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. This ambiguity is the foundation of tritone substitution in jazz.

In 31-ET, there are four distinct tritones, each of which implies a clear root. Each of these intervals lacks the ambiguity associated with the single tritone in 12-ET. As such, tritone substitution in 31-ET is not possible without altering the width of the interval by changing one or more of the notes forming the tritone.

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. This ambiguity is related to the fact that 31-ET is a meantone tuning, with a whole tone intermediate in width between the 9:8 and 10:9 intervals in just intonation.

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, which functions identically in 12-ET, 31-ET, and just intonation. 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.

Examples of chords with implied dissonance usable in 31-ET but not 12-ET

Examples of chords with implied dissonance analogous to that in the minor chord can be reached by looking at other intervals in the harmonic series and constructing 3-note chords while inverting the top and bottom interval. However, the analogies break down somewhat because these chords can behave in more complex ways than minor chords. A minor chord has a strong implied root of the bottom note, a secondary, weaker implied root of the middle note, and a third, even weaker implied root a whole step beneath the bottom note. These other chords can have three or more different implied roots, which may or may not be in the chord. More than three possible implied roots can arise due to 31-ET lumping together two just intervals into a single interval, such as 10:9 and 10:8 being lumped into a single whole tone.

An example that utilizes the 7th harmonic is stacking a septimal minor third on top of a regular minor third to create a lesser septimal tritone corresponds to the 5:6:7 ratios. Inverting these by lowering the middle note by a diesis produces one such chord. The strongest interval in this chord would be the minor third, which implies a root a major third below the bottom note (which is now the middle note of the chord), which would be a chromatic semitone (2 steps) below the bottom note of the chord. However, the other intervals imply alternate tonal centers: the bottom interval, the septimal minor third, implies a root a perfect fifth below the bottom note of the chord. And the lesser septimal tritone implies a root a major third below the bottom note of the chord.

A different example, utilizing the 11th harmonic, is stacking a neutral second on top of a major third to make a lesser undecimal tritone, and then switching the intervals by lowering the middle note by four steps, or a neutral second. In this chord, the major third is much stronger than the neutral second, so, unless the overall harmonic context suggests otherwise, the chord is likely to be interpreted as having the middle note as root. An alternate tonal center for the chord would be the bottom note of the chord, implied by the lesser undecimal tritone. However, neither of these tonal centers mesh with the neutral second. In 31-ET, the neutral second is closest to the 12:11 ratio, which implies a root a lesser undecimal tritone below the bottom note, or a perfect fourth above the top note (which here is the middle note of the chord). Both of these implied roots are weak and unlikely to be perceived unless implied by context. The complexity of this chord in 31-ET though, may be even greater, as the tuning does not distinguish between the 12:11 (lesser neutral second) and 11:10 (greater neutral second) ratios and the switching of the intervals decouples the neutral second from the context which would normally strongly imply the 12:11 ratio, so it is thus possible that it could be heard or interpreted as 11:10, an interval that would imply a root a major third below the bottom note. Thus this single chord has one dominant tonal center and up to three weak implied alternate tonal centers.