GeCo Triadic Generator System
Traditional jazz improvisation is often taught through scales, modes, and chord-scale relationships. In this approach, the improviser learns to associate a specific collection of notes with a particular chord or harmonic situation and develops melodic lines by moving through that collection.
While this method remains fundamental, many modern improvisers also employ an intervallic approach. Instead of thinking primarily in terms of scales, they organize musical material into smaller structures such as triads, triad pairs, and other interval-based patterns. These structures provide a different way of navigating harmony and often produce lines with greater intervallic variety and a more contemporary sound.
A triad pair may be viewed as a compact representation of a larger pitch collection. Rather than using all the notes of a parent scale, the improviser selects two complementary triads whose combined notes capture the characteristic color of the underlying harmonic context.
The objective is not to replace scalar improvisation, but to complement it. Scale-based and interval-based approaches can coexist and enrich one another. In practice, many musicians move freely between these two perspectives depending on the musical situation.
GeCo extends the concept of triad pairs by introducing a cyclic generation process. Two triads are not treated merely as a collection of notes, but as a structured path through pitch-class space. By traversing the notes of each triad according to a predefined rule, GeCo generates closed melodic cycles that can be explored, transposed, and transformed systematically.
The purpose of this document is to present the triadic generator systems currently implemented in GeCo and to provide practical examples of their use in modal, tonal, and contemporary harmonic environments. GeCo-Tool can generate melodic paths through pairs of triads. Rather than constructing melodic material from scales, the system uses two interacting triads that generate a cyclic path. Generation terminates when the starting note is encountered again.
For example, the triad pair
contains the two triads
The resulting GeCo path is
The last note is included in parentheses since it closes the cycle when the starting pitch is reached again.
Although triads have always been a fundamental component of Western music, their systematic use as melodic generators became particularly important in modern jazz improvisation.
John Coltrane was among the first major improvisers to explore highly structured intervallic concepts. During the late 1950s and 1960s his improvisational language increasingly relied on triadic structures, pentatonic cells, interval cycles, and harmonic superimpositions. Many of the melodic ideas found in works such as So What, Impressions, Mr. P.C., Giant Steps, and Countdown can be interpreted through intervallic and triadic thinking rather than purely scalar approaches.
The pedagogical formalization of these ideas was later developed by educators such as Jerry Bergonzi. Through his work on Hexatonics and Triad Pairs, Bergonzi demonstrated how two triads can be combined to create compact melodic resources applicable to modal, tonal, altered, and symmetrical harmonic situations.
Another important figure in the development of intervallic improvisation is George Garzone. Through his Triadic Chromatic Approach (TCA), Garzone demonstrated how triads can function as independent melodic building blocks capable of moving freely across harmonic environments. His work further reinforced the idea that triads can serve not only as harmonic structures but also as generators of melodic vocabulary.
The GeCo Triadic Generator System is inspired by these intervallic traditions. However, GeCo extends the concept by introducing explicit cyclic traversal rules. In this framework, a pair of triads is transformed into a closed melodic path, allowing systematic generation, transposition, and exploration of intervallic material.
Given two triads T₁ and T₂, a GeCo path is obtained by traversing the notes of the first triad followed by the notes of the second triad. The sequence is generated in pitch-class space and is therefore independent of octave placement.
The final note is shown in parentheses because it is not part of the generated collection itself. Its function is to indicate the closure of the cycle by returning to the starting pitch.
In general:
where R indicates the root, T indicates the third and F indicates the fifth. The note in parentheses marks the point at which GeCo encounters the starting pitch again and terminates the generation process.
A triadic generator is defined by three elements:
Different combinations of these parameters produce distinct melodic and harmonic environments. The following families represent the core generator models currently used in GeCo. In the examples below, most generators are built on the fourth and fifth degrees of the reference collection. This makes the relationship between the parent scale and the triadic generator immediately visible.
For clarity, note names within triadic generators are written according to their triadic function rather than strict scalar spelling. This choice makes the underlying triadic structure immediately visible to the reader.
The following table summarizes the principal generator families currently implemented in GeCo. Only one reference collection is shown for each family. Since GeCo supports automatic transposition, all remaining keys can be generated directly from the corresponding reference model.
| Family | Formula | Reference Collection | Generator | Application |
|---|---|---|---|---|
| Church Modes | M + M | C Major | F + G | Dm7, G7, Cmaj7, Fmaj7#11, Bm7b5 |
| Melodic Minor | M + Aug | A Melodic Minor | D + Eaug | AmMaj7, D7#11, E7alt |
| Harmonic Minor | m + M | A Harmonic Minor | Fm + G | AmMaj7, E7(b9) |
| Harmonic Major | M + Aug | C Harmonic Major | G + Abaug | Cmaj7(b6), G7(b9) |
| Whole Tone | Aug + Aug | C Whole Tone | Caug + Daug | C7#5, C7#11 |
| Diminished | M + M | C Diminished | C + F# | Cdim7, C7(b9) |
The generator column identifies the reference triad pair used by GeCo. All remaining keys can be obtained through transposition.
| Reference | Pair | GeCo Directional Sequence | Sound | GeCo Path |
|---|---|---|---|---|
| C Major | F + G | 4 3 −5 4 3 3 | Church Modes | F–A–C–G–B–D–(F) |
| A Melodic Minor | D + Eaug | 4 3 −5 4 4 2 | Melodic Minor | D–F#–A–E–G#–C–(D) |
| A Harmonic Minor | Fm + G | 3 4 −5 4 3 3 | Harmonic Minor | F–Ab–C–G–B–D–(F) |
| C Harmonic Major | G + Abaug | 4 3 −6 4 4 3 | Harmonic Major | G–B–D–Ab–C–E–(G) |
| C Whole Tone | Caug + Daug | 4 4 −6 4 4 2 | Whole Tone | C–E–G#–D–F#–A#–(C) |
| C Diminished | C + F# | 4 3 −1 4 3 −1 | Diminished | C–E–G–F#–A#–C#–(C) |
The GeCo directional sequence is calculated within a single octave. When the root of the second triad lies below the last note of the first triad, the corresponding interval is written as a negative value. This keeps both triads inside the same octave rather than treating the path as a purely ascending pitch-class cycle.
The triadic generator system can also be applied to altered dominant chords. A common jazz practice consists of approaching a dominant seventh chord through the melodic minor scale built a minor second above its root.
For example, over
an altered sound can be obtained using
whose notes are
This collection contains the characteristic altered tensions:
relative to the dominant chord. Consequently, triadic generators derived from the parent melodic minor collection can be used to create melodic material over the altered dominant.
One possible generator is
which produces the sequence
where the note in parentheses indicates closure of the cycle.
This example illustrates how GeCo generators can be used not only as abstract pitch collections but also as practical improvisational tools for modern harmonic environments.
Select the desired triad pair and enter the notes in GeCo-Tool. The generated path is cyclic and terminates automatically when the starting pitch is reached again. By changing the starting note or using transposition, the same generator can be applied in any key.
Possible future developments include: