# Rahn Atonal Prime Form Confusion

A review via the mark I eyeball of the prime form tables in "Basic Atonal Theory" (Rahn, 1980) versus modern software calculations of the prime form yields some differences; are these errors in publication, the software, or something else?

``````7-Z18  0, 1, 2, 3, 5, 8, 9      # BAT
0, 1, 4, 5, 6, 7, 9      # software

7-20   0, 1, 2, 4, 7, 8, 9      # BAT
0, 1, 2, 5, 6, 7, 9      # software

8-26   0, 1, 2, 4, 5, 7, 9, 10  # BAT
0, 1, 3, 4, 5, 7, 8, 10  # software
``````

For `7-Z18` and `7-20` Rahn is in agreement with what Forte published, but not software that uses the Rahn method? `8-26` I've no idea.

They are both right and here's why.

There are two different algorithms for determining the prime form which are Forte's and Rahn's. In most cases they are the same, however there are a handful that are not the same. The one's you've noted are not the same with both algorithms. The breakdown of them are as follows:

```
Pitch Class Set     Forte Prime           Rahn Prime
7-Z18              (0,1,2,3,5,8,9)      (0,1,4,5,6,7,9)
7-20               (0,1,2,4,7,8,9)      (0,1,2,5,6,7,9)
8-26             (0,1,2,4,5,7,9,10)    (0,1,3,4,5,7,8,10)

```

You can even try it out yourself on this calculator.

Here's a more in depth explanation of why this is and what the actual difference is bewteen the two from the What is this? button next to the two algorithms.

There are two algorithms for computing the prime form of a Pitch Class Set. The first was introduced by Allen Forte in The Structure of Atonal Music and the second is used by John Rahn in his book Basic Atonal Theory and is also used by Joseph N. Straus in his Introduction to Post-Tonal Theory.

The difference between the two algorithms is apparent when examining Pitch Class Set 6-31. The Prime Form using the Forte algorithm is (0,1,3,5,8,9), and the prime form using the Rahn algorithm is (0,1,4,5,7,9). As you can see, the Forte algorithm puts a priority on making the small numbers smaller (i.e. 3 instead of 4), whereas the Rahn algorithm wants the larger numbers to be smaller (i.e. 7 instead of 8).

Which is better? Well, it depends on who you ask. Computer programmers and computer music people will typically prefer the Rahn algorithm because it is computationally more elegant. However, the Forte algorithm has the more established pedigree, and so it tends to be preferred by academics.

• So why is the Rahn book using Forte-algorithm derived prime forms for three of its entries? The rest of the prime forms in the Rahn book agree with the Rahn algo. Sep 29, 2015 at 18:00
• @thrig The software's output is showing Rahn's results. The book on those three is not which may be considered a typo, but there're not wrong in a sense of prime form since it's valid for Forte's.
– Dom
Sep 29, 2015 at 18:08
• Hmm, David H. Smyth in "Perspectives of New Music" Vol 22 No 1/2 (Autumn, 1983 - Summer 1984) pp. 549-555 reviews "Basic Atonal Theory" and points out errata and food for folks implementing code to do this sort of thing. Oct 2, 2015 at 22:25