# Wolf interval

When the twelve notes within the octave are tuned using meantone temperament, one of the fifths will be much sharper than the rest. The notes that meantone temperament was normally based around meant that the interval G# to Eb will be this anomalous interval. This interval is known as a diminished sixth, which is meant to be the enharmonic equivalent to a fifth, however the interval did not sound the same as the other fifths: it was severely augmented and dissonant, and seemed to howl like a wolf. This anomalous interval thus came to be called the wolf fifth. By extension, any interval which is regarded as in like manner howling may be called a wolf.

## Temperament and the wolf

The average value of the twelve fifths must equal the 700 cents of equal temperament. If eleven of them have a flattened meantone value of 700-ε cents, the wolf will equal 700+11ε cents. In terms of frequency ratios, the product of the fifths must be 128, and if f is the size of the meantone fifths, 128/f11 will be the size of the wolf. In 1/4-comma meantone, the meantone fifth is of size 51/4, 3.422 cents flatter than 700 cents, and so the wolf is 37.637 cents sharper than 700 cents, which is 35.683 cents sharper than a just fifth of size exactly 3/2, and this is the original howling wolf fifth. A fifth of the size Mozart favored, at or near the 55-equal fifth of 698.1818 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.

We likewise find varied tunings for the thirds. Major thirds must average 400 cents, and to each pair of meantone thirds of size 400-4ε cents we have a sharp third (or diminished fourth) of 400+8ε cents, leading to eight thirds of size 400-4ε cents and four of size 400+8ε cents. Three of these form major triads with meantone fifths, and one triad is the wolf major triad, with a wolf fifth and a sharp major third. Similarly, we obtain nine minor thirds of 300+3ε cents and three flat minor thirds (or augmented seconds) of 300-9ε cents.

In 1/4-comma meantone, the flat minor thirds are only 2.335 cents sharper than a subminor third of size 7/6, and the sharp major thirds, of size exactly 32/25, are 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.

In Pythagorean tuning, we have eleven justly tuned fifths sharper than 700 cents by 1.955 cents, and hence one fifth will be flatter by eleven times that, which is a Pythagorean comma flatter than a just fifth. A fifth this flat can also be regarded as howling like a wolf. We also now have eight sharp major thirds, and four major thirds only 1.954 cents flat.

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 interval; depending on the size of the meantone fifth it can be made to be exactly 20/13 or 17/11.

## Sound files demonstrating the wolf

Wolf_fifth.ogg (33.1KB) is a sound file demonstrating the flat Pythagorean wolf fifth. The first two fifths are perfectly tuned in the ratio 3:2, the third is the Pythagorean wolf. It may be useful to compare this to Et_fifths.ogg (38.2KB), which is the same three fifths tuned in equal temperament, each of them tolerably well in tune.

With meantone fifths running from Eb to G#, the I-IV-ii-V-I cadence in the key of G# major becomes the chord progression G#-C-Eb, C#-F-G, Bb-C#-F, Eb-G-Bb, G#-C-Eb. The Ogg Vorbis file wolf gives this cadence in the key of G# major in a meantone with fifths of 695.838 cents, which is nearly the same as the 2/7-comma meantone of Gioseffo Zarlino. The tonic chord is the wolf triad, the subdominant C# chord is a supermajor triad, the supertonic Bb chord is a subminor triad, and the dominant Eb chord is an ordinary major triad. All four chords of the progression therefore have a different character, and the auditor can make up his or her own mind about the nature of the wolf tonic chord. It is instructive to compare this to the same progression in equal temperament, equal, and in a meantone tuned to G# major, meantone.

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