Skip to content

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
v2 · 2026-09-19 · History
Domain-specific #
1267
Domain group
Arts & Aesthetic Practice
Origin domain
Music & Musicology
Subdomain
Pitch Class Set Theory → Music & Musicology
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.

How would you explain it like I'm…

Four Notes, Every Gap Once

Pick four different notes. Every pair of those notes has a certain kind of gap between them, and with four notes there are six pairs. In an all-interval tetrachord, the six pairs each have a different kind of gap, so every kind of gap shows up exactly once. There are only two basic ways to do this.

Four Notes, Six Different Gaps

In music, the gap between two notes is called an interval. Music theorists who study the twelve notes of the octave group intervals into six basic 'interval classes', from the smallest step up to the halfway point of the octave. Four notes make six pairs, and an all-interval tetrachord is a set of four notes where those six pairs use each of the six interval classes exactly once. If you count flipping a set upside down or sliding it up or down as the same, there are exactly two such sets. You can play the notes in any order or octave and it's still the same set.

Complete Interval-Class Tetrachord

In pitch-class set theory, which treats the twelve notes of equal temperament as numbers 0 to 11 counted modulo 12, the distance between two notes reduces to one of six interval classes (1 to 6). A set of four pitch classes, a tetrachord, has exactly 6 unordered pairs, so it is possible for them to cover all six interval classes once each. A tetrachord that does this is an all-interval tetrachord, and its interval vector, the count of each interval class, is ⟨1,1,1,1,1,1⟩. Up to transposition and inversion there are exactly two such set classes: Forte 4-Z15, prime form [0,1,4,6], and 4-Z29, prime form [0,1,3,7]. They are 'Z-related': they have the same interval vector but cannot be turned into each other by transposing or inverting. 'All interval' means all interval classes, not every one of the eleven directed intervals.

 

An all-interval tetrachord is a set of four distinct pitch classes in twelve-tone equal temperament whose six unordered pairs realize each of the six interval classes exactly once. The counting fits exactly: four elements give C(4,2) = 6 unordered dyads, and pitch-class space modulo 12 has six interval classes, so the defining interval vector is ⟨1,1,1,1,1,1⟩. Up to transposition and inversion, exactly two set classes have this vector: Forte 4-Z15 with prime form [0,1,4,6], and 4-Z29 with prime form [0,1,3,7]. These are Z-related, sharing an interval vector while not being equivalent under transposition or inversion, and each has its own inversionally related forms. Register, voicing, ordering, duration, and doubling yield many audible realizations of the same set, none of which affect membership. The phrase 'all interval' refers to interval classes after reduction, not to all directed intervals 1 through 11, nor to compound intervals or tuning-specific acoustic distances.

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

Local relationship map for All-Interval TetrachordParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.All-IntervalTetrachordDOMAINPrime abstraction: Constraint — includesConstraintPRIME

Current abstraction All-Interval Tetrachord Domain-specific

Parents (1) — outgoing typed relations

  • All-Interval Tetrachord includes Constraint Prime

    the all-ones vector is an exact membership condition.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

All-Interval Tetrachord sits in a sparse region of the domain-specific corpus (86th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Musical Intervals & Voice Texture (9 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25