Wolf interval

Wolf fifth on C  Play

Pythagorean wolf fifth as eleven just perfect fifths

In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including most meantone temperaments.

When the twelve notes within the octave of a chromatic scale are tuned using the quarter-comma meantone tuning system, one of the twelve intervals spanning seven semitones (classified as a diminished sixth) turns out to be much wider than the others (classified as perfect fifths). Typically, this interval is from G♯ to E♭. The eleven perfect fifths sound almost perfectly consonant. Conversely, the diminshed sixth is severely dissonant and seems to howl like a wolf, because of a phenomenon called beating. Since the diminished sixth is meant to be enharmonically equivalent to a perfect fifth, this anomalous interval has come to be called the wolf fifth.

Besides the above-mentioned quarter comma meantone, other tuning systems may produce severely dissonant diminished sixths. Conversely, in 12-tone equal temperament, which is currently the most commonly used tuning system, the diminished sixth is not a wolf fifth, as it has exactly the same size as a perfect fifth.

By extension, any interval which is perceived as severely dissonant and may be regarded as howling like a wolf may be called a wolf interval. For instance, in quarter comma meantone, the augmented second, augmented third, augmented fifth, diminished fourth and diminished seventh may be considered wolf intervals, as their size significantly deviates from the size of the corresponding justly tuned interval (see Size of 1/4-comma meantone intervals).

.

Temperament and the wolf

In 12-tone scales, the average value of the twelve fifths must equal the 700 cents of equal temperament. If eleven of them have a value of 700−ε cents, as in quarter-comma meantone and most other meantone temperament tuning systems, the other fifth (more properly called a diminished sixth) will equal 700+11ε cents. The value of ε changes depending on the tuning system. In other tuning systems (such as Pythagorean tuning and 1/12-comma meantone), eleven fifths may have a size of 700+ε cents, thus the diminished sixth is 700−11ε cents. If 11ε is very large, as in the quarter-comma meantone tuning system, the diminished sixth is regarded as a wolf fifth.

In terms of frequency ratios, the product of the fifths must be 128, and if f is the size of eleven fifths, (128/f)¹¹, or (f/128)¹¹, will be the size of the wolf.

We likewise find varied tunings for the thirds. Major thirds must average 400 cents, and to each pair of thirds of size 400−4ε (or +4ε) cents we have a third (or diminished fourth) of 400+8ε (or −8ε) cents, leading to eight thirds 4ε cents narrower or wider, and four diminished fourths 8ε cents wider or narrower than average. Three of these diminished fourths form major triads with perfect fifths, but one of them forms a major triad with the diminished sixth. If the diminished sixth is a wolf interval, this triad is called the wolf major triad.

Similarly, we obtain nine minor thirds of 300+3ε (or −3ε) cents and three minor thirds (or augmented seconds) of 300−9ε (or +9ε) cents.

Quarter comma meantone

In quarter-comma meantone, the fifth is of size 5^1/4, about 3.42157 cents (or exactly one twelfth of a diesis) flatter than 700 cents, and so the wolf is about 37.637 cents sharper than 700 cents, or 35.682 cents sharper than a perfect fifth of size exactly 3/2, and this is the original howling wolf fifth.

The flat minor thirds are only about 2.335 cents sharper than a subminor third of size 7/6, and the sharp major thirds, of size exactly 32/25, are about 7.712 cents flatter than the supermajor third of 9/7. Meantone tunings with slightly flatter fifths produce even closer approximations to the subminor and supermajor thirds and corresponding triads. These thirds therefore hardly deserve the appellation of wolf, and in fact historically have not been given that name.

Pythagorean tuning

In Pythagorean tuning, there are eleven justly tuned fifths sharper than 700 cents by about 1.955 cents (or exactly one twelfth of a Pythagorean comma), and hence one fifth will be flatter by eleven times that, which is 23.460 cents (one Pythagorean comma) flatter than a just fifth. A fifth this flat can also be regarded as howling like a wolf. There are also now eight sharp and four flat major thirds.

Five-limit tuning

Five-limit tuning determines one diminished sixth of size 1024:675 (about 722 cents, i.e. 20 cents sharper than the 3:2 Pythagorean perfect fifth). Whether this interval should be considered dissonant enough to be called a wolf fifth is a controversial matter.

Five-limit tuning also creates two impure perfect fifths of size 40:27 (about 680 cents; less pure than the 3:2 Pythagorean perfect fifth), sometimes also called imperfect fifths. These impure perfect fifths are not diminished sixths, but relative to the Pithagorean perfect fifth they are less consonant (about 20 cents flatter) and hence they might be considered to be wolf fifths. The corresponding inversion is an impure perfect fourth of size 27:20 (about 520 cents), also known as imperfect fourth. For instance, in the C major diatonic scale, an impure perfect fifth arises between D and A, and its inversion arises between A and D.

Taming the wolf

A fifth of the size Mozart favored, at or near the 55-equal fifth of 698.182 cents, will have a wolf of 720 cents, 18.045 cents sharper than a justly tuned fifth. This howls far less acutely, but still very noticeably.

The wolf can be tamed by adopting equal temperament or a well temperament. The very intrepid may simply want to treat it as a xenharmonic music interval; depending on the size of the meantone fifth it can be made to be exactly 20/13 or 17/11, or less commonly to 32/21 or 49/32.