IX. Post-tonal music
- Set theory often relies on the distinction between versus .
- Pitch classes are best represented with , where C=0.
Pitches are discrete tones with individual frequencies.
The concept of pitch, then, does not imply . C4 is a pitch, and it is not the same pitch as C3.
Throughout set theory, the word “class” means “group.” So a pitch class is a group of pitches. The group is a pitch and all other pitches related by and . You have probably encountered both of these concepts before, even if not by name.
Our system of letter names for notes implies : equivalence between pitches that are spelled the same but any number of octaves apart. C4 is the same as C3 is the same as C9, and so on, because they are all Cs.
When someone says “there are 12 notes,” they are implying . Two notes are enharmonically equivalent if you would press the same key on the piano to play them—even if the spelling is different. Enharmonic equivalence is the sense in which A♭ and G♯ are “the same.” While tonal music nearly always distinguishes between enharmonic pitches—A♭ as leads to G, but G♯ as leads to A. Post-tonal music is often different. Because many composers no longer felt constrained by a tonal center, the relationships amongst scale degrees and spellings aren’t important.
In summary, pitch classes are groups of pitches related by octave and enharmonic equivalence. A♭4, A♭3, G♯2, etc. are all members of the same pitch class.
If notes are enharmonically equivalent, then the system of seven letter names does not work well to describe the twelve pitch classes. Instead, in set theory, we use integer notation. All Cs, and any notes that are enharmonically-equivalent to C (B♯, for example), are pitch class 0. All C♯s, and any notes that are enharmonically-equivalent to C♯ (D♭, for example) are pitch class 1. And so on. To summarize with a few of the most common note names:
- C (B♯, etc.)
- C♯, D♭
- D (C𝄪, etc.)
- D♯, E♭
- E (F♭, etc.)
- F (E♯, etc.)
- F♯, G♭
- G (F𝄪, etc.)
- G♯, A♭
- A (G𝄪, etc.)
- A♯, B♭
- B (C♭, etc.)
Test your recall of integer names by filling in the blanks.
- Straus, Joseph Nathan. 2016. Introduction to Post-Tonal Theory. 4th ed. Upper Saddle River, NJ: Prentice Hall.
- Set Theory Quick Reference Sheet: summarizes the definitions of pitch vs. pitch class, intervals vs. interval classes, and sets vs. set classes.
- Pitch Space
- Post-tonal music is extremely various. Composers have individual compositional styles, aesthetic goals, and unique conceptions of pitch. All this is to say that you must approach a composition with flexibility. For example: because it is quasi-tonal, Debussy’s music often benefits from a view that does not assume enharmonic equivalence. But sometimes it does. You must rely on your musical intuitions when analyzing this music, and you should also be willing to approach pitch in these compositions from multiple perspectives until you find one that seems most appropriate. ↵
Refers to how "high" or "low" a sound is
All pitches that are equivalent enharmonically and which exhibit octave equivalence
A system of naming pitch classes that treats C as 0, C♯ as 1, D as 2, etc.
The assumption that pitches separated by one or more octaves are musically equivalent (e.g. an octave above "A" is "A")
Notes, intervals, or chords that sound the same but are spelled differently