Music Performance
Compute interval class vectors from a pitch-class set.
What this calculator does
Interval class vectors are a tool from pitch-class set theory, an advanced compositional analysis system developed to study atonal and twelve-tone music. An interval class represents the shortest distance between any two pitches (counting up or down), reducing it to a class 1-6 (semitones 1-6, with class 6 being the tritone). The interval-class vector displays how many of each interval class appear in a musical collection or chord. For example, a major chord has different interval class frequencies than a diminished chord. Understanding interval-class vectors helps composers analyze and design harmonic density, consonance-dissonance profiles, and systematic pitch relationships for modern and experimental music.
How it works
The calculator takes four pitch classes (notes specified as 0-11, where C=0, C#=1, etc.) and examines all pairwise combinations (6 total pairs from 4 notes). For each pair, it calculates the shortest chromatic distance (the interval class, 1-6). It then counts how many times each interval class appears, generating six counts that form the interval-class vector. This vector reveals the chord's harmonic character: vectors rich in class-1 and class-2 intervals sound consonant; those emphasizing class-3 and class-6 sound more dissonant or unstable.
Formula
For each pair of pitch classes, calculate interval class = min(|pc₁-pc₂|, 12-|pc₁-pc₂|). Count occurrences for each interval class value 1-6. The resulting vector is [count₁, count₂, count₃, count₄, count₅, count₆], revealing the harmonic density and intervallic symmetry of the pitch set.
Tips for using this calculator
- Pitch classes ignore octave and register; C3 and C7 are both pitch class 0, allowing abstract harmonic analysis independent of instrumental context
- Symmetric pitch sets (like augmented triads: 0-4-8) produce symmetric interval-class vectors, useful for creating unified harmonic spaces in composition
- Compare interval vectors to identify chords with similar harmonic character; different chords may share identical vectors despite different pitch arrangements
- Use interval-class vector analysis to systematically explore musical rows in twelve-tone composition or to construct chords with specific sonic properties
- Resources like pitch-class set theory databases catalog all possible 4-note sets and their interval vectors, useful for composition planning
Frequently asked questions
Why is interval class calculated as the shortest distance between pitches?
Interval class represents perceptual interval identity independent of direction. The distance between C and G (7 semitones up) is perceptually equivalent to the distance from G down to C (5 semitones), both representing 'fifths.' By taking the minimum of both directions (5 semitones), interval class creates a unified measure. This abstraction is central to pitch-class set theory, treating pitch relationships symmetrically.
How do I interpret an interval-class vector?
An interval-class vector like [1,2,1,2,1,1] means: one semitone interval (class 1) appears 1 time, class 2 appears 2 times, class 3 appears 1 time, class 4 appears 2 times, class 5 appears 1 time, and tritone (class 6) appears 1 time. Higher counts indicate denser harmonic character. Vectors with many class-3 and class-6 intervals (tritones, minor seconds) tend to sound modern or dissonant, while major consonance emphasizes class-2, class-3, and class-4 intervals.
Can different chords have identical interval vectors?
Yes, this is called 'vector equivalence.' For example, a C major chord (C-E-G = 0-4-7) and F# major (F#-A#-C# = 6-10-1) have different pitches but identical interval vectors because they have the same interval relationships. This concept is powerful in composition: designing a vector creates a harmonic family of structurally equivalent chords that can be transposed or rearranged for variety while maintaining harmonic unity.
Is interval-class vector analysis useful for tonal music?
Interval-class vectors originated in atonal and twelve-tone analysis, but they apply to any music using twelve-tone equal temperament. For tonal music, vectors reveal underlying harmonic structure and can help composers understand why certain chords have specific characters. However, tonal analysis traditionally uses functional harmony (dominant, tonic, etc.), which interval vectors don't represent. Use interval vectors as a supplementary analytical tool for tonal music, not a replacement.
