Wolf interval
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 Pythagorean and most meantone temperaments.
When the twelve notes within the octave of a chromatic scale are tuned using the quarter-comma meantone systems of temperament, one of the twelve intervals apparently spanning seven semitones is actually a diminished sixth, which turns out to be much wider than the in-tune genuine fifths,
In mean-tone systems, this interval is usually from C to A or from G to E but can be moved in either direction to favor certain groups of keys.
The eleven perfect fifths sound almost perfectly consonant. Conversely, the diminished sixth used as a substitute is severely dissonant: It sounds like the howl of a wolf, because of a phenomenon called beating. Since the diminished sixth is nominally enharmonically equivalent to a perfect fifth, but in meantone temperament, enharmonic notes are only nearby ; the discordance of substituted interval is 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 regarded as "howling like a wolf" is called a wolf interval. For instance, in quarter comma meantone, the augmented second, augmented third, augmented fifth, diminished fourth, and diminished seventh may be called wolf intervals, as their frequency ratio significantly deviates from the ratio of the corresponding justly tuned interval.
Temperament and the wolf
The reason for "wolf" tones in meantone tunings is the bad practice of performers pressing the key for an enharmonic note as a substitute for a note that has not been tuned on the keyboard; e.g. pressing the black key tuned to G when the music calls for A. In all meantone tuning systems, sharps and flats are not equivalent; a relic of which, that persists in modern musical practice, is to fastidiously distinguish the musical notation for two notes which are the same pitch in equal temperament and played with the same key on an equal tempered keyboard, despite the fact that they are the same in all but theory.In order to close the circle of fifths in 12 note scales, twelve fifths must average out to.
Each of the first eleven fifths has a value of, where is some small number of cents that all fifths are detuned by.
In meantone temperament tuning systems, the twelfth and last fifth does not exist in the 12 note octave on the keyboard.
The actual note available is really a diminished sixth: The interval is, and is not a correct meantone fifth, which would be The difference of between the available pitch and the intended pitch is the source of the "wolf". The "wolf" effect is particularly grating for values of that approach.
A simplistic reaction to the problem is: "Of it sounds awful: You're playing the wrong note!"
With only 12 notes available in a conventional keyboard's octave, in meantone tunings there must always be omitted notes. For example, one choice for tuning an instrument in meantone, to play music in the key of C, would be
with this set of chosen notes in bold face, and some of the omitted notes shown in grey.
This limitation on the set meantone notes and their sharps and flats that can be tuned on a keyboard at any one time, was the main reason that Baroque period keyboard and orchestral harp performers were obliged to retune their instruments in mid-performance breaks, in order to make available all the accidentals called for by the next piece of music.
Some music that modulates too far between keys cannot be played on a single keyboard or single harp, no matter how it is tuned: In the example tuning above, music that modulates from C major into both A major and C minor is not possible, since each of the two meantone notes, G and A, both require the same string in each octave on the instrument to be tuned to their different pitches.
For expediency, keyboard players substitute the wrong diminished sixth interval for a genuine meantone fifth, or neglect retuning their instrument. Though not available, a genuine meantone fifth would be consonant, but in meantone tuning systems the sharp of any note is always different from the flat of the note above it. A meantone keyboard that allowed unlimited modulation theoretically would require an infinite number of separate sharp and flat keys, and then double sharps and double flats, and so on: There must inevitably be missing pitches on a standard keyboard with only 12 notes in an octave.
The value of changes depending on the tuning system. In other tuning systems, each of the eleven fifths may have a size of thus the diminished sixth is If their difference, is very large, as in the quarter-comma meantone tuning system, the diminished sixth is used as a substitute for a fifth, it is called a "wolf fifth".
In terms of frequency ratios, in order to close the circle of fifths, the product of the fifths' ratios must be of leading to eight thirds narrower or wider, and four diminished fourths wider or narrower than average. Three of these diminished fourths form major triads with perfect fifths, but one of them forms a major triad substituting the diminished sixth for a real fifth. If the diminished sixth is a wolf interval, this triad is called the wolf major triad.
Similarly, we obtain nine minor thirds of and three minor thirds of
Quarter comma meantone
In quarter-comma meantone, the frequency ratio for the fifth is, which is about flatter than an equal tempered and so the wolf is about or sharper than a perfect fifth of ratio exactly and this is the original "howling" wolf fifth.The flat minor thirds are only about sharper than a subminor third of ratio, and the sharp major thirds, of ratio exactly are about flatter than the supermajor third of 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.
The wolf fifth of quarter-comma meantone can be approximated by the 7-limit just interval, which has a size of
Pythagorean tuning
In 12-tone Pythagorean temperament, there are eleven justly tuned fifths sharper than by about , and hence one fifth will be flatter by twelve times that, which is 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
was designed to maximize the number of pure intervals, but even in this system several intervals are markedly impure. 5-limit tuning yields a much larger number of wolf intervals with respect to Pythagorean tuning, which can be considered a 3-limit just intonation tuning. Namely, while Pythagorean tuning determines only 2 wolf intervals, the 5-limit symmetric scales produce 12 of them, and the asymmetric scale 14.It is important to note that the two fifths, three minor thirds, and three major sixths marked in orange in the tables, even though they do not completely meet the conditions to be wolf intervals, deviate from the corresponding pure ratio by an amount large enough to be clearly perceived as dissonant.
Five-limit tuning determines one diminished sixth of size . 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.
Five-limit fifths are about less pure than the Pythagorean and/or just
They are not diminished sixths, but relative to the Pythagorean perfect fifth they are less consonant and hence, they might be considered to be wolf fifths. The corresponding inversion is an impure perfect fourth of size . 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.
Since in this context the term perfect is interpreted to mean 'perfectly consonant', the impure perfect fourth and perfect fifth are sometimes simply called the imperfect fourth and fifth. However, the widely adopted standard naming convention for musical intervals classifies them as perfect intervals, together with the octave and unison. This is also true for any perfect fourth or perfect fifth which slightly deviates from the perfectly consonant or ratios. For instance, those tuned using 12 tone equal or quarter-comma meantone temperament. Conversely, the expressions imperfect fourth and imperfect fifth do not conflict with the standard naming convention when they refer to a dissonant augmented third or diminished sixth, e.g. the wolf fourth and fifth in Pythagorean tuning.
"Taming the wolf"
Wolf intervals are a consequence of mapping a two-dimensional temperament to a one-dimensional keyboard.The only solution is to make the number of dimensions match. That is, either:
- Keep the piano keyboard, and shift to a one-dimensional temperament, or
- Keep the two-dimensional temperament, and shift to a two-dimensional keyboard.
Keep the piano keyboard
Because of the compromises forced on meantone tunings by the one-dimensional piano-style keyboard, well temperaments and eventually 12-tone equal temperament became more popular.
A fifth of the size Mozart favored, at or near the 55 equal temperament fifth of 698.182 cents, will have a wolf of sharper than a justly tuned fifth. This howls far less acutely, but is still noticeable.
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, the wolf fifth can be tuned to more complex just ratios 20:13, 26:17, 17:11, 32:21, or 49:32.
With a more extreme meantone temperament, like 19 equal temperament, the wolf is large enough that it is closer in size to a sixth than a fifth, and sounds like a different interval altogether rather than a mistuned fifth.