Why use Pitch Class Sets to describe music?

(related to Pitch-class set theory ?)

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.)

FURTHER READING

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

Perspectives of New Music
Music Theory Spectrum

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.