VIII. 20th- and 21st-Century Techniques
Analyzing with Set Theory (or not!)
Mark Gotham and Megan Lavengood
Key Takeaways
- Be aware of why you are grouping some sets together and not others (segmentation).
- Create analytical unity by relating sets to one another.
- Remember that composers wrote free atonal music before set theory was invented.
- Be aware of the limitations of set theory: it does not discuss any non-pitch aspects of the music, and it even neglects to address many pitched aspects.
There are many issues with the analysis of music using pitch-class sets—some generic to many forms of analysis, some more specifically relevant to set theory.
The flexibility of set theory is a double-edged sword. Because any group of pitches can be a set, analysts can move beyond triadic harmony and analyze music based on any sort of pitch organization. But by the same token, because any group of pitches can be a set, analysts could “prove” anything they want to. For instance, consider a motivic analysis in which the motive is a single interval. By locating this one interval wherever possible, an analyst could claim that a work has motivic unity. While that interval may indeed be found throughout the piece, the frequency of the interval’s appearance is unlikely to be meaningful on its own, so it’s up to the analyst to set out how and why it assumes musical meaning for them in the given context. In short, analysts should always be aware of the basis, significance, and perceptibility of their observations.
Following are several potential pitfalls of using set theory, along with ways to bolster any claims you make using set theory.
Segmentation
The emphasis on numbers (via pc integers) can make set theory seem more “scientific” than it really is. And because set theory relationships are factual—for example, a pitch-class set is related transpositionally to another set, or it isn’t—it’s easy to end up with an unfalsifiable analysis that may still be ineffective in describing the music. For example, every single piece with notes in it could be described in terms of one-note “sets,” all of which are equivalent to each other. Clearly this is not a helpful analytical statement!
The best way to ensure that your set theory claims are useful is to be aware of your process of segmentation: how you’ve divided up the piece into pitch-class sets. Here are just some of the reasons why an analyst might group notes together:
- The contiguity of the grouping. There’s a strong case for grouping together notes that are simultaneous or that happen in direct succession. You could make a case for uniting notes that are far away from each other and excluding many other notes in between, but it will take more explaining.
- Shared rhythmic profile. It’s common to segment sets based on what is set off by rests, or based on a shared rhythmic motive, or any other number of rhythmic connections.
- Shared metric placement
- Shared timbre and texture
- Shared articulation
- Shared register (high range, low range)
Joseph Straus (2016, 70) summarizes the importance of segmentation eloquently:
In all of your musical segmentations, strive for a balance between imaginative seeking and musical common sense. On the one hand, do not restrict yourself to the obvious groupings (although these are often a good place to start). Interesting relationships may not be apparent the first, second, or third time through, and you need to be thorough and persistent in your investigations. On the other hand, you have to stay within the boundaries of what can be meaningfully heard. You can’t pluck notes out in some random way, just because they form a set that you are interested in.
Relationships Between Sets
Simply identifying sets is not analytically interesting on its own. Here are some ways to relate sets to one another and create a sense of unity among them.
- Find multiple instances of the same set class, and then relate them to each other by transposition and inversion. You might calculate transpositional/inversional relationships from one set to the following set, or from the first set to all the others, or any other configuration. Your relationships should all involve notes that actually appear in the score—don’t bother relating sets to the normal form that corresponds to the prime form, for example, unless that normal form actually appears in the piece.
- See if different sets belong to the same superset. That superset might be a familiar mode or collection.
- Relate pairs of notes across sets by symmetry across an axis of inversion or emphasis on a pitch class center.
Theory Following Practice
On a broader level, much if not most music theory is retrospective: the terms and techniques for dealing with a repertoire emerge after the fact; they exist to make sense of music that exists. This is by no means always true: many composer-theorists have developed ideas about how music might be organized, and then implemented those ideas. A significant example of this is serialism. But set theory is a retrospective theory, popularized by Allen Forte decades after the repertoire it is meant for was composed. As such, we should take care to separate set theory from the compositional acts and intentions of those composers. Arnold Schoenberg (1950), for instance, felt that a structure for free atonal music was impossible, hence his subsequent development of the serial technique.
Just because a theory was developed after the fact does not make the theory any more or less valid (composers historically have been at least as active in obfuscating their methods as clarifying them!), but it’s worth keeping in mind.
What Set Theory Won’t Tell You
Be sure to consider all the things that set theory does not address in any capacity:
- History and context
- Any non-pitch parameters (though this depends on how the analyst approaches questions like segmentations)
- Pitch parameters like voice leading, spacing, or order (again, depending on the analytical approach)
This list includes many aspects of music that are important to listeners and performers. You may want to take care to address these issues in your analysis through other means, because a set theory analysis will often ignore them.
- Buchler, Michael. 2017. “A Case Against Teaching Set Classes to Undergraduates.” Engaging Students: Essays in Music Pedagogy 5. http://flipcamp.org/engagingstudents5/essays/buchler.html.
- Schoenberg, Arnold. 1950. Style and Idea. New York: Philosophical Library.
- Straus, Joseph N. Introduction to Post-Tonal Theory. 4th ed. Upper Saddle River, NJ: Prentice Hall, 2016.
- Guided analysis of “Wie bin ich Froh!” by Anton Webern (.pdf, .docx). Recording
- Segmentation worksheet (.pdf, .docx). Asks students to justify the given segmentations by explaining what the grouped pitches have in common.
- Atonal analysis using pc sets (.pdf, .mscx). Open-ended prompt asks students to use set theory to analyze an excerpt.
The process of dividing a passage or piece of music into its component parts.
Music that is atonal, avoiding a traditional pitch center and harmonic hierarchy, but is not serial.
A methodology for analyzing pitch in atonal music. Pitch classes are given an integer name (0–11, where C is 0, C♯ is 1, etc.). Groups of pitches are considered together as "sets." Sets may be related by inversion or transposition.
A group of pitch classes.
A regularly recurring unit of music that's smaller than an idea, and which is typically transformed across a work. The word "motive" usually refers to pitch material, but other kinds of motives such as rhythmic or contour also exist.
A system of naming pitch classes that treats C as 0, C♯ as 1, D as 2, etc.
The most compressed way to write a given collection of pitch classes.
A name for a set class. The prime form is the version of the set class that is most compact to the left and transposed to begin on 0.
A larger set that contains other smaller sets. For example, a superset of (037) is the diatonic collection, (013568t).