# Twelve-Tone Music — Invariance

*Invariance* refers to the preservation of something: intervals, dynamics, rhythms, pitches, and so on. In elementary twelve-tone theory, we are mostly concerned with *intervallic* invariance and *pitch class segmental* invariance.

**Intervallic Invariance**

Any time a row is transposed, the ordered intervallic content of the row is unchanged. Thus, transposition always results in intervallic invariance. Retrograde inversion creates retrograde intervallic invariance.

**Segmental Invariance**

When a pitch-class segment of a row is unchanged when that row is transformed, we say that the segment is “held invariant.” Consider the following example, from Webern’s String Quartet, Op. 28:

The brackets show the discrete tetrachords of the row. Notice that these tetrachords are the *same* amongst the to different rows. That is, the tetrachords are *invariant segments.* These segments are held invariant because of they share the same *relationship* with one another that the rows share. Because the tetrachords are related by *T8*, when the row *as a whole* is transposed by *T8*, those tetrachords are “held invariant.” (Think of the process like this: when the first tetrachord [6789] is transposed by *T8*, it becomes the last tetrachord [2345]. And therefore, when the whole row is transposed by *T8, _the last tetrachord _becomes* the first tetrachord.)

To determine when and if a pitch-class segment of a row will be held invariant:

(1) Find an equivalent set-class elsewhere in the row. This may be a dyad, trichord, tetrachord, etc.

(2) Determine the transpositional or inversional relationship between them.

(3) When the row is transposed or inverted by that *same* relationship a segment will be held invariant.