Wolf interval
Encyclopedia
In music theory
, the wolf fifth is a particularly dissonant
musical interval
spanning seven semitone
s. 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 temperament
s.
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 semitone
s (classified as a diminished sixth
) turns out to be much wider than the others (classified as perfect fifth
s). 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).
.
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
ratio
s, 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 third
s 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 triad
s 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 third
s of 300+3ε (or −3ε) cents and three minor thirds (or augmented seconds) of 300−9ε (or +9ε) cents.
, the fifth is of size 51/4, 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.
, 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 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.
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.
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
, the wolf fifth is a particularly dissonant
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
musical interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
spanning seven semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
s. 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
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including most meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
s.
When the twelve notes within the octave of a chromatic scale
Chromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
are tuned
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
using the quarter-comma meantone tuning system, one of the twelve intervals spanning seven semitone
Semitone
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically....
s (classified as a diminished sixth
Diminished sixth
In classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning...
) turns out to be much wider than the others (classified as perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
s). 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
Beat (acoustics)
In acoustics, a beat is an interference between two sounds of slightly different frequencies, perceived as periodic variations in volume whose rate is the difference between the two frequencies....
. 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 second
In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning...
, augmented third
Augmented third
In classical music from Western culture, an augmented third is an interval produced by widening a major third by a chromatic semitone. For instance, the interval from C to E is a major third, four semitones wide, and both the intervals from C to E, and from C to E are augmented thirds, spanning...
, augmented fifth
Augmented fifth
In classical music from Western culture, an augmented fifth is an interval produced by widening a perfect fifth by a chromatic semitone. For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C to G, and from C to G are augmented fifths,...
, diminished fourth
Diminished fourth
In classical music from Western culture, a diminished fourth is an interval produced by narrowing a perfect fourth by a chromatic semitone. For example, the interval from C to F is a perfect fourth, five semitones wide, and both the intervals from C to F, and from C to F are diminished fourths,...
and diminished seventh
Diminished seventh
In classical music from Western culture, a diminished seventh is an interval produced by narrowing a minor seventh by a chromatic semitone. For instance, the interval from A to G is a minor seventh, ten semitones wide, and both the intervals from A to G, and from A to G are diminished sevenths,...
may be considered wolf intervals, as their size significantly deviates from the size of the corresponding justly tuned
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
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 centsCent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
of equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
. If eleven of them have a value of 700−ε cents, as in quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
and most other meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
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
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
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
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
s, the product
Product (mathematics)
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...
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
Third (music)
In music and music theory third may refer to:*major third*minor third*augmented third/perfect fourth*diminished third/major second*Third , chord member a third above the root*Mediant, third degree of the diatonic scale...
. Major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
s 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 triad
Triad (music)
In music and music theory, a triad is a three-note chord that can be stacked in thirds. Its members, when actually stacked in thirds, from lowest pitched tone to highest, are called:* the Root...
s 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 third
Minor third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major...
s 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 meantoneQuarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
, the fifth is of size 51/4, about 3.42157 cents (or exactly one twelfth of a diesis
Diesis
In classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...
) 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
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
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
Septimal minor third
In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5...
of size 7/6, and the sharp major thirds, of size exactly 32/25, are about 7.712 cents flatter than the supermajor third
Septimal major third
In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal...
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 tuningPythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
, there are eleven justly tuned fifths sharper than 700 cents by about 1.955 cents (or exactly one twelfth of a Pythagorean comma
Pythagorean comma
In musical tuning, the Pythagorean comma , named after the ancient mathematician and philosopher Pythagoras, is the small interval existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B , or D and C...
), and hence one fifth will be flatter by eleven times that, which is 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.
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
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
or a well temperament
Well temperament
Well temperament is a type of tempered tuning described in 20th-century music theory. The term is modelled on the German word wohltemperiert which appears in the title of J.S. Bach's famous composition, The Well-Tempered Clavier...
. 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.