In
music theory
Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, the wolf fifth (sometimes also called Procrustean fifth,
or imperfect fifth)
[
Paul, Oscar (1885). ]
A manual of harmony for use in music-schools and seminaries and for self-instruction
', p.165. Theodore Baker, trans. G. Schirmer.
is a particularly
dissonant
In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...
musical
interval spanning seven
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.
It is defined as the interval between two adjacent no ...
s. Strictly, the term refers to an interval produced by a specific
tuning system
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
Tun ...
, widely used in the sixteenth and seventeenth centuries: the
quarter-comma meantone
Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80 ...
temperament. More broadly, it is also used to refer to similar intervals (of close, but variable magnitudes) produced by other tuning systems, including Pythagorean and most
meantone temperament
Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. M ...
s.
When the twelve notes within the octave of a
chromatic scale
The chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the ...
are
tuned using the quarter-comma mean-tone systems of temperament, one of the twelve intervals spanning seven
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.
It is defined as the interval between two adjacent no ...
s (classified as a
diminished sixth
In classical music from Western culture, a diminished sixth () is an interval produced by narrowing a minor sixth by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an d6 not ...
) turns out to be much wider than the others (classified as
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
s). 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 is severely dissonant and seems to howl like a wolf, because of a phenomenon called
beating
Beat, beats or beating may refer to:
Common uses
* Patrol, or beat, a group of personnel assigned to monitor a specific area
** Beat (police), the territory that a police officer patrols
** Gay beat, an area frequented by gay men
* Battery ...
. Since the diminished sixth is meant to be
enharmonically equivalent
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written no ...
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
Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resultin ...
, 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
In classical music from Western culture, an augmented second is an interval that, in equal temperament, is sonically equivalent to a minor third, spanning three semitones, and is created by widening a major second by a chromatic semitone.Ben ...
,
augmented third
In classical music from Western culture, an augmented third () is an interval of five semitones. It may be produced by widening a major third by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . For ...
,
augmented fifth
In classical music from Western culture, an augmented fifth () is an interval produced by widening a perfect fifth by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . For instance, the interval ...
,
diminished fourth
In classical music from Western culture, a diminished fourth () is an interval produced by narrowing a perfect fourth by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an d4 ...
and
diminished seventh
In classical music from Western culture, a diminished seventh () is an interval produced by narrowing a minor seventh by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an d7 ...
may be considered wolf intervals, as their size significantly deviates from the size of the corresponding
justly tuned interval (see
Size of quarter-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
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
. If eleven of them have a value of 700 − ''ε'' cents, as in
quarter-comma meantone
Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80 ...
and most other
meantone temperament
Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. M ...
tuning systems, the other fifth (more properly called a diminished sixth) will equal 700 + 11''ε'' cents. The value of ''ε'' changes depending on the tuning system. In other tuning systems (such as
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
and twelfth-comma meantone), eleven fifths may have a size of 700 + ''ε'' cents, thus the diminished sixth is 700 − 11''ε'' cents. If 11''ε'' 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 is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s, the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of the fifths must be 128, and if ''f'' is the size of a fifth, 128:''f''
11, or ''f''
11:128, will be the size of the wolf.
We likewise find varied tunings for the thirds.
Major third
In classical music, a third is a musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
s must average 400 cents, and to each pair of thirds of size 400 ∓ 4''ε'' cents we have a third (or diminished fourth) of 400 ± 8''ε'' cents, leading to eight thirds 4''ε'' cents narrower or wider, and four diminished fourths 8''ε'' 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 third
In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...
s of 300 ± 3''ε'' cents and three minor thirds (or augmented seconds) of 300 ∓ 9''ε'' cents.
Quarter comma meantone
In
quarter-comma meantone
Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80 ...
, the fifth is of size , about 3.42157 cents (or exactly one twelfth of a
diesis) flatter than 700 cents, and so the wolf is about 737.637 cents, or 35.682 cents sharper than a
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
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.
The wolf fifth of quarter-comma meantone can be approximated by the 7-limit just interval 49:32, which has a size of 737.652 cents.
Pythagorean tuning
In
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
, there are eleven