# How was the ordering of Allen Forte's list of prime forms determined?

I'm familiar with how "prime forms" are determined from a pitch class set using the Forte (or Rahn) systems. A good list is at http://en.wikipedia.org/wiki/Forte_number

However I am puzzled by how Allen Forte created his original list of prime forms. Specifically, how was the order of the sequence determined? For example here is the beginning of the list of four note prime forms:

``````4-1 [0,1,2,3]
4-2 [0,1,2,4]
4-3 [0,1,3,4]
4-4 [0,1,2,5]
``````

This order cannot be created by simply incrementing the final number. Why does [0,1,3,4] intervene between [0,1,2,4] and [0,1,2,5]?

Another example is the beginning of the list of three note prime forms:

``````3-1 [0,1,2]
3-2 [0,1,3]
3-3 [0,1,4]
3-4 [0,1,5]
3-5 [0,1,6]
3-6 [0,2,4]
``````

Here it appears that the last note in the sequence is being incremented until it reaches the tritone (6) in [0,1,6]. Subsequently [0,2,3] is omitted as it is a mode of [0,1,3], and [0,2,4] follows as the sixth form in the list.

If the list of three note forms followed the same pattern as the list of four note forms then [0,2,4] would follow [0,1,4] and precede [0,1,5]. This pattern would be to increment the highest note, then the next highest etc., skipping over sets which are not prime. However this is demonstrably not the case.

There must be an ordering principle he used to create the list but I can't see it. Any answers?

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I've searched and searched and I can't find the reason for some of the sets being seemingly out of order. I can say though that he has a contact link on his website. allenforte.com I suggest just emailing him and asking why he ordered them the way he did. : ) – ecline6 Apr 11 '13 at 22:38

Allen Forte's list of prime forms is ordered by a descending sort on the interval vector of each set. For example:

``````4-1 [0,1,2,3] <3,2,1,0,0,0>
4-2 [0,1,2,4] <2,2,1,1,0,0>
4-3 [0,1,3,4] <2,1,2,1,0,0>
4-4 [0,1,2,5] <2,1,1,1,1,0>
``````

Another example:

``````3-1 [0,1,2] <2,1,0,0,0,0>
3-2 [0,1,3] <1,1,1,0,0,0>
3-3 [0,1,4] <1,0,1,1,0,0>
3-4 [0,1,5] <1,0,0,1,1,0>
3-5 [0,1,6] <1,0,0,0,1,1>
3-6 [0,2,4] <0,2,0,1,0,0>
``````

This ordering principle applies for the entire list, excepting the Z-related sets. When two sets are Z-related they share the same interval vector. In this case the set with the "most prime" prime form is allocated to its natural position in the above sequence, and the other set in the pair gets relegated to the end of the sequence. It seems like the ordering of the Z-related forms was an afterthought as it would have been more logical to allocate them adjacent sequential positions in the list, as per their interval vector ordering.

After sleeping on it the answer became clear to me.

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Z-related sets don't share the same prime form, they are a different prime form, but with the same interval vector. Like (0146) and (0137), which are completely different sets and, thus, different prime forms, but both have interval vector <111111>. – Pat Muchmore Jan 22 at 3:27
Thanks @PatMuchmore - good spot. I have corrected this. – Richard Padley Jan 22 at 9:49