IX. Twelve-Tone Music

Mark Gotham

Key Takeaways

  • This chapter goes through the different ways of representing pitches, rows, and transformations in twelve-tone serial music.
  • When reading other writing on twelve-tone music, be prepared for any of these conventions to be used.
  • In your own work, simply choose one you feel comfortable with and use it consistently.

There are different conventions for labeling rows, transformations, and even pitches and intervals. This chapter compares the main approaches that you’re most likely to encounter in analytical writings. The focus is on rows and matrices, but before we get to that, let’s deal first with the pitches themselves.

Pitch

As we’ve seen earlier in the book, it is useful in some analytical contexts to use pitch-class notation (integers from 0 for C to 11 for B) as an alternative to spelling out those pitches (e.g., C♯ vs D♭). This convention is mostly associated with non-tonal music (including most twelve-tone music), where it can be handy for performing the kinds of mathematical operations we’ve seen (in both pitch-class set analysis and twelve-tone music) and for sidestepping questions of pitch spelling. There’s often still a logic to the pitch spellings used in a twelve-tone piece, but that logic is often of a different and perhaps less generalizable kind. For instance, using specific pitch spellings in a row-form representation usually does not stand for the music in the same way that the pitches of a scale do in tonal music.

Beyond this, there’s a specific convention (probably most common in twelve-tone music) that sees pitch classes 10 and 11 swapped in one of two ways: replacing them with either A and B or else T and E (for “ten” and “eleven”) respectively (and in either UPPER or lower case). This is simply a matter of convenience and/or style. For instance, it is common in contexts where it’s preferable to express rows succinctly, with one character per note and no separators (e.g., 0e378426519t). We’ll stick with the long-form in this book (that is, 10 and 11).

Rows

For , the main difference in notation and labeling centers on a single choice: to organize our rows around either:

  1. the same pitch in all contexts (conventionally, that pitch is C)
  2. a pitch that’s important to the musical context in question

For instance, in Elisabeth Lutyens’s Motet, we set the row out starting on C, so with the pitch-class sequence [0, 11, 3, 7, 8, 4, 2, 6, 5, 1, 9, 10]. Alternatively, we could set out the P0 starting on D, as the first voice to enter (alto) starts on D4 and proceeds to set out the first of this prime-form row on that pitch level.[1] That would give us a P0 of [2, 1, 5, 9, 10, 6, 4, 8, 7, 3, 11, 0].

Option 1: P0 starts on C

In this convention, whatever you decide the prime form to be, the transposition of that form starting on C is P0. This is probably the most common convention today, and sometimes called “zero-centered” or “fixed-do” (by analogy to tonal solfège systems).

As we have set P0 to begin on C, I0 also begins on C, and R0 and RI0 will end (sic) on C. This separation of P0 and I0 from R0 and RI0 makes sense in so far as we prefer P0 and R0 to be exact retrogrades of one other (and likewise I0 and RI0). We could theoretically have an even more consistently “zero-centered” system in which all of P0, I0, R0 and RI0 begin on C, but that’s not a convention that people have widely adopted.

In summary:

  • P0 starts with C
  • I0 starts with C (same note as P0)
  • R0 starts with the last note of P0 (by definition, not C)
  • RI0 starts with the last note of I0 (by definition, not C)

Option 2: P0 starts wherever we chose

The main alternative contention sees the P0 form assigned to the first, or other “most meaningful” form of the row, whatever pitch level that happens to be on. Depending on the context, this may be evident from the piece, deduced from the analysis, or allocated semi-arbitrarily. Transpositions and other operations are then worked out in the same way, in relation to that P0 form. This convention is sometimes called “original-centered” or “movable-do” (to continue the solfège analogy).

In summary:

  • P0 takes a transposition (and thus starts with a pitch) chosen by the analyst
  • I0 still starts with the same note as P0
  • R0 still starts with the last note of P0
  • RI0 still starts with the last note of I0

Same? Different? Better? Worse?

As the two summaries suggest, these naming conventions are actually not so different. It bears repeating that for all naming systems, transposition and the other operations all work in the same way, so it’s mostly just a matter of where you start: which row form you use as the referential form to relate others to.

And as is so often the case when multiple parallel naming conventions emerge, there are benefits to (and detractions from) each approach. If you’re analyzing music that makes you want to assign P0 in a musically sensitive way, then the “original-centered” (“movable-do”) convention may suit your purposes. But if you go down that route, then you’ll probably feel compelled to come up with a “good” reason for the pitch level of P0 in all your analyses, and that may not always be appropriate, even across the movements of a single work, for instance. At least the “zero-centered” system has the benefit of clarity and consistency. That’s probably why it’s become more common in recent scholarship, but that doesn’t necessarily make it “better.”

Indeed, in many cases, it won’t even be clear which orientation should be P and which I (or R for that matter). Unfortunately, there’s no widely recognized system for hedging on that as yet!

