31 Tone Equal Temperament

Harmonic Ambiguity

Harmonic ambiguity is a state in which the musical intervals in a chord or the overall harmonic context provide conflicting signals about a root or tonal center. An interval or chord is said to be harmonically ambiguous when it does not imply a clear root.

The intervals that are harmonically ambiguous in 12-ET are not necessarily harmonically ambiguous in 31-ET, and vice versa. For example, in 12-ET, a tritone on its own is harmonically ambiguous. In 31-ET, there are four distinct tritones, each of which implies a clear root. The whole tone in 31-ET, however, is more ambiguous than the whole tone in 12-ET, which unambiguously implies the bottom note as root.

Harmonic Ambiguity in Chords

Harmonically unambiguous intervals can be stacked in ways that do not correspond to the natural sequence of intervals in the harmonic series, such that the whole chord provides conflicting signals about root or tonal center. The simplest example of such ambiguity is a minor chord. The top interval of a minor chord, a major third, implies the middle note of the chord as a root. Yet the perfect fifth implies the bottom note of the chord as root. The perfect fifth, being the stronger interval, dominates the harmonic context and defines the root of the chord, but the other implied root provides some tension or richness.

While this example sounds trivial, 31-ET opens up a tremendous possibility for new chords and combinations of intervals. Many of these combinations exhibit the same sort of harmonic ambiguity evident in a minor chord: that in which a dominant interval implies one note as a tonal center, whereas other intervals in the chord suggest other notes as tonal center. In 12-ET, there are four intervals and their inversions which can imply tonal centers, and only one of them, the minor third (or major sixth) implies a center not in the interval. In 31-ET, by contrast, eleven intervals and their inversions clearly imply tonal centers, and six of these intervals imply notes not in the interval. Understanding the subtle tensions resulting from the slight harmonic ambiguities resulting by stacking intervals in 31-ET is the key to understanding the harmonic nuances offered by this tuning system.