# Why use Pitch Class Sets to describe music?

Can someone please elaborate on the significance and relation of pitch class sets to music? I understand the labeling and nomenclature methods but still can’t grasp how and WHY it relates to music other than giving chords an alternative naming system. Please help!

• The topic is a broad one. I suggest finding a book on "post-tonal music theory" to better understand the types of music and analysis facilitated by pitch-class-based interpretation. You could also seek out much material on line. Start perhaps with Wikipedia's explanation of twelve-tone music: en.wikipedia.org/wiki/Twelve-tone_technique. Jul 20, 2020 at 0:45
• Just revisiting this post. Is there anything more you're looking for from an answer? Apr 2, 2023 at 18:09

In a sense, you're correct that it's just an alternate nomenclature. However, that nomenclature facilitates the abstraction of musical analysis and understanding. It's much like the difference between Roman and Arabic numerals. You can do arithmetic with both, but Arabic numerals are better at making clear the patterns and relationships between numbers.

That is to say...

``````   15
+  35
-----
50
``````

...better expresses the base-10 system than does...

``````XV + XXXV = L
``````

Similarly, the tone row...

``````0 1 4 3 2 5 6 9 8 7 t e
``````

...more clearly expresses the compositional basis of Anton Webern's Variations for Orchestra, Op. 30, than does a literal listing of the actual pitches involved.

OVERVIEW

Consider the following chords:

``````X: 1
T: Enharmonically equivalent chords
M: 4/4
L: 1/4
K: C
[C_E_G__B]4|[__D_E_G__B]4|[CD^F^A]4|[^B,_E^^EA]4|[__B,C_E_G]4

``````

(Imagine the above are labeled (1) - (5) from left to right.)

All are enharmonically equivalent, but they have very different interpretations in the context of Tonality.

``````(1) = Cdim7
(2) = Eb dim[6-5]
(3) = D# dim[6-5]
(4) = ???
(5) = Cdim7[6-5]
``````

Further, the interpretation of each of those chords is different depending on the key we're in and the musical context.

``````C Major: (1) = i dim7
Bb Minor: (1) = ii dim7
Db Major: (1) = vii dim7
``````

This all comes about because in Tonal music, pitches and collections of pitches have a hierarchy and a meaning within that hierarchy. We need a nomenclature that describes the function and meaning of each chord.

On the other hand, post-tonal music looks to operate outside of that hierarchy and the meanings associated with it. The relationships between pitches (i.e., intervals) become the defining characteristic. For example, in Tonality, in the key of C major, `B` moving to `C` has one meaning, `E` to `F` another, and `F` to `E` yet another. In pitch class analysis, these are all equivalent — movement by a half step.

So, in pitch class terms, `(1) - (5)` are all simply the set `[0369]`. This is always true no matter the spelling, pitch order, or musical context. If the specific pitch classes and their order are important, then you can use function notation to express the relationships.

`T_n`: Transposition by `n` half-steps, where `n = {0, ..., 9, t = 10, e = 11}`

`(1) = (2) = (3) = (4) = T_3[(5)]`

Especially when analyzing music where pitches are stripped of their Tonal meaning, the ability to express relationships mathematically is very convenient (and equally meaningful once you understand the language).

AND BEYOND

Not only is this notation handy for pitch/interval relationships, but it opens up similar kinds of notation for, say, rhythm. For example see Tom Johnson's article, "Tiling in My Music" (First published in Perspectives of New Music Vol. 49, No. 2, 9 - 21; the whole issue, in fact, is devoted to articles on "Tiling Rhythmic Canons" and the math that arises from the underlying idea.)

On this site: Pitch-class set theory ?)

Dr. Justin Henry Rubin has published a short paper on using pitch classes in composition ("Composing with Sets", 2005; accessed 21 July 2020). It may provide a useful example.

Of course, entire books (or at least, chapters) have been written just to introduce the subject. With that in mind, here are a few frequently cited sources...

• Books

Forte, Allen. The Structure of Atonal Music.
Lewin, David. Generalized Musical Intervals and Transformations.
Morris, Robert. Composition with Pitch-Classes: A Theory of Compositional Design.
Rahn, John. Basic Atonal Theory. (Out of print, but copies turn up from time to time.)

• Journals

Wikipedia offers a reasonable explanation of the basics: And this article also gives as concise a description of the basic concepts as I've seen.

• Thank for your response Aaron! I will check the link out to the article that you posted now. Please continue to share your feedback I actually do read and follow up with good and thorough responses.
– BLG
Jul 20, 2020 at 1:33
• @BLG: I'm looking for an example of how pitch classes are used analytically. I've found one and have requested permission to post. Jul 20, 2020 at 7:07
• @Aaron you can find easily in books, like Kostka's Materials of Post Tonal Music or Straus' Introduction to Post-tonal Theory. Plenty of analyses of different structures and points. And different composers among 20th century. Those ones you referenced here are somewhat denser and less "user friendly" for those not used to terminology and these math operations (but must read). Oct 10, 2020 at 23:04
• @RodrigoB.Furman Do you have access to an analytical example from one of those books? I wasn't able to get the permission I was looking for, so I'm looking for something else I could post, but can't access Kostka or Straus right now. Oct 10, 2020 at 23:10
• Yes, I have them both. But it's not too hard to find some examples. Kostka also has companion websites for his books. But you may search some scientific papers on analysis. And we have Open Music Theory: openmusictheory.com/postTonalAnalysis.html Oct 17, 2020 at 14:59

One of the main strengths of pitch-class sets is that they show what we call inversional equivalence.

When we think of something like chordal analysis (like saying something is a "C-major triad"), that labeling system shows octave equivalence. In other words, no matter what octave(s) the notes C, E, and G appear, they combine to create a C-major triad. All octaves are viewed as equivalent, hence the term.

Pitch-class sets take this one step further. In addition to octave equivalence, they treat inversion as equivalent—but inversion in a way different than first and second inversions of chords (with E and G in the bass, say). With regard to pitch-class sets, inversion invert the direction of intervals within a set.

As an example, let's take this C-major triad. From C, we go up a major third to E, then up a minor third from E to G. When we invert the direction of these intervals, C does down a major third to A♭, and this A♭ goes down a minor third to F.

Because these are the same intervals in inverted directions, pitch-class sets view them as the same due to inversional equivalence. This means—believe it or not—that from a pitch-class set standpoint the major triad (C, E, G) and the minor triad (like F, A♭, and C) are actually equivalent (!): both are members of the (037) set class.

And because of this inversional equivalence, using pitch-class sets can help you identify relationships between sets that would otherwise be viewed as entirely different constructs.