A tritriadic scale is a 7-note just intonation scale generated from a chain of three T:M:D triads whose roots are separated by the D/T interval (where T stands for "tonic", M for "mediant" and D for "dominant")[1]. These three chords can be interpreted as (pseudo)subdominant, root and (pseudo)dominant. Since a tritriadic scale is generally assumed to be octave-repeating, it is obtained by octave-reducing the notes from all three chords so that they fit within an octave.

A tritriadic scale is a special case of a cross-set scale. We can express a T:M:D tritriadic scale as CrossSet({1/1, M/T, D/T}, {T/D, 1/1, D/T}).

Also, this scale type is a kind of a generator sequence scale: GS(T:M:D)[7], or GS(M/T, D/M)[7].

The concept of tritriadic scales was first developed by John Chalmers in 1986[2].

Example

To build a tritriadic scale, the first step is to choose a triad. For this example, the 6:7:9 triad will be used.

The root chord gives the first three notes of the scale. In this example, these notes are {6/6, 7/6, 9/6}, which becomes {1/1, 7/6, 3/2} after simplification and reduction.

Next, to build the triad on the (pseudo)dominant, multiply all the notes from the root chord by D/T. In this example, the dominant is 3/2, so the triad is {3/2, 21/12, 9/4}, which becomes {3/2, 7/4, 9/8} after simplification and reduction.

Then, to build the triad on the (pseudo)subdominant, multiply all the notes from the root chord by T/D. In this example, the subdominant is 2/3, so the triad is {2/3, 14/18, 1/1}, which becomes {4/3, 14/9, 1/1} after simplification and reduction.

Finally, assemble all the notes in ascending order (without repeating the tonic and the dominant) to build the scale. The unison can be replaced by the octave for compatibility with the Scala scale file format. In this example, we get {9/8, 7/6, 4/3, 3/2, 14/9, 7/4, 2/1}, which is the Tritriad69 scale.

References

1. "tritriadic". www.tonalsoft.com. Retrieved July 24, 2021.
2. "Xenharmonikôn Tables of Contents". xh.xentonic.org. Retrieved 24 July, 2021.