These scales can be built either with the repetition of the tonic as the root for the second octave, or not. The easiest way of building such scales is by combining two different scales, like so:
There are many harmonic and melodic possibilities to be explored here but probably one of the most obvious ones is by simply writing a melody that uses the different notes present in each octave:
Melody is harmonized with chords derived from each tonal set; C Mixolydian from F major tonality (in red); and C Phrygian from Ab major tonality (in orange)
Chords with extensions that belong to the second octave of the scale; and/or polychords in which each chord is derived from its respective octave scale material are just two more options to explore the possibilities offered by this type of scale construction.
Same melody harmonized with extensions from each tonal set
Same melody harmonized with polychords from each tonal set
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