
major third up becomes major third down)
takes the form:
E C B G F# D# C# G# A D F A#

Note that I always use sharps-notation.
This is not universal, but it is VERY useful when using computers,
wherein the closest to a sharp is the well-matching # sign,
and the closest to a flat( without serious character-mapping )
is the lower-case 'b', which causes confusion.
Also, it is very clear whether or not one has re-used pitches in
the row - it would be easy to miss a collision of F-sharp and
G-flat, which in equal temperament and dodecaphony can be considered
the same thing.

For the score of 'Fairy Lights', however, some of the keys I
have chosen are flat, simply because while A-Flat is a perfectly
everyday key, G Sharp Major is highly eccentric, and need not be
employed.(The piece observes the equal temperament of the piano.)

I am sure there are many more resources regarding the more traditional
application of twelve-note method elsewhere on the internet,
but this brief description will be enough to familiarise the student
with the concept and allow meaningful understanding of the piece.