# VIII. 20th- and 21st-Century Techniques

This section introduces some common methodologies for the analysis of 20th- and 21st-century music, including .

# Prerequisites

This section assumes a familiarity with the topics covered in Fundamentals.

# Organization

The first five chapters introduce students to the building blocks of set theory: , pitches vs. , intervals and , and , , and interval class vectors.

The chapter Analyzing with Set Theory (or not!) is an important conclusion to these earlier chapters: it discusses the philosophy of segmentation and also turns a critical eye to set theory as a methodology. What makes us group notes together and call them sets? What are we ignoring when we use set theory?

The final three chapters discuss collections, including modes, the octatonic collection, and others. Students are also taught here how tonic pitches can be heard within a composition using these collections.

A methodology for analyzing pitch in atonal music. Pitch classes are given an integer name (0–11, where C is 0, C♯ is 1, etc.). Groups of pitches are considered together as "sets." Sets may be related by inversion or transposition.

A system of naming pitch classes that treats C as 0, C♯ as 1, D as 2, etc.

All pitches that are equivalent enharmonically and which exhibit octave equivalence

The smallest possible distance between two pitch classes. The largest interval class is 6, because if order is disregarded, the tritone is the largest possible interval. A P5 can be inverted to a smaller P4, m6 to M3, and so on.

A group of pitch classes.

A group of pitch class sets related by transposition or inversion. Set classes are named by their prime forms. E.g., (012) is a set class.

In set theory, "operations" refers to transposition and inversion.