| To supplement pingouin's description above: the basis of any twelve-tone composition is the so-called tone row, a sequence of notes of any duration in any octave that run through the twelve notes of the chromatic scale without repetition. Once established in a piece, the tone row becomes the building block for the rest of the composition. Typically the tone row is inverted temporally, that is, played in reverse, as well as inverted tonally, that is, "flipping" the intervals of the tone row (see music theory.)
More advanced compositions involve transposing the tone row into a different key and applying all of the inversions above, as well as layering the original tone row, transposed tone rows, inverted tone rows and transposed-and-inverted tone rows on top of one another.
This is hard to do. Very very hard. Only the most skilled composers can even attempt a full length twelve-tone work. What's more interesting is that it doesn't sound all that . . . musical. To the classically trained ear (in fact to most people's ears) twelve-tone sounds like crap at best, random noise at the worst. It is the preoccupation of those to whom western music is boring and bland.
Why would a twelve-tone composer want to put him/herself through such arbitrary and rigid musical contortions -- just to produce music that most people don't even like?
The point of having such a rigid and esoteric set of rules is mainly to challenge the composer to create something cogent and beautiful despite the limitations imposed on him. The idea is that the artistic essence of the composer's musical ideas should be able to transcend the medium. The analogy I've heard is this: take a gifted sculptor who works in marble and bronze. Give him a pile of mashed potatoes and a rusted out VW bug and tell him to create something beautiful. A sculptor that truly understands his art will be able to do this -- thus, a composer should be able to create a meaningful, evocative work using the constraints of twelve-tone music based on his or her understanding of what it means for music to be beautiful.
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