31 Tone Equal Temperament

Intervals: 14 : Lesser Undecimal Tritone

The lesser undecimal tritone is the simplest interval involving the 11th harmonic. It is somewhere between a perfect fourth and major third in width. The match to this interval in 31-ET is somewhat close, but not perfect: the interval is narrower than its natural size by 9.38 cents, an amount large enough to make the interval sound slightly out of tune to trained ears.

The lesser undecimal tritone has a distinct sound that makes it relatively easy to distinguish both from the perfect fourth and the septimal tritone, even though it is quite close to each of these intervals in width. This interval tends to sound more consonant, colder, and much more stable than the septimal tritone. Compared to the perfect fourth, it sounds more dissonant, and perhaps more importantly, it implies its bottom note, rather than its top note, as root. Thus, while it is very close to the perfect fourth in size, it makes a radically different contribution to the harmonic texture. The lesser undecimal tritone can function either as dissonant or consonant, depending on the context.

Of all the intervals in 31-ET, the lesser undecimal tritone is arguably the one that makes a relatively clean break with 12-ET (and most Western music) while being relatively easy to hear and comprehend.

Unlike the tritone in 12-ET, or the two septimal tritones in 31-ET, the undecimal tritone is quite distinct from its inversion, the greater undecimal tritone.

Root of the Undecimal Tritone

The undecimal tritone implies its bottom note as the root. Because this note occurs in the interval, this interval has a stronger effect for defining a tonal center than one would expect from 11's high position in the harmonic series.

The 15:11 Ratio

The 15:11 ratio of the harmonic series is also a very close match to this interval, 536.95 cents wide, closer even than the 11:8 ratio. This similarity enables the 14-step interval in 31-ET to function in either role, depending on context.

When two 14-step intervals are stacked, the resulting chord thus corresponds quite closely to the 15:11:8 harmonics, which implies the bottom note as root.