Exercise 4-1. Naming sets using normal form


By definition, a set's normal form is the one that lists the set's pcs in the most compact "ascending" (clockwise) order. Here, to review, are the steps for finding a set’s normal form.

  1. List the pc content of the segment, eliminating all repetitions.

  2. Arrange the pc integers in "ascending" (clockwise) orders.
    Remember: there are always as many possible "ascending" orders as the set has pcs.

  3. Choose the most compact of the "ascending" orders.
    Do this by subtracting the first integer from the last. The one whose overall interval span is smallest is the set’s normal form.

    With some sets, two or more orders tie for overall compactness. If so,

  4. Choose the set that is most compact towards the left. Measure the intervals from first to second-last pc. Still tied? Measure from first to third-last pc, to fourth-last, and so on, until one set wins in compactness.

    With sets of great interval regularity, no amount of measuring will break the compactness tie. In such a case

  5. Choose the order that begins with the lowest number.

The following exercise has five parts, a-e. In parts a-d, you’ll be guided through the steps for finding the normal forms of four sets. In part e, you’re on your own for four more sets (though answers and comments are available).

Begin the exercise.


Page last modified 27 July 2001 / GRT

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