Wolf Intervals in Truncated Tunings

The main problem with truncated tunings is that they tend to produce some awful sounding intervals. These are called wolf intervals and are those that are between pitches whose classes span a wolf fifth. For example, we have seen that in a size 12 truncated Pythagorean tuning, $\ensuremath{\underline{\Phi_W}} \approx -19.6$ mil. This means that the P5 between pitches like and is -19.6 mil flatter than just, a very large error. Such a fifth is said to ``howl'' and cannot be held for any appreciable amount of time without inducing a queasy feeling in listeners. Not all wolf intervals sound bad, though. For example, we have seen above that the wolf M3 in a size 12 truncated Pythagorean tuning deviates from just by only -1.6 mil, a very small amount. In contrast, the normal Pythagorean M3 deviates from just by 17.9 mil! For size 12 truncated tunings, wolf intervals become less severe as approaches $\ensuremath{\underline{C_P}}/12$, which is the value of for 12TET. Of course, for 12TET, no truncation is necessary.

Truncated tunings work well for music that avoids wolf intervals. This type of music typically visits a bounded range of pitch classes that does not span a wolf fifth. Indeed, the vast majority of early baroque keyboard music obeys such bounds since it was intended for instruments using truncated meantone tunings [3, 179-87].