Tone Clusters in 31 Equal Temperament

A tone cluster is a chord of closely-spaced musical intervals. Although there is no clear definition of what constitutes or does not constitute a cluster, the cutoff is usually the distinction between seconds and thirds, with chords made mostly from stacked thirds normally being considered normal chords, and chords made mostly from stacked seconds normally being considered tone clusters. Here, we use the term tone cluster to refer to chords made out of stacked intervals smaller than the septimal minor third.

New opportunities for using tone clusters, relative to 12-ET

As 31-ET has six distinct "seconds", or intervals smaller than a third, this tuning provides a rich vocabulary from which to construct tone clusters. The inclusion of harmonies involving the 7th and 11th harmonics allows for the construction of tone clusters that have a number of different tonal centers, both within or outside the cluster. There is also greater consonance possible in many of these tone clusters.

To compare, 12-ET has only four distinct types of three-note tone clusters, whereas 31-ET has thirty-six.

Although tone clusters in 31-ET often do unambiguously imply a tonal center, these effects are not always evident to the untrained ear. Even the most consonant tone clusters typically sound at least somewhat dissonant, and, in the absence of strong harmonic intervals like fourths, fifths, and thirds, their root or tonal center is usually only subtly evident.

Simple (3-note) Example Tone Clusters in 31-ET

Here are some tone clusters in 31-ET that unambiguously imply a tonal center:

• Stacked diatonic semitones - Two stacked diatonic semitones in 31-ET, adding up to a septimal whole tone, provide a close match to the 14:15:16 harmonics, thus implying the top note as root.
• Stacked neutral seconds - Two stacked neutral seconds add up to a minor third, and provide a close match to the 10:11:12 harmonics, thus implying a root a major third below the bottom note, or a perfect fifth below the top note.
• Whole tone beneath a neutral second - These intervals add up to a neutral third, and this tone cluster, matching the 9:10:11 harmonics, implies a root a whole tone beneath the bottom note, or a lesser undecimal tritone beneath the top note.
• Stacked whole tones - Just as in 12-ET, this tone cluster implies the bottom note as root, roughly matching the 8:9:10 harmonics. Although the 9:8 ratio is less in-tune than in 12-ET, the stronger 8:10 ratio (a major third) is more in-tune, so this tone cluster sounds more consonant and more strongly implies its root than in 12-ET, where the top interval's match to the 9:8 ratio imparts a slight dissonance and ambiguity.
• Septimal whole tone beneath a whole tone - The intervals of this cluster add up to a septimal major third, and correspond to the 7:8:9 harmonics closely, implying the middle note as root.

Some more ambiguous tone clusters:

• Diatonic semitone beneath a whole tone - This tone cluster matches the 15:16:18 harmonics, implying the middle note as root, but this effect is often overridden by the harmonically stronger 15:18 (or 5:6) ratio, a minor third, which implies a root a major third beneath the bottom note, or a perfect fifth beneath the top note.
• (Not technically a tone cluster although close) Septimal whole tone beneath a septimal minor third - If these intervals are stacked in the opposite order, the intervals appear in the same order as in the harmonic series. This out-of-order cluster is thus a lot like the minor triad, in that it contains two sequential intervals from the harmonic series, but in a different order. The effect is ambiguity...the septimal whole tone, the stronger interval, implies the top note as root. However, there is a weaker effect of the septimal major third implying a root one whole tone beneath the top of the chord (a half-step above the middle note of the chord), and also the possibility that the whole tone could imply the bottom note as root. The subtle switching of these intervals thus produces a tone cluster markedly more dissonant than the other ordering.

Understanding larger tone clusters

These example of three note tone clusters (two interval stackings) can serve as a starting point for understanding more complex tone clusters. When the harmonic ratios of larger tone clusters fit consistently into a single context, like stacking two neutral seconds on top of two whole tones, it will tend to sound more consonant. When the different combinations of notes imply different roots, which corresponds to when the intervals occur out of order from their corresponding order in the harmonic series, it creates more dissonance.

The more different implied roots there are in the cluster, and the more distant the relationships between the implied roots, the more dissonant and tonally ambiguous the cluster will sound. So for example, a tone cluster with two implied tonal centers or roots a fifth apart will sound relatively consonant, but one with many implied roots, or with two strongly implied tonal centers separated by a tritone or by a second or semitone will tend to sound more dissonant and ambiguous.