Friday, July 20, 2007

Journal Club: Math and Ozzy Osbourne

This post was originally written on November 4, 2004. I've reposted it so Guy can correct my ass to the amusement of you readers.

Chad Orzel has been writing up reviews of research articles in a weekly feature he calls Journal Club. Inspired by this, I'm offering my own music theory/music cognition journal club, starting with an article in the latest Music Theory Spectrum, "Neo-Riemannian Theory and the Analysis of Pop-Rock Music" by Guy Capuzzo. In the last 10 years it has become quite acceptable for theorists to write about popular music, using tools of voice-leading analysis, block-chord analysis, and other analytical methods originally designed for Baroque/Classical/Romantic/20th-century art music. Guy continues this tradition with the use of Neo-Riemannian operations to explain why Pop-rock musicians use the chords they use.

First, what the hell are Neo-Riemannian operations? They are mathematical transformations of chords to make other chords. There are seven operations, which can be used together to make compound operations. The first, Identity (I), leaves the chord as it is, just as any decent mathematical identity function does. The Leading-tone exchange (L) moves the bottom note of a major triad down a half-step, so a C major triad (C E G) becomes an E minor triad (B E G -> E G B). If the starting chord is a minor triad, the top note is moved up a half step to reverse the process (E minor becomes C major again). The opposite of this operation, L', moves the top two notes up a half step. The C major triad woud become an F minor triad (C F Ab -> F Ab C). The Parallel operation (P) moves the middle note of the triad so a major triad would become a minor triad, or a minor triad becomes a major triad. With C major, the E becomes Eb under the P operation to create the C minor triad. The opposite operation, P', moves the top and bottom notes to shift the mode from major to minor or vice versa. With the C major triad, C and G are moved up a half step to create a C# minor triad (C# E G#). The C minor triad would become a Cb major triad under the P' operation. Finally, the Relative operation (R) flips a chord between relative relationships. This is done by moving the upper note of the major triad up a whole step, so C major becomes A minor (C E A -> A C E), and A minor would become C major by moving the bottom note down a whole step. The opposite operation, R', shifts C major to G minor by moving the bottom two notes down a whole step (C E G becomes Bb D G).

Note that all these operations involve moving notes by step only, and at least one note is held in common through any operation. To explain chords that don't have any notes in common, a combination of operations is needed. So moving from C major to B major involves two operations, PP'. P(C E G) = (C Eb G) P'(C Eb G) = (B Eb F#) Eb and D# are the same pitch on the piano, so the final result can be written as (B D# F#) for the B major triad.

These operations are used to model chord progressions. The author shows examples from Depeche Mode's "Shake the Disease," Ozzy Osbourne's "Flying High Again," Frank Zappa's "Easy Meat," Radiohead's "Morning Bell," Bob Dylan's "Lay, Lady, Lay," Beck's "Lonesome Tears," King Crimson,'s "Dinosaur," "The Council of Elrond" from The Lord of the Rings: FOTR soundtrack, and a full analysis of Soundgarden's "Blow Up the Outside World." Patterns of operations, including the basic operation of transposition, create networks of connected chords, which can be used to trace the progression of harmonies.

The argument is that clearly defined networks for chord progressions explain the logic of the progression, why it works well. The weakness of this model is that it is too flexible. Any two chords can be linked by countless compound operations. C major to D major can be reached by RR', R'L', or PP'PR, among many others. There should be some type of cognitive restraint, that only a certain number of operations can be combined and still provide a salient progressional logic. Guy does stick with single or double operations for most of his analyses, with the exception of three three-operation compounds in the network for the introduction to "Dinosaur." This progression is theoretically more complex, because it involves seventh chords and triads together. This needs an extra operation, the inclusion operation, to shift from a three note chord to a four note chord. I don't believe this extra complexity required for the model reflects the complexity of the progression, since shifts between triads and seventh chords is a normal activity in all tonal-based music. Thus the inclusion operation could be considered negligible in the judgement of model complexity.

