Every trichord can be a generator except for {0, 3, 6} (diminished triad).
Select Three Pitches:
(Win: Control-Select) (Mac: Command-Select)

Pitch Name Pitch Number

This row calculator creates a 'derived series' or a twelve tone row composed of trichords all of which are from the same class. In this case, the calculator uses the prime, inverted, retrograde and retrograde inverted ordering of the trichord to derive a unique twelve pitch series. The results are presented in a matrix presenting all the transpositions of the prime form read from left to right and retrograde form read from right to left. The inverted form is read from top to bottom and the retrograde inverted form is read from bottom to top.

Webern's Concerto for Nine Instruments, Op. 24 is often cited as an example of the use of this technique in a composition:
Prime (0): {G, B, Bb}
Retrograde Inversion (11): {D#, D, F#}
Retrograde (6): {E, F, C#}
Inversion (5): {C, Ab, A}
In this example, each of the trichords belongs to {0, 1, 4}.

Currently, this calculator only presents one possible ordering of the trichord sets that compose the derived series. An example of this limitation can be seen by trying to enter the trichord given above {G, B, Bb}. The entered trichord becomes {Bb, B, G} or {0,1,4} resulting in transpositions for each trichord different than Webern's solution. Future enhancements of this calculator will attempt to return all possible derived series.

Tetrachords can also be used to generate unique twelve tone rows All tetrachords that exclude interval class 4 (Major third) can be a generator. Future enhancements of this calculator will include tetrachord derived series.

Reference: Joseph N. Straus: Introduction to Post-Tonal Theory, pp. 130-132, Prentice-Hall, 1990.

Comments and questions are always welcome. Please email them to Sounding Stone Management