Trichords and Tetrachords

Trichords Overview

The Trichord Ear Trainer and Trichord Filter encompass a comprehensive collection of 3,960 unique trichords. These three-note combinations are derived from all possible intervals within specific pitch-class sets, providing a foundation for analysing symmetry, transpositions, permutations, and intervallic relationships. Each trichord is assigned a unique code representing its transformations, including majorisations, rotations with octave shifts, permutations of rotations, and transpositions.

Representation

For the trichords, we can utilise a numerical shorthand system to describe semitone distances between notes, with Roman numeral equivalents serving as primary reference points:

I (1+1): C → C# (1 semitone) → D (1 semitone)
II (1+2): C → C# (1 semitone) → D# (2 semitones)
III (1+3): C → C# (1 semitone) → E (3 semitones)
IV (1+4): C → C# (1 semitone) → F (4 semitones)
V (1+5): C → C# (1 semitone) → F# (5 semitones)
VI (2+2): C → D (2 semitones) → E (2 semitones)
VII (2+3): C → D (2 semitones) → F (3 semitones)
VIII (2+4): C → D (2 semitones) → F# (4 semitones)
IX (2+5): C → D (2 semitones) → G (5 semitones)
X (3+3): C → D# (3 semitones) → F# (3 semitones)
XI (3+4): C → D# (3 semitones) → G (4 semitones)
XII (4+4): C → E (4 semitones) → G# (4 semitones)

Symmetrical Trichords

Symmetrical trichords are characterised by equal interval pairs or rotational symmetry: I (1+1): C, C#, D. VI (2+2): C, D, E. X (3+3): C, D#, F#. XII (4+4): C, E, G#.

The above trichords are easy to spot given the numbers are the same, however the remaining symmetrical trichord, IX (2+5: C, D, G, presents a special case of symmetry. Whilst its initial numbers differ, its first rotation creates perfect fourths (5+5). For example, IX in prime form (C, D, G) becomes D, G, C in R1, displaying symmetrical intervals 5+5. Here’s the full list of symmetrical trichords:

I (1+1): C, C#, D,
VI (2+2): C, D, E,
X (3+3): C, D#, F#,
XII (4+4): C, E, G#,
IX R1 (2+5): D, G, C.

Majorisation

The following asymmetrical trichords can undergo majorisation:

II (1+2) → IIM (1+2)
III (1+3) → IIIM (1+3)
IV (1+4) → IVM (1+4)
V (1+5) → VM (1+5)
VII (2+3) → VIIM (2+3)
VIII (2+4) → VIIIM (2+4)
XI (3+4) → XIM (3+4)

Majorisation involves reversing the interval order. For instance, XI (3+4), representing a minor triad (C, Eb, G), becomes a major triad (C, E, G) when majorised to 4+3. This transformation is denoted by “M” in the trichord code.

You will be likely very familiar with e.g. X, XI and XII, these are the diminished (X), minor and major (XI, XIM) and the augmented triad (XII).

Clock Visualisation

The trichords can be visualised on a clock face, the trichord becomes a triangle, the points meet it’s intervallic position on the hours I-XII. This is a great way to display how the trichords can be steered into twelve-tone rows too (Tone Clock Theory). This visualisation method, developed by composers Peter Schat and Jenny McLeod, offers an intuitive approach to understanding trichord relationships. This website’s trichord and tetrachord ear training exercises utilise this clock formation. (Tetrachords use the same labelling system, the extra note is created via repetition of the first interval in the trichords, e.g. I (1+1) becomes I4 (1+1+1). I made a clock visualisation page for you to recreate rows from the tone clock or make your own, visit the Tone Clock Visualiser page.

The Collection

The complete collection comprises 3,960 unique trichords generated from the 12 prime form trichords. It incorporates:

The mentioned majorisations of asymmetrical trichords
Rotations + octave shift (first note moved to end and pitched up one octave)
The 6 possible permutations of the trichords, their majorisations and their rotations.
Transpositions across all 12 semitones for trichords I-XI and 4 transposition for XII (the augmented triad).

I’ve assigned every trichord and tetrachord a code made up of these elements.

Trichord Code Breakdown:

Parent Trichord (I-XII)
Majorisation indicator (M)
Rotation number (R)
Permutation designation (P)
Transposition (T0 is C, T1 is C#, etc.)

e.g. IIM R1P2T6 (resulting in G#5, A4, F#4)

Try it out on the trichord filter page

These codes are annotated on the trichord filter notation and displayed during feedback when you respond to intervals on the trichord ear trainer.

Listening/Links

Check out Jenny Mcleod’s 24 Tone Clocks