Consonance in 31-ET

Consonance is a musical concept describing the stability of a chord or harmony. Consonance is a subjective notion, and varies culturally, although there are certain well-established trends of harmonies or intervals that are considered consonant across various types of music. The opposite of consonance is dissonance.

In general, intervals corresponding to ratios between smaller numbers in the harmonic series are more likely to be perceived as more consonant, and intervals corresponding to larger numbers are more likely to be perceived as dissonance. As Western culture developed, the perception of consonance broadened, and intervals like thirds which were at one point considered dissonant came to be considered consonant.

New possibilities for consonance

The 31-ET tuning opens up new possibilities for consonance. Because it matches both the 7th and 11th harmonics, there are a number of new potentially consonant intervals in 31-ET that are not found in 12-ET:

• Septimal minor third
• Septimal whole tone
• Septimal major third
• The lesser septimal tritone, and its inversion the greater septimal tritone
• The lesser undecimal tritone, and its inversion, the greater undecimal tritone
• Neutral Second

The potential for consonance in 31-ET is also greatly enhanced by the fact that two of the main consonant intervals in western music, the major and minor third, are much more in-tune in this system. This gives these intervals a purer, more stable sound, which paves the way for new contrasts and harmonic nuances between the highly consonant major and minor thirds, and the slightly more dissonant consonances involving higher harmonics.

Building new, more consonant chords

Here are a few examples of chords in 31-ET that utilize the new intervals and produce consonances not possible in 12-ET:

• Harmonic seventh chord - By making the seventh of a typical dominant seventh chord slightly flatter, one can match the natural harmonics 4:5:6:7, producing a much more consonant chord than a dominant seventh chord in 12-ET.
• Septimal minor triad - The triad produced by taking two notes a perfect fifth apart and adding a note a septimal minor third above the root is more consonant than the regular minor chord, because it occurs in the harmonic series whereas the minor chord does not.
• Septimal minor third inside a perfect fourth - Starting with a perfect fourth and adding a note that is a septimal minor third above the bottom note and a septimal whole tone below the top note produces a chord that is more consonant than any chord in 12-ET produced by adding a note in between a perfect fourth.
• Neutral second inside a minor third - Splitting a minor third into two neutral seconds, adding the middle note, produces a tone cluster that is more consonant than any similar tone cluster in 12-ET, because it matches the ratio of the 10:11:12 harmonics quite closely.
• Neutral second on top of a major third, forming a lesser undecimal tritone - This chord is also more consonant than either of the two closest chords in 12-ET produced by stacking a half- or whole-step on top of a major third. This chord matches the 8:10:11 harmonics.