Symmetric and Asymmetric Trichord Pairs Over Common Chord Progressions
By Bruce Arnold — Muse Eek Publishing Inc. — ISBN 978-1-59489-180-9
Two trichords. One hexachord. Every key.
Sonic Twins takes trichord superimposition to the next level — pairing two trichords that together cover six notes of the chromatic scale, forming a hexachord with specific harmonic properties over each chord type. Systematic, audible, and immediately musical. This course contains all usable triad pairs but expands to include all 3 note combination that are not built in 3rds.
What’s included with the book
The full book in PDF, with access to all downloadable files: chord voicings, arpeggios, scale runs, étude melodies, étude melodies with rests, vamp accompaniments, vamp étude files, and dyad sequencing — for all 328 trichord pairs, across all twelve keys.
In Sonic Single, you work with one trichord at a time — learning to hear and execute each three-note structure over each chord type in every key. Sonic Twins takes the next step: instead of one trichord at a time, you work with pairs.
A trichord pair is two trichords chosen so that together they cover six distinct pitch classes — a hexachord. It is the same concept as two triad pairs but expands to include any 3 note combination. The two trichords share no notes. When you alternate between them over a chord progression, you produce a six-note sound world that is harmonically complete, intervalically rich, and recognizably different from any single trichord.
Symmetric pairs use two transpositions of the same trichord prime form. The two trichords share no notes but have identical interval structures. Their combined hexachord is inversionally symmetric — it maps onto itself under inversion. Sonic Twins covers 69 symmetric pairs across all twelve trichord types.
Asymmetric pairs combine two different trichord prime forms. The hexachord produced may or may not be inversionally symmetric. Sonic Twins covers 259 asymmetric pairs. Together the 328 pairs represent the complete curated set of musically useful trichord pairings across all eight chord types.
Within the symmetric pairs, 249 produce inversionally symmetric hexatonic scales when considered over specific chord types. The table below shows the count of symmetric hexatonic pairs available for each trichord type over each chord type:
| Trichord | Maj7 | min7 | dom7 | 7sus4 | minMaj7 | min7b5 | dim7 | Maj7#5 | Total |
|---|---|---|---|---|---|---|---|---|---|
| 012 | — | — | 9 | 9 | — | — | — | — | 18 |
| 013 | 1 | 1 | 7 | 7 | 2 | — | 2 | — | 20 |
| 014 | — | — | 7 | 5 | 1 | — | 2 | — | 15 |
| 015 | — | — | 5 | 8 | — | — | — | — | 13 |
| 016 | — | — | 12 | 8 | — | — | 4 | — | 24 |
| 024 | 2 | 2 | 1 | 9 | — | — | — | — | 14 |
| 025 | — | — | 7 | 6 | — | — | 2 | 1 | 16 |
| 026 | — | — | 12 | 9 | 3 | — | 4 | — | 28 |
| 027 | 3 | 3 | 9 | 15 | — | — | — | 1 | 31 |
| 036 | — | — | 16 | 9 | 4 | 1 | 16 | — | 46 |
| 037 | 1 | 1 | 7 | 8 | 2 | 1 | 2 | — | 22 |
| 048 | — | — | 1 | 1 | — | — | — | — | 2 |
| Total | 7 | 7 | 93 | 94 | 12 | 2 | 32 | 2 | 249 |
The fastest way into the sound of a trichord pair is to listen to music built from it. The compositions below use specific trichord pairs consistently throughout — the melodies and harmonies derive from the two trichords moving through the chord changes.
Trichord 015 over Rhythm Changes (I Got Rhythm). The available tones for each chord determine which 015 pair positions appear — watch how the pair shifts as the harmony moves through the A and B sections.
Trichord 015 over the changes to Back Home in Indiana. A contrafact built entirely from 015 pair positions — the trichord pair is the sole pitch source throughout.
Trichord 015 over Rhythm Changes with a bass groove feel. Three compositions using the same trichord over the same progressions — each one demonstrates a different approach to deploying the pair.
Each file type in the Sonic Twins library is shown below with a representative example. The complete library contains files for all 328 trichord pairs across all twelve keys — these examples show the format and sound of each type.
Hexachord: C Db Eb E F Ab | Works over: dom7, 7sus4
Six closed-position voicings of the trichord pair cycle through ascending and descending inversions across all twelve keys. Learn the shapes first, then transpose. The goal is automatic recognition of each voicing by ear and by touch.
Same 6 inversions as chord voicings — arpeggiated in 6/8, all 12 keys
The same six inversions as the chord voicings, presented as flowing 6/8 melodic lines. The shapes are identical — but your ear now hears them as melody rather than harmony. This is the bridge between chordal and linear use of the pair.
Hexachord: C D E Gb G B | 2 octaves up + down, all 12 keys
The six-note hexachord as a two-octave ascending and descending scale, starting from middle C. Each key occupies one line. Final note is a dotted half. This is the most direct way to hear the hexachord as a single pitch collection before breaking it into its component trichords.
Drums + bass + chords | 12 bars loopable | Key of C
A loopable drum, bass, and chord track — the backing groove for practicing the étude material. Twelve bars, six ascending inversions and six descending, loops cleanly. Use this as your practice backing track while working through the étude files.
Hexachord: C Eb E Gb G Bb | Works over: dom7, dim7 | Key of C
A 32-bar trombone melody using the trichord pair as its pitch source — quarters, eighths, and eighth triplets. The melody alternates freely between the two trichords. Available as melody only, or merged with the vamp accompaniment for full band context.
30% quarter rests on beats | Key of C
The same 32-bar melody with approximately 30% of the quarter-note beats replaced by quarter rests. The rests create phrasing and breathing room. Use this version after the standard étude feels automatic — the rests reveal whether you are truly hearing the material or just executing it mechanically.
Hexachord: C Db E F A B | Key of C
An asymmetric pair combining trichords 015 (C Db F) and 027 (E A B). The hexachord C Db E F A B has a wide interval range — both a half step (C–Db) and a major seventh (Db–C at the octave) — giving the melody a dramatic, wide-ranging character.
Works over: min7 | Dyad pair: C-Eb / D-F | All 12 keys
Each trichord has three dyads; a pair has nine combinations. Dyad sequencing works one combination through three octaves ascending and back down, across all twelve keys. This is the microscope of the Sonic Twins system — isolating each two-note subset to make the pair’s interval content fully transparent.
Every file type is available as a complete zip download from right inside the PDF. These downloads are extensive and covering all 328 trichord pairs across all twelve keys. Various types of exercises are presented. Below is a list:
| File Set | Contents | |
|---|---|---|
| Chord Voicings — Symmetric | 69 pairs × all 12 keys. PDF + MIDI. | |
| Chord Voicings — Asymmetric | 259 pairs × all 12 keys. PDF + MIDI. | |
| Trichord Pair Arpeggios — Symmetric | 69 pairs × all 12 keys. PDF + MIDI. | |
| Trichord Pair Arpeggios — Asymmetric | 259 pairs × all 12 keys. PDF + MIDI. | |
| Hexachord Scale Runs | All 328 pairs. PDF + MIDI. | |
| Étude Melodies | All 328 pairs × 12 keys. PDF + MIDI. | |
| Études With Rests | All 328 pairs × 12 keys. PDF + MIDI. | |
| Vamp Accompaniments | All 328 pairs × 12 keys. MIDI. | |
| Vamp Études with Melody | All 328 pairs × 12 keys. MIDI. | |
| Vamp Études with Melody + Rests | All 328 pairs × 12 keys. MIDI. | |
| Dyad Sequencing | All 249 symmetric pairs, 9 combinations each, all 12 keys. PDF + MIDI. |
| Level | Resource | What It Builds |
|---|---|---|
| 1 | Ear Training One Note Complete | Identify any of the twelve chromatic pitches against a key center — by ear, not by counting intervals. |
| 2 | Contextual Ear Training | Hear pitches in context of moving chord progressions — how function changes what a note sounds like. |
| 3 — Critical | SerialEar | Hear all twelve trichords in every inversion and transposition against a drone. Without this, the Sonic Twins étude material is mechanical rather than audible. |
Not sure where to start? The free Ear Training Guided Tour maps the full curriculum.
| Book | How It Relates |
|---|---|
| Sonic Single | Recommended by not required. Complete Sonic Single before starting Sonic Twins especially if you are new to using trichords or pivoting between two groups of three notes. |
| Sonic Resource Guide | The complete theoretical reference for every possible use of every three-note combination across every chord type. Plus a lot more. Think of this book as your Thesaurus Of Scales which shows you every chord that works over every scale. Massive 800 page book. |
| SerialEar | Trains the ear to hear all possible three-note structures in every inversion and transposition. Essential for making the Sonic Twins material audible. This course is a life changing method to develop your ability to hear all three note structures instantly. A must for anyone who wants to truly play by ear! |
| The Sound Cells Series | 78 volumes covering all possible trichord pair combinations with compositions, étude files, and reference material. A profound collection of exercises that give you the big picture and a deep dive into hexatonics and trichord pairs. |
| MetroDrone® | Combined drone and metronome essential for key-center ear training. Every file in this book is designed to be practiced against a drone. MetroDrone can be used for ear training but also to develop the ability to “feel time not count time.” |
Do I need to complete all of Sonic SingleSonic Single before starting Sonic Twins?
You need a solid foundation — comfortable with several trichords across multiple chord types in all twelve keys. You do not need to have completed all twelve trichords. The minimum is a genuine ear-level fluency with the trichords you plan to work with in pairs.
How many trichord pairs does Sonic Twins cover?
328 pairs in total — 69 symmetric and 259 asymmetric. Each pair has files for all twelve keys across all applicable chord types.
What is the difference between symmetric and asymmetric pairs?
A symmetric pair uses two transpositions of the same trichord prime form whose combined hexachord is inversionally symmetric. An asymmetric pair combines two different trichord prime forms. Both are fully covered in Sonic Twins.
What instrument is this written for?
The book and étude files are instrument-agnostic. The étude melodies are notated in treble clef for trombone range but can be played on any pitched instrument. The chord voicings are most directly useful for pianists and guitarists. All MIDI files work with any DAW or MIDI player.
How does Sonic Twins relate to the Sound Cells series?
The Sound Cells series covers all 78 possible trichord pair combinations in individual volumes with compositions and recordings. Sonic Twins is the comprehensive systematic course — étude files, position references, and reference tables for all 328 pairs across all chord types and keys.
The principle of hearing music from a tonal center was passed forward through three centuries of teaching. The core chain: Scarlatti → Porpora → Haydn → Beethoven → Czerny → Vengerova → Madame Chaloff → Banacos → Bruce Arnold.
Bruce Arnold studied with Charlie Banacos and Jerry Bergonzi. That insistence on hearing every note in relationship to a single key center is the foundation on which the entire Sonic Twins system is built.
Questions? sales@muse-eek.com — muse-eek.com
