To follow up on pitch-class sets, previously talked about in Part 1:

First, a clarification that may help you to better understand inversions of pitch-class sets in general. The easiest way to visualize the inversion of a pitch-class set is to start with the top pitch-class member (right-most) of the pitch-class set, and working your way backwards, go up from that set member the same distance as you go down in the original set. For example:

To invert the set [5,6,9], start with pitch-class 9. Working backwards gives an ordered pitch-class interval of -3 (9 down to 6). In the inverted set, add +3 to 9 to arrive at the second inverted set member, 0. Continuing to work backward in the original set, 6 down to 5 is -1, so the third inverted set member is 0 + 1, or 1. In this way, [5,6,9] inverts to [9,0,1].

It is important to note, however, that this is not an inversion at T0I. This inversion relationship, or index number, is actually 6 (9 + 9; 6 + 0; 5 + 1).

The final two ideas that we need to discuss relating to pitch-class sets involve *set classes* and the *prime form* of a set.

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 starts with the smallest interval. In the above case, the inversion [0,1,5] starts with a smaller interval 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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