Pitch Set Calculator

Introduction to Pitch Set Theory

In the realm of contemporary and atonal music, traditional harmonic analysis often falls short. This is where Pitch Set Theory, largely formalized by music theorist Allen Forte, becomes an invaluable tool. It provides a systematic way to classify and relate collections of pitches, offering insights into the underlying structures of compositions that deviate from conventional tonality.

At its core, pitch set theory treats musical pitches as abstract mathematical entities called "pitch classes." By abstracting away octave and enharmonic equivalence, we can focus on the fundamental intervallic relationships within a collection of notes, regardless of their specific registral placement or notation.

Key Concepts in Pitch Set Analysis

Pitch Classes

A pitch class represents all pitches that are an octave apart (or multiples of an octave). In Western music, there are 12 distinct pitch classes, typically numbered 0 through 11. This numbering system is based on C=0, C#=1/Db=1, D=2, and so on, up to B=11.

• C / B# / Dbb = 0
• C# / Db = 1
• D / C## / Ebb = 2
• D# / Eb = 3
• E / Fb / D## = 4
• F / E# / Gbb = 5
• F# / Gb = 6
• G / F## / Abb = 7
• G# / Ab = 8
• A / G## / Bbb = 9
• A# / Bb = 10
• B / Cb / A## = 11

A "pitch set" is simply an unordered collection of unique pitch classes. For example, the notes C, E, G (a C major triad) form the pitch set {0, 4, 7}.

Normal Form

The Normal Form of a pitch set is its most compact possible ordering. It helps us understand the intervallic "span" of a set and is crucial for comparing different sets. To find the normal form, we consider all possible rotations of the set and transpose each rotation so it starts on 0. We then choose the rotation that has the smallest interval between its first and last element. If there's a tie, we look for the one with the smallest interval between the first and second element, and so on.

For example, for the set {0, 1, 6}:

• {0, 1, 6} (span 6)
• {1, 6, 0} → {0, 5, 11} (span 11)
• {6, 0, 1} → {0, 7, 8} (span 8)

The normal form is [0, 1, 6] because it has the smallest span (6).

Prime Form

The Prime Form is the most fundamental representation of a pitch set. It's the most compact ordering of the set, considering both its original form and its inversion. It allows us to identify sets that are related by transposition and/or inversion as belonging to the same "set class."

To find the prime form:

1. Determine the normal form of the original set.
2. Determine the normal form of the inverted set (each pitch class p becomes 12 - p mod 12).
3. Compare these two normal forms. The one that is "more compact" (using the same criteria as normal form: smallest span, then smallest intervals from the beginning) is the prime form.
4. The prime form is always transposed to start on 0.

For example, the set {0, 4, 7} (C major triad):

• Normal Form (original): [0, 4, 7]
• Inversion: {0, 8, 5} → sorted {0, 5, 8}
• Normal Form (inverted): [0, 3, 7] (derived from {5,8,0} transposed to {0,3,7})

Comparing [0, 4, 7] (span 7) and [0, 3, 7] (span 7). Both have span 7. Next, compare the second element: 4 vs 3. Since 3 is smaller, [0, 3, 7] is the prime form. This means a major triad {0,4,7} and a minor triad {0,3,7} are inversionally equivalent within set theory.

Interval Vector

The Interval Vector is a six-digit number that summarizes the intervallic content of a pitch set. Each digit represents the count of a specific interval class (1-6) present in the set. Interval classes are symmetrical: an interval of x semitones is equivalent to an interval of 12-x semitones. For instance, a major 7th (11 semitones) belongs to interval class 1 (12-11=1).

• Digit 1: Count of interval class 1 (minor 2nd, major 7th)
• Digit 2: Count of interval class 2 (major 2nd, minor 7th)
• Digit 3: Count of interval class 3 (minor 3rd, major 6th)
• Digit 4: Count of interval class 4 (major 3rd, minor 6th)
• Digit 5: Count of interval class 5 (perfect 4th, perfect 5th)
• Digit 6: Count of interval class 6 (tritone)

The interval vector provides a concise way to characterize the sonority or "sound" of a set, as sets with similar interval vectors tend to have similar sonic qualities.

Forte Names

Allen Forte assigned unique names to each prime form, known as Forte Names. These names consist of two numbers separated by a hyphen, e.g., "3-11". The first number indicates the cardinality (number of pitches) in the set, and the second number is an index assigned by Forte. These names provide a standardized way to refer to specific set classes in theoretical discussions.

For example, the prime form [0, 3, 7] (minor triad) is known as 3-11. The prime form [0, 4, 8] (augmented triad) is 3-12.

Using the Pitch Set Calculator

This calculator simplifies the process of pitch set analysis. Simply enter your desired pitches, separated by spaces or commas, using either pitch class numbers (0-11) or standard note names (e.g., C, C#, D, Eb, F#, G, Bb). The calculator will then:

• Parse your input into unique, sorted pitch classes.
• Determine the Normal Form of the set.
• Calculate the Prime Form, the most compact representation.
• Identify the Forte Name for the set class.
• Generate the Interval Vector, showing the intervallic content.
• Indicate the T/I operations required to transform the original set into its prime form.

This tool is useful for composers experimenting with atonal sonorities, music theorists analyzing complex scores, and students learning the fundamentals of 20th-century music analysis.

Conclusion

Pitch set theory offers a powerful framework for understanding and manipulating musical collections outside of traditional tonal contexts. By systematically analyzing pitch classes, normal forms, prime forms, interval vectors, and Forte names, musicians can gain deeper insights into the structural and sonic properties of modern music. Use this calculator as a practical aid in your exploration of this fascinating area of music theory.