All-Interval Tetrachord

A four-note pitch-class set whose six unordered dyads realize each of the six interval classes exactly once, yielding interval vector ⟨1,1,1,1,1,1⟩.

Version

v1 · 2026-08-30 · History

Domain-specific #

1267

Origin domain

music

Subdomain

pitch class set theory

Aliases

AIT, All-interval four-note chord, All-interval tetrachord set class

Core Idea

An All-Interval Tetrachord is a set of four pitch classes in twelve-tone equal-tempered pitch-class set theory whose six unordered note pairs realize the six interval classes exactly once. Four elements have 4 choose 2 = 6 unordered pairs, and the modulo-12 system has six interval classes. The defining interval vector is therefore ⟨1,1,1,1,1,1⟩.

Up to transposition and inversion, exactly two four-note set classes have this vector: Forte classes 4-Z15, with prime form [0,1,4,6], and 4-Z29, with prime form [0,1,3,7]. They are Z-related: they share the same interval vector but cannot be mapped onto one another by transposition or inversion.

Scope of Application

The abstraction belongs to post-tonal and twelve-tone music theory, especially pitch-class set analysis and twentieth-/twenty-first-century composition. Analysts use it to identify harmonies with maximally distributed interval-class content for four notes. Composers can use either set class as a harmonic reservoir, a source of melodic cells, or material related through transposition, inversion, complement, partition, or voice-leading transformations.

The concept appears in analyses of music by Elliott Carter, George Perle, Frank Bridge, and other composers whose harmonic languages exploit interval-class collections.

Clarity

For [0,1,4,6], unordered differences reduce as follows: 0–1→1, 0–4→4, 0–6→6, 1–4→3, 1–6→5, and 4–6→2. Each class 1–6 appears exactly once. The vector is therefore all ones. The same computation for [0,1,3,7] also yields all ones.

Manages Complexity

A four-note chord can be voiced in many registers and orders, creating a large surface family. Pitch-class set reduction factors out octave, transposition, inversion, and order so the analyst can compare interval content. The all-interval label then compresses six dyadic facts into one exact property.

At the same time, retaining the pair of Z-related classes prevents overcompression. The common interval vector captures total interval-class inventory; the class label captures relational arrangement.

Abstract Reasoning

1. Any four-note pitch-class set has exactly six unordered dyads, so containing all six interval classes forces each to appear once. 2. Transposing an all-interval tetrachord preserves every modular difference and therefore the interval vector. 3. Inverting it replaces directed intervals with complements but preserves interval classes. 4. Two sets sharing ⟨1,1,1,1,1,1⟩ need not be Tn/TnI-equivalent; 4-Z15 and 4-Z29 are the canonical counterexample pair.

Knowledge Transfer

Exact transfer occurs across harmonic, melodic, and analytical contexts when the underlying unordered four-class set retains the all-ones interval vector. A vertical chord, distributed orchestral sonority, or temporally unfolded cell can instantiate the same set class.

The deeper pattern—use the smallest pair-count matching the number of categories to cover each exactly once—transfers combinatorially. In nonmusical systems it instantiates Coverage, Difference Set, or Constraint Satisfaction rather than this node.

Relationships to Other Abstractions

Current abstraction All-Interval Tetrachord Domain-specific

Parents (1) — more general patterns this builds on

• All-Interval Tetrachord is part of Constraint Prime

  the all-ones vector is an exact membership condition.

Hierarchy path (1) — routes to 1 parentless root

• All-Interval Tetrachord → Constraint

Neighborhood in Abstraction Space

All-Interval Tetrachord sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Viennese Trichord — 0.88
• A-Flat Major — 0.81
• Diminished Triad — 0.81
• Atonality — 0.79
• Diatonic and Chromatic — 0.79

Computed from structural-signature embeddings · 2026-09-08