A *set class* contains all the pitch-class sets (in normal order) that are related by transposition, and all the transpositions of the inversion of the set. For example, start with the randomly chosen set, [9,1,2]. The first twelve sets show all the transpositions of [9,1,2].

[9,1,2]

[10,2,3]

[11,3,4]

[0,4,5]

[1,5,6]

[2,6,7]

[3,7,8]

[4,8,9]

[5,9,10]

[6,10,11]

[7,11,0]

[8,0,1]

Now you add all the transpositions of the previous twelve sets.

[2,3,7]

[3,4,8]

[4,5,9]

[5,6,10]

[6,7,11]

[7,8,0]

[8,9,1]

[9,10,2]

[10,11,3]

[11,0,4]

[0,1,5]

[1,2,6]

The twelve transpositions and the twelve transpositions of the inversion form a single *set class*. We refer to a set class by its *Prime Form*. The prime form a set always starts on pitch class 0, and travels clockwise or counter-clockwise around the clock face by virtue of which direction is the most compressed and compact. In the above case, the inversion [0,1,5] is more compact than [0,4,5], so [0,1,5] is the prime form of this set class.

Prime forms are usually written in parentheses, without commas. In this system, use “t” and “e” for 10 and 11. Using this system, the prime form of the above set class would be written (015). Using this system we can distinguish between a specific pitch-class set as it appears in a piece of music and the set class to which it belongs.

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