33.3: Prime Form

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33.3 Prime Form

Whereas normal form deals with the exact pitches as they occur in the music, prime form is usually a transposition and possibly an inversion of the normal form to its most essential form, much in the way an E♭ major triad in second inversion belongs to the category “major triad,” or a G7/F belongs to the more general idea “dominant seventh chord.” Perhaps because of the primacy of C in music theory—many ideas are demonstrated in their relation to the C major scale—all prime forms are transposed to and start on C (pitch integer 0). Following is the process for determining prime form.

Transpose the normal form—[11, 2, 3, 7] from the normal form example in the previous section—so it starts on C (pitch integer 0): 0, 3, 4, 8
Invert the transposed normal form (what went up now goes down).
Write this inverted form in ascending form (4, 8, 9, 0), then do one of the following:
1. If there were no ties when determining normal form, proceed to the step 4.
2. If there were ties, put this inverted version through every ascending “scale” ordering to determine which is the most compact form from first note to penultimate note. In the example below, we see that the second ordering (the “tie loser” from normal form) is the most compact of the reorderings of the inverted normal form.
Compare the normal form (transposed to 0) to the most compact inverted form (transposed to 0). The most compact form is the prime form. Prime form is written in parentheses with no commas: (0148).
1. In the event the prime form reaches pitch integers 10 or 11, use T for 10 and E for 11; for example (013568T)

33.3.1 Application of Normal Form and Prime Form

Let’s determine normal form and prime form of the first set from the Webern excerpt. The first chord contained E♭, B, and D (3, 11, and 2).

The normal form is [11, 2, 3]. Below is the calculation to determine prime form.

set-theory-webern-op5no3-reordering-of-invNF-of-1st-set.svg

The prime form is (014). Now let’s determine the normal form and prime form for the third set we encountered: G♯, C, and A, or 8, 0, and 9.

The normal form is [8, 9, 0]. In the example below, we transpose the normal form to zero, then invert it.

In the following example, we put the inverted normal form through the reorderings to find the most compact form, then compare it to the normal form.

We see that the third set has the same prime form—(014)—as the other sets in the opening measures of Webern’s Op 5, No. 3. Prime form can allow us to see relationships that may not be apparent on the surface of the music.

33.3.2 Segmentation

What about the C♯ in the cello part? Should it be included with the three notes from the chords? Will another similarity be revealed? Segmentation is the term for “segmenting” or determining which notes to group together and analyze in a passage. Usually, segmentation is based on the music—notes sounding together as a chord, or notes in a melodic line. However, analysts may look at every possible combination of notes to search for deeper layers of connection. Below, we examine the first two chords with the C♯ included in each.

The prime form of the first set, when including the C♯ from the cello, is (0124). Here is the second chord with the C♯ added to it.

The prime form of the second set, when including the C♯ from the cello, is (0236). We do not see any relationship between these first two sets after including the C♯ with each three-note set. One doesn’t know this until one examines this new segmentation.