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What's the purpose of pitch-class set theory? How would you directly relate it to a CMAJ (C, E, G) chord? I know pitch-class set theory and the basis of it. Does it give you more options of sound or help you create a unique type of sound or something?

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Pitch class set theory is just an analytical tool that gives you a different way to look at sets of notes. It enumerates the notes in the 12 ET system and allows certain calculations to be carried out on sets of notes that would you not be able to easily able to do otherwise.

One example of a common calculation that uses pitch class set theory is prime form which is useful for finding highly related sets of notes. This calculation is very good at finding inversions of chords, transpositions of motifs/ideas, and closely related chords such as C6 and Am7.

For your example, the normal form of C major would be to [0, 4, 7] with the prime form being [0, 3, 7] (which is the same as minor due to the intervals that make up both chords). Any major chord form can be transformed into this along with any inversion of a major chord. This may seem trivial, but if you were dealing with a less common set of notes such as C, D♭, F, G, and A♭, you could quickly reduce the set which in this case would reduce to [0, 1, 5, 7, 8] and you could find subsets and supersets that are related to this if you like the sound with calculations.

  • So, is C always 0, or is 0 the first/lowest in a set of notes? Where/when did the concept first arise? – Tim Feb 20 '17 at 15:33
  • @Tim any note can be considered 0, although C is typically 0 due to typicall music theory ideas. The idea of the set is more important than the actual values chosen as [0, 4, 7] is the same set as [1, 5, 8] it's just reduced. – Dom Feb 20 '17 at 15:37
  • So, a bit like playing with numbers?! – Tim Feb 20 '17 at 15:39
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    @Tim yup it allows computations you wouldn't be able to do otherwise. The only downside is it only works for ET systems. – Dom Feb 20 '17 at 15:40
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    Just a note, the prime form of a C-major triad is actually [037] not [047]. The latter is the normal form of a C-major triad, but the prime form also takes into account potential inversions. Major and minor triads belong to the same set class, and thus have the same prime form. – Pat Muchmore Feb 20 '17 at 17:02
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In addition to Dom's answer, pitch-class set theory often deals with the equivalence between transposed and inverted forms of pitch material.

For instance: you mentioned a C-major triad. We intuitively agree that all major triads are equivalent in some way to this C-major triad, because they are all transpositions of each other.

But in pitch-class set theory, we can also understand that items that are inversions of each other are also equivalent. But here we don't mean an inverted triad, but rather a major triad literally read from top to bottom. Whereas this C-major triad is (ascending) a major third followed by a minor third, we can invert this to show that (again ascending) a minor third followed by a major third---also known as a minor triad---is inversionally equivalent to the major triad.

Thus one of the advantages (or disadvantages, depending on your viewpoint) to pitch-class set theory is that we can compare pitch material that is otherwise not viewed as equivalent. In this case, we show the equivalence between major and minor triads.

  • Sort of C-E M3, E-G m3, but backwards, C-A m3, A-F M3? Never used pcst, so only guessing! – Tim Feb 20 '17 at 15:38
  • @Tim Ha, I misread your earlier comment. In your scenario, the M3 is still on the bottom, so that's just transpositional equivalence among major triads. Backwards (aka, "inverted"), that C-major triad would be C-Af (M3), Af-F (m3), showing an inversional equivalence between CM and Fm. – Richard Feb 20 '17 at 16:08
  • I see, thanks. However, I can't see how useful it is to know, for instance, that CM and Fm are equivalent in anything but a particular number system... – Tim Feb 20 '17 at 16:19
  • This particular case isn't that helpful for seeing the value of the system; my example was just to show easily the capabilities of the system. When you're dealing with more complex pitch-class sets, the value can become more obvious. – Richard Feb 20 '17 at 18:03
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I would say that the primary purpose of musical set theory is to have a system capable of labeling, describing and manipulating any combination of pitches, rather than just a limited collection of triads and seventh chords.

If no one were interested in extending either the vocabulary of common-practice music (triads, major/minor scales, seventh chords, further triadic extensions) or the grammar of common-practice music (functional, sequential, voice-leading, etc.), then conventional tonal theory would be able to both analyze all existing music and give clear instructions on how best to approach new compositions. Set theory wouldn't add much to the equation.

For those composers generally interested in the same vocabulary but new grammars (jazz, rock/pop, fin du siecle music, many types of Minimalism, etc.), set theory starts to have some potential uses because of its focus on different kinds of connections (or non-connections) between harmonies. Connecting or avoiding harmonies that are superficially different but bear deeper connections is somewhat easier to visualize in set theory—especially through the algorithm that reduces any collection of pitches to its most essential elements known as "finding the prime form." Still, an offshoot of set theory such as neo-Riemannian theory is probably also sufficient.

Where set theory really comes into its own is when composers are interested in exploring new vocabularies and inventing grammars to go along with the new vocabularies. (As others have noted, the system is still pretty much confined to equal temperament, though it could theoretically be extended.) For example, if we're looking at every possible combination of three pitches, and we reduce them to their most fundamental interval content (i.e. embracing octave and inversional equivalence), set theory shows that there are only twelve possibilities. One of them, dubbed (037), contains all major and minor triads, and is, as such, quite familiar to Western listeners. Others, such as (016) and (014), are less familiar to most listeners and have their own soundworlds.

Many composers are interested in the sonic possibilities of such less-well-known combinations. For example, Elliott Carter—who had a passion for as much diversity as possible—loved two particular four-note collections, or "tetrachords", known as (0137) and (0146). These collections can be discovered by set theory to contain one of every possible equal-temperament interval between their pitches. This is pretty astonishing considering they are only four distinct pitches in them. They are called "all-interval tetrachords" and an example of such a collection would be C–C#–E–F#.

Carter was also interested in a now-famous hexachord (six-pitch collection) called (012478) or "the all-trichord hexachord", a collection of only six notes that nevertheless contains at least one example of each of the 12 possible trichords discussed above. An example would be C–C#–D–E–G–G#. Set theory makes it much easier to discover such structures in composer's music and makes it easier to compose using such structures.

Of course, whether any of these vocabularies or grammars are actually interesting varies from person to person. I love them, but I know it's sort of a niche enterprise. For those of us interested in writing music like this, or interested in better understanding music like this by other composers, set theory can be an invaluable tool.

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A pitch class set is a way to describe the intervallic structure (how the notes are distributed in pitch-space) of a simultaneity (a bunch of notes played at once) without respect to centrism (meaning within a key or tonality).

If you are dealing with atonal music like Weber or late Schoenberg, you are dealing with composers who purposefully avoided any notion of tonality or pitch centrism. To call a chord "C major" wouldn't make sense in their musical language; instead, such a chord is composed of a perfect fifth (between C and G), a minor third (between E and G), and a major third (between C and E). But major, minor, and perfect are all tonal concepts, so it is preferable to describe it as 7 half-steps, 4 half-steps, and 3-half steps. Thus its pitch class set would (in normal form) be {0, 4, 7}. Notice the 3 doesn't appear in this set explicitly; it is inferred from the difference between 4 and 7.

A pitch class set merely tells you how the frequencies of the notes will combine with each other within that particular chord (i.e. if it is dissonant or consonant with itself, or somewhere in between). It doesn't you a thing about the chord's centrism (what note is its root), inversion, or how the chord functions within a scale or key; tonic and dominant, for example, are both going to be {0, 4, 7} no matter what. In fact some tonal works are entirely made up of a single pitch class set expressed at different scale degrees. For this reason, it would be highly unusual to use pitch class set theory in the analysis of tonal music.

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