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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⟩.[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. Each class has its own inversionally related presentations, while register, voicing, order, duration, and doubling can produce many audible realizations.[2]

The locked identity is: four distinct pitch classes modulo 12 + all six unordered dyads + interval-class reduction -> exactly one occurrence of each class 1 through 6. “All interval” refers to interval classes, not every directed interval from one to eleven, and not every compound or tuning-specific acoustic distance.

Structural Signature

  • the twelve pitch classes — octave-equivalent chromatic positions modeled as integers modulo 12;
  • the tetrachord — an unordered set of four distinct pitch classes, not necessarily a tertian chord;
  • the six dyads — every unordered pair among the four members;
  • the directed pitch-class distance — modular difference between pair members;
  • interval-class reduction — map distance d to min(d,12−d), producing classes 1–6;
  • the interval vector — six-component count of dyads in interval classes 1 through 6;
  • the all-ones criterion — every component equals one;
  • transpositional equivalence — adding a constant modulo 12 preserves the set class;
  • inversional equivalence — negating pitch classes, then transposing, preserves the Forte set class;
  • the two set classes — 4-Z15 and 4-Z29 exhaust the solutions under standard equivalence;
  • the Z relation — equal interval vector without Tn/TnI equivalence;
  • the realization layer — ordering, spacing, register, orchestration, rhythm, and emphasis shape musical use without changing membership.

Recognition requires computing all unordered dyads. Hearing “varied intervals” or locating several interval types is insufficient.

What It Is Not

  • Not an all-interval twelve-tone row. Such a row orders all twelve pitch classes so adjacent intervals exhibit a specified pattern; this node is an unordered four-class set.
  • Not every tetrachord. Most four-note sets repeat some interval classes and omit others.
  • Not a traditional tetrachord genus. Ancient and modal theory use “tetrachord” for four-note spans with scalar organization.
  • Not a tuning system. It assumes a pitch-class universe; All Fourths Tuning, the semantic neighbor, configures strings rather than a set class.
  • Not all directed intervals. Inversionally equivalent distances are collapsed into six classes.
  • Not one unique voicing. Transpositions, inversions, registers, doublings, and orderings create many sonorities.
  • Not a guarantee of a particular sound or function. Context and voicing influence perception.
  • Not identical to its interval vector alone. Two Z-related classes share the vector but differ in membership structure.

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. Childs studies structural and transformational properties of the two all-interval tetrachords and their relations rather than treating the vector as the end of analysis.[2]

The node is scoped to twelve pitch classes with octave equivalence and standard interval-class equivalence. Generalizing “all interval” to other equal divisions changes both the number of interval classes and the combinatorial conditions. Such generalized objects are related constructions, not unqualified instances.

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.

An interval vector discards which specific notes create each interval and how the dyads interlock. This is why 4-Z15 and 4-Z29 can be intervallically equivalent in aggregate without being transpositionally or inversionally equivalent. Z-relation is not a bookkeeping anomaly; it identifies information lost by the vector.

No current catalog node represents this music-theoretic condition. prime:constraint covers satisfaction of a declared restriction, while prime:complete_enumeration covers exhaustive accounting. Neither supplies modulo-12 pitch classes, interval vectors, Forte classes, or Z-relation.

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. Together they separate “what intervals occur” from “how pitch classes organize those intervals.”

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.
  5. Doubling a pitch class in a voiced chord does not change the underlying set but adds sounding dyads, so acoustic interval counts differ from set-class counts.
  6. Register changes preserve pitch class and interval class while changing compound intervals and voice-leading distances.
  7. An ordered arpeggiation can emphasize only some adjacency intervals even though the unordered set contains all six classes.
  8. Removing one pitch class produces a trichord with only three dyads, so it cannot satisfy the same six-class coverage property.
  9. Adding a fifth distinct pitch class creates ten dyads and necessarily repeats interval classes; “all-interval pentachord” would need a different definition.

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.

Examples

  • 4-Z15: prime form [0,1,4,6], with one dyad in each interval class;
  • 4-Z29: prime form [0,1,3,7], sharing the vector but not the set class;
  • transposition: adding four modulo 12 to [0,1,4,6] yields [4,5,8,10], still 4-Z15;
  • inversional presentation: an inversion of a representative remains inside its same Forte class;
  • orchestrated realization: the four pitch classes spread across multiple octaves remain the same set despite different sonority;
  • non-example: [0,1,2,3] repeats small interval classes and omits others.

Structural Tensions

  • aggregate equivalence vs. relational difference — the interval vector unites the two Z classes while their internal mappings differ;
  • set identity vs. audible realization — register and order do not change class but strongly change musical effect;
  • analytical compression vs. contextual function — a class label clarifies inventory while omitting rhythm, timbre, and syntax;
  • symmetry quotient vs. compositional choice — transposition/inversion equivalence aids analysis but composers may treat forms differently;
  • formal completeness vs. perceptual salience — all classes occur once even when some dyads are more prominent.

Structural–Framed Character

The all-interval tetrachord is predominantly structural. Once the modulo-12 pitch-class model and equivalence conventions are fixed, membership is exact and enumerable. The choice of analytical model is framed by musical practice, but community judgment does not change the vector.

Structural Core vs. Domain Accent

The core is six pair relations -> six equivalence categories -> one of each. The domain accent is twelve-tone pitch class, tetrachord, interval-class reduction, Forte notation, and Z-relation. Removing it yields a combinatorial Difference-Covering Set.

  • Constraint — the all-ones vector is an exact membership condition.
  • Complete Enumeration — every unordered dyad is counted once.
  • Equivalence Class — transposition and inversion quotient surface forms.
  • Information Loss — identical interval vectors do not determine one set class.

The prospective DAG uses composition under prime:constraint.

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 — is part ofConstraintPRIME

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

Neighborhood in Abstraction Space

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

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

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

Not to Be Confused With

  • all-interval twelve-tone row;
  • all-trichord hexachord;
  • traditional scalar tetrachord;
  • All Fourths Tuning;
  • one particular chord voicing;
  • interval vector as a complete invariant;
  • generalized all-interval sets in other tuning systems.

References

[1] Joseph N. Straus, Introduction to Post-Tonal Theory, 4th ed., W. W. Norton, 2016. registry ↩

[2] Adrian P. Childs, “Structural and Transformational Properties of All-Interval Tetrachords,” Music Theory Online 12(4), 2006, https://mtosmt.org/issues/mto.06.12.4/mto.06.12.4.childs.html. registry ↩a ↩b

[3] Arnold Whittall, The Cambridge Introduction to Serialism, Cambridge University Press, 2008. registry

[4] “All-interval tetrachord,” Wikipedia, frozen evidence packet, https://en.wikipedia.org/wiki/All-interval_tetrachord. registry