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The extended augmented scale
(Messiaen mode 3)

The basic augmented scale is a six-note mode created by alternating the interval of a minor third with semitone steps. However, it offers fairly limited harmonic resources, and can be extended by the addition of a third augmented triad. By this means the scale retains its transpositional possibilities - it can be transposed four times, like any augmented triad - but gains harmonic flexibility.

The resulting mode is listed as one of the possible "modes of limited transposition" by Messiaen. It has three modes. Here are the conventional chord shapes available in the three modes. In this example, mode I has root C, mode II has root D, and mode III has root Eb. These roots are reflected at major third intervals.

Another way of thinking about the harmonic possibilities in the three modes is to notice what tones are lacking in relation to the bass notes:

Mode 1 lacks b2 (b9), 4 (11) and 6 (13)

Mode 2 lacks maj7, m3 (#9) and 5

Mode 3 lacks b7, 2 (9) and #4 (#11).

Below, I have extrapolated the parallel chord sequences derived from various chord-types at mode I.

What appears to be strings of chords is actually a linear representation of a circular phenomenon. Here is one of the parallel sequences represented as a circle.