The Rahn (1980) vs. Forte (1973) Normal Form Algorithms

The normal form (p. 36) and prime form (p. 58) algorithms given in Straus (2005) are based on Rahn (1980). Forte (1973) employs a slightly different algorithm for determining normal form.The difference lies in the method for breaking next-most-interval ties in the ascending orderings. Rahn's algorithm breaks ties in right-to-left order, whereas Forte's breaks ties in left-to-right order. Rahn's procedure has been more widely adopted. Amazingly, this results in only five discrepancies between the Rahn and Forte prime forms: see 5-20, 6-29, 6-31, 7-18, 7-20 and 8-26. As you are aware of by now, these algorithms can be a pain to execute on certain sets.

The Short Cut Method

In the second edition of his book, Straus provides a "slightly simpler" method for determining prime form that I call the short-cut method. (It is interesting to note that this section was removed in the third edition his book. It was replaced by an intuitive geometric analogy for determining prime form using a pc clockface. The short-cut method is discussed in Rahn (1980) on p. 82.

When calculating prime forms by hand, the short-cut method can save you time and I encourage you to use it. However, you should be aware that the short-cut method does not work for all pc sets. Be sure to use the Straus (2005) algorithm, or a program like Atonal Assistant, for such sets.

Example 1

Let's take a look at a set for which the short-cut method will give you the wrong answer. In Straus, Ch. 3, Theory Exercise: II. 3. Straus asks you to find the prime form of the pc set [0,2,3,6,9]. The short-cut method for determining the prime form of this pc set is summarized in Fig. 1.

The short-cut method yields the prime form: (02369). However, an examination of all pentachordal set classes in Straus Appendix 1 reveals that this is not a valid prime form. You might ask, "What's the deal?" The reason the short-cut method fails is because 03679, the retrograde of the normal form of the pc set, is not a normal form. In Fig. 1, the normal form of (0,3,6,7,9) is actually [6,7,9,0,3]. Thus, this pc set belongs to set class 5-31 (01369).

For many pc sets, the retrograde of the normal form is the normal form of the T_nI forms. When this is not the case, the short-cut method will not work. Though the short-cut method saves time by skipping the step of calculating the actual normal form of the inverted forms, it also produces invalid answers on occasion. Nonetheless, it is worth knowing because it can save you time in your hand calculations.

Example 2

In Straus, Ch. 2, Theory Exercise: VI. 1. e. Straus asks you to find the prime form of the pc set [4,6,9,10,1]. The short-cut method for determining the prime form of this pc set is summarized in Fig. 2.

The short-cut method yields the prime form: (02569). However, an examination of all pentachordal set classes in Straus Appendix 1 reveals that this is not a valid prime form. The reason the short-cut method fails is because 03479, the retrograde of the normal form of the pc set, is not a normal form. In Fig. 2, the normal form of (0,3,4,7,9) is actually [3,4,7,9,0]. Thus, this pc set belongs to set class 5-32 (01469).

Example 3

Fig. 3 provides an example of the hand-calcuation method working just fine.

[0,2,3,4,5] is a member of set class 5-2 (01235).

Example 4

Finally, one last interesting example. Like Examples 1 and 2, the retrograde of the normal form is not the normal form. However, it doesn't matter here, because the inverted form was eliminated from consideration by the most-packed-to-the-left rule.

The normal form of (0,2,5,7,9) is actually [5,7,9,0,2], yet it doesn't matter. It should be mentioned that the major pentatonic collection [0,2,4,7,9] is a member of set class 5-35 (02479).

Updated: Septebmer 21, 2006

© 2006 Reginald Bain
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