The analyses are all determined both by mathematical logic and by musical logic. The musical logic, comparisons of the given progression to expected tonal or blues progressions to judge the "functionality" of a given chord pair, helps to limit the NR operation possibilities. Guy often doesn't explain these applications of musical logic, since all theorists reading the article have these logical rules ingrained in their musical souls. The main conclusion: most progressions involve very small movements of notes from chord to chord, called parsimonious voice-leading. The small motions give the aural impression of closely connected chords, and hence logical chord progressions.


Sonia and Guy's Blog said...

Thanks for your interest in my article, Scott. I will respond to two points you made.

First, it's not always the case that a transformation from a triad to a seventh chord (or vice versa)requires an "extra" operation. For example, the transformation between G major and G dominant seventh is / (using / to designate the inclusion sign). Likewise, the transformation between G minor and E half-diminished seventh is /. The "extra" operation creeps in depending on the roots of the chords, not the cardinalities. For instance, the transformation from G major to E half-diminished seventh is P/ (first do P, then do /).

Second, I find the ability of Neo-Riemannian operations to model a move between two chords in numerous ways to be a boon, not a bane. Similar things happen in common-practice harmony. For instance, the opening chord of Beethoven's First Symphony is V7/IV if you're Walter Piston and Ib7 if you're Heinrich Schenker. I understand your concern that Neo-Riemannian theory has the potential to spin out of control, heedlessly labeling every triadic transformatin in its path, but it's the analyst's responsibility to prevent that, not the system's.

Let's keep the dialog going! Great minds don't think alike.



Anonymous said...

Just out of wild curiosity... Would you say that these techniques are more useful for after-the-fact analysis, or are these used in actual composition?

Scott said...

Guy, thanks for the clarification on the transformations from triads to seventh chords. I was trying to state that another operation, besides L P and R, is needed when dealing with both triads and seventh chords. How do you feel about my statement that the inclusion operation doesn't add much to the perceived complexity of a model?

As for the flexibility, you have a point. However, one can make a perceptual distinction between V7/IV and Ib7, as well as a logical/systematic distinction. The V7/IV places that chord lower in the tonal hierarchy, relying upon the following IV chord (and eventual V and I) to give it identity. Ib7 does not rely upon the following IV chord, as it is the first statement of the top dog tonic chord, albeit chromatically altered. Thus these different descriptions of the chord have different perceptual meanings. What is the difference in perception between RR', R'L', and PP'PR?

Scott said...

Anonymous, I won't speak for Guy but I place these techniques as analysis, not compositional strategy. However, I have seen attempts to use similar operations as a method to teach improvisation. I don't see this method as successful.

Scott said...

Hey Guy, while I have you on the phone, how do you feel about having to rely upon enharmonic equivalence for these operations to work? Is there any way to make a closed system without using enharmonic equivalence? And does it make any difference, especially as applied to music that is clearly performed in just or Pythagorean tuning (orchestral music, non-piano chamber music, and a cappella choral music, especially)?

Sonia and Guy's Blog said...

Sorry so long in responding!

To answer anonymous's question, Thom Yorke of Radiohead consciously uses NROs in his songwriting, though he does not refer to them by name. He alludes to them in a discussion of common-tone chord progressions in an interview with Alex Ross (New Yorker, August 20 & 27, 2001).

To answer Scott's question about the perceptual difference between "synonyms" such as PLR and RLP (both of which map C major onto F minor), Richard Cohn (JMT, 1997) cites this as an open research question. It depends on how much ontological weight you want to place on the Tonnetz moves. Does the C major to F minor move have three distinct stages, each of which deserves audibility, or is it one move that happens to pass through three locations on the Tonnetz?

As for Scott's question about enharmonic equivalence, most theorists state that enharmonic equivalence is what makes the wraparound Tonnetz possible, which in turn makes the elegant mathematical group structure of NROs possible. You could construct a Tonnetz that lacks enharmonic equivalence--it's just going to be "messy" from a mathematical point of view.