Equal Divisions of the Octave
In the 19th century (and even as early as the late 18th century), composers became more interested in creating ambiguity in their music; avoiding chord progressions that explicitly confirmed a tonal center allowed them to thwart listeners’ expectations. One way of achieving this ambiguity was to use chord progressions in which the roots of the chords divide the octave equally. In a purely diatonic system, chord progressions tend to divide the octave unequally–think about the tritone leap in a descending-fifths sequence, or the half steps in an ascending-step sequence, for instance. In a tonal system, the asymmetry of major and minor scales serves to pull chords and scale degrees towards a tonic. Conversely, root progressions that divide the octave equally quickly bring the music outside of the diatonic scale. For example, consider a root progression by major thirds: C–E–G♯/A♭–C. While the chords built upon C and E could certainly be harmonized by chords within the C-major scale, the chord with a root of G♯/A♭ cannot. Further, the root progression from G♯/A♭ to C subverts the harmonic functions that define the tonal system: this chord progression does not contain any sort of dominant to tonic motion at all. The earliest examples of this are found as short passages in larger works that otherwise abide by the norms of common-practice tonality. Nevertheless, these examples are remarkable in that they foreshadow an unhinging of the tonal system, making way for compositional practices in the 20th century and beyond to new systems of pitch organization.
Ways to divide the octave equally
There are only five ways to divide a single octave equally. Such root progressions would move by:
This is somewhat trivial, as it simply produces the chromatic scale. Chord progressions in which roots move by ascending or descending minor seconds do exist, though.
Plenty of music has been written using the whole-tone scale, which divides the octave equally into major seconds. Much of this music is not tonal, or even triadic. There are a handful of examples in which tertian chord roots outline the whole-tone scale.
Roots progressing by minor third outline the diminished-seventh chord. These chord progressions are most often found in examples where each of these chord roots outlining the division of the octave are tonicized, or otherwise interrupted by intervening chords. More rarely, they’re found in progressions in which the chord roots outlining the division are presented in immediate succession. These kinds of progressions also tie in with the octatonic scale, which contains overlapping diminished-seventh chords.
Chord progressions in which the roots move by major third will outline the augmented triad. These chord progressions can be found both in examples where the chords are in immediate succession, and with intervening chords. These progressions go hand-in-hand with the hexatonic scale, which contains two overlapping augmented triads.
The last way in which one might divide the octave equally is by tritone. A chord progression that does this would simply be two chords, so repertoire examples would be somewhat trivial.
Of course, you can create chord progressions in which roots progress consistently by other intervals, like perfect fourths and fifths. These will create tonal ambiguity, but they will not come to completion within a single octave in the way the five listed above do.
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