PC-Set

Pitch Class Set Theory for Python

A library for analyzing and manipulating pitch class sets using musical set theory. Useful for atonal music analysis, composition, and music theory education.

Installation

pip install pc-set

Quick Start

from pc_set import PCSet

# Create a pitch class set (C=0, C#=1, ..., B=11)
s = PCSet([0, 4, 7])  # C major triad: C-E-G

# Get prime form (canonical representation)
print(s.prime_form())  # (0, 3, 7) - minor triad is prime form

# Get Forte number (set class identifier)
print(s.forte_number())  # "3-11"

# Get interval vector (harmonic content)
print(s.interval_vector())  # (0, 0, 1, 1, 1, 0)

# Transpose
print(s.transpose(5))  # PCSet([5, 9, 0]) - F major

# Invert
print(s.invert())  # PCSet([0, 8, 5]) - C-Ab-F

What are Pitch Class Sets?

In music theory, a pitch class groups all notes with the same name across octaves:

• C4, C3, C5 → all belong to pitch class C (or 0)
• We use integers 0-11: C=0, C#=1, D=2, ..., B=11

A pitch class set is an unordered collection of these pitch classes, used to analyze atonal music and discover harmonic relationships.

Features

Basic Operations

from pc_set import PCSet

# Create sets
major = PCSet([0, 4, 7])    # C-E-G
minor = PCSet([0, 3, 7])    # C-Eb-G
diminished = PCSet([0, 3, 6, 9])  # C-Eb-Gb-A

# Transposition
print(major.transpose(7))   # G-B-D

# Inversion
print(major.invert())       # C-Ab-F

# Normal order (most compact arrangement)
print(PCSet([8, 0, 4]).normal_order())  # (8, 0, 4)

# Prime form (canonical version)
print(major.prime_form())   # (0, 3, 7)

Set Analysis

# Forte number (set class classification)
print(major.forte_number())  # "3-11"

# Interval vector (interval content)
print(diminished.interval_vector())  # (0, 3, 0, 3, 0, 0)
# Interpretation: 3 minor thirds, 3 tritones

# Subset relationships
print(PCSet([0, 4]).is_subset_of(major))  # True
print(major.is_superset_of(PCSet([0, 4])))  # True

# Get all subsets of a given size
for subset in major.subsets(2):
    print(subset)  # {C, E}, {C, G}, {E, G}

Z-Relations

Z-related sets have the same interval vector but different prime forms:

# Two hexachords that are Z-related
s1 = PCSet([0, 1, 4, 5, 7, 8])
s2 = PCSet([0, 2, 3, 5, 8, 9])

print(s1.interval_vector())  # Same as s2
print(s2.interval_vector())  # Same as s1
print(s1.is_z_related_to(s2))  # True

Integration with twelve-tone

PC-Set works great with the twelve-tone package for serial composition:

from twelve_tone import Composer
from pc_set import PCSet

# Generate a twelve-tone row
c = Composer()
c.compose()
row = c.get_melody_integers()

# Analyze hexachords
hexachord1 = PCSet(row[:6])
hexachord2 = PCSet(row[6:])

print(f"First hexachord: {hexachord1.prime_form()}")
print(f"Second hexachord: {hexachord2.prime_form()}")
print(f"Z-related: {hexachord1.is_z_related_to(hexachord2)}")

API Reference

PCSet Class

Initialization

• PCSet(pitch_classes): Create from list of integers 0-11

Properties

• .pitch_classes: Tuple of pitch classes
• .prime_form(): Canonical form starting at 0
• .forte_number(): Forte classification (e.g., "3-11")
• .interval_vector(): Interval content tuple

Operations

• .transpose(semitones): Transpose by semitones
• .invert(): Invert around pitch class 0
• .normal_order(): Most compact rotation
• .is_subset_of(other): Check subset relationship
• .is_superset_of(other): Check superset relationship
• .subsets(size): Get all subsets of given size
• .is_z_related_to(other): Check Z-relation

Background

Pitch class set theory was developed in the 1960s-70s by music theorists including Allen Forte, John Rahn, and others to analyze atonal music (Schoenberg, Webern, Berg, Babbitt). It provides a mathematical framework for understanding harmonic relationships in music without traditional tonality.

Key concepts:

• Normal Order: Most compact arrangement of a set
• Prime Form: Canonical representation (transposed to start at 0)
• Forte Number: Classification system (e.g., "3-11" for major/minor triads)
• Interval Vector: Distribution of interval classes in the set
• Z-Relation: Sets with same interval vector but different prime forms

License

BSD 2-Clause License - see LICENSE for details.

Contributing

Contributions welcome! This is a learning project exploring music theory through code.

Related Projects

• twelve-tone - Twelve-tone matrix generator
• music21 - Comprehensive music analysis toolkit
• teoria - Music theory library (JavaScript)