Matrices

Before we wrap this up, there’s one final confusion to add to the pile: how to set out these conventions on the row matrix. Here are three types.

Type 1

First, here’s a reminder of the matrix we saw for the Lutyens example in the last chapter (P0 starts on C and is in the top row). This is probably the most common and standard form.

I0 I11 I3 I7 I8 I4 I2 I6 I5 I1 I9 I10
P0 0 11 3 7 8 4 2 6 5 1 9 10 R10
P1 1 0 4 8 9 5 3 7 6 2 10 11 R11
P9 9 8 0 4 5 1 11 3 2 10 6 7 R7
P5 5 4 8 0 1 9 7 11 10 6 2 3 R3
P4 4 3 7 11 0 8 6 10 9 5 1 2 R2
P8 8 7 11 3 4 0 10 2 1 9 5 6 R6
P10 10 9 1 5 6 2 0 4 3 11 7 8 R8
P6 6 5 9 1 2 10 8 0 11 7 3 4 R4
P7 7 6 10 2 3 11 9 1 0 8 4 5 R5
P11 11 10 2 6 7 3 1 5 4 0 8 9 R9
P3 3 2 6 10 11 7 5 9 8 4 0 1 R1
P2 2 1 5 9 10 6 4 8 7 3 11 0 RI0
RI2 RI1 RI5 RI9 RI10 RI6 RI4 RI8 RI7 RI3 RI11 RI0

Type 2

Now here’s the same matrix, with P0 still on the top row, but with that P0 starting on D. Note how the lists of row forms stay the same (P0, P1, P9…), but the pitches have moved around.

I0 I11 I3 I7 I8 I4 I2 I6 I5 I1 I9 I10
P0 2 1 5 9 10 6 4 8 7 3 11 0 R10
P1 3 2 6 10 11 7 5 9 8 4 0 1 R11
P9 11 10 2 6 7 3 1 5 4 0 8 9 R7
P5 7 6 10 2 3 11 9 1 0 8 4 5 R3
P4 6 5 9 1 2 10 8 0 11 7 3 4 R2
P8 10 9 1 5 6 2 0 4 3 11 7 8 R6
P10 0 11 3 7 8 4 2 6 5 1 9 10 R8
P6 8 7 11 3 4 0 10 2 1 9 5 6 R4
P7 9 8 0 4 5 1 11 3 2 10 6 7 R5
P11 1 0 4 8 9 5 3 7 6 2 10 11 R9
P3 5 4 8 0 1 9 7 11 10 6 2 3 R1
P2 4 3 7 11 0 8 6 10 9 5 1 2 RI0
RI2 RI1 RI5 RI9 RI10 RI6 RI4 RI8 RI7 RI3 RI11 RI0

Type 3

Perhaps most confusing of all is a kind of hybrid version where we still have the D version on the top row, but now we label it P2. So:

  • We organize the row class around a chosen pitch/transposition (here D).
  • We still label the row forms around the alternative option (P0 starts on C).

Note how this time, comparing it with the version above, the pitches have stayed the same, but the lists of row forms have changed (Px, Py…).

I2 I1 I5 I9 I10 I6 I4 I8 I7 I3 I11 I0
P2 2 1 5 9 10 6 4 8 7 3 11 0 R2
P3 3 2 6 10 11 7 5 9 8 4 0 1 R3
P11 11 10 2 6 7 3 1 5 4 0 8 9 R11
P7 7 6 10 2 3 11 9 1 0 8 4 5 R7
P6 6 5 9 1 2 10 8 0 11 7 3 4 R6
P10 10 9 1 5 6 2 0 4 3 11 7 8 R10
P0 0 11 3 7 8 4 2 6 5 1 9 10 R0
P8 8 7 11 3 4 0 10 2 1 9 5 6 R8
P9 9 8 0 4 5 1 11 3 2 10 6 7 R9
P1 1 0 4 8 9 5 3 7 6 2 10 11 R1
P5 5 4 8 0 1 9 7 11 10 6 2 3 R5
P4 4 3 7 11 0 8 6 10 9 5 1 2 R4
RI2 RI1 RI5 RI9 RI10 RI6 RI4 RI8 RI7 RI3 RI11 RI0

Summary

In summary, the first row can read:

  • P0, starting on 0
  • P0, starting on n (here 2)
  • Pn, starting on n

All of these naming and matrix-generating conventions are out there. It’s best simply to be aware of these options and check that you have the right convention in mind when you come across one (especially where the matrices neglect to explicitly label the row names).

Assignments
  1. Chose any row from the Twelve-Tone Anthology that interests you and write out the row matrix with all 48 row forms (i.e., with numbers on the grid as shown above) in each of the three ways shown above. (Then choose your favorite method and never do this again!)

  1. The actual row distribution is a bit more complicated. See Parsons 1999 for an analysis and discussion.

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OPEN MUSIC THEORY by Mark Gotham is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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