31 Tone Equal Temperament

Intervals: 18 : Perfect Fifth

Musical Staff with Treble Clef, C, and G

The perfect fifth is one of the most fundamental musical intervals. It corresponds to the difference between the 2nd and 3rd harmonics.

Intonation of the Fifth in 31-ET:

In 31-ET, the fifth is slightly less in-tune than in 12-ET. In both tunings, the interval is slightly narrower than the just fifth. Although the difference between this interval and its true ratio in the harmonic series is small, close to the perceptual limits of most people's identification of pitch, it is large enough to cause problems, especially when the interval is stacked. Those with a good ear may notice a very faint out-of-tune quality to this interval in 31-ET. However, the larger problem is when musicians, such as string players, tune an instrument based on fifths. In 12-ET the match to this interval is so good that stacking 3 of them (as with 4 strings of most string instruments), the difference is still negligible. In 31-ET, however, stacking 3 fifths results in a difference, of 15.57 cents, which is a full 40% of the width of the diesis, the building-block of 31-ET. Tuning in this manner would result in a serious problem.

A more stable way to tune by ear in 31-ET is to use major thirds. Major thirds are so close to their true interval that they can be effectively stacked indefinitely while remaining more or less in-tune. Even cycling through 8 fifths, the difference would add up to only 6.32 cents, right at the edge of most people's perception of intonation. However, since most stringed instruments use fifths for the spacing of most strings, it is safest to use a tuner to match the exact frequency for each string.

Root of a Perfect Fifth:

The perfect fifth strongly implies the bottom note as root, and this effect is strong enough that the fifth usually overpowers any other intervals implying other notes. However, perfect fifths appear other places in the harmonic series besides just the 3:2 ratio, such as the 9:6 ratio, which would imply a note a fifth below the lower note in the interval. An example of such a context would be a septimal minor triad, which matches the 6:7:9 harmonics. A chord containing multiple fifths and fourths can also sometimes have an ambiguous tonal center.

Although perfect fifths are arguably the strongest musical interval in terms of implying a tonal center, they still have some ambiguity in them, which can be evident when the context implies their interpretation as a higher ratio, such as 9:6. In the case of this ratio, the implied root is one fifth below the bottom note of the fifth. This concept is more important in 31-ET than 12-ET, because there are more musical intervals in 31-ET that can imply this alternate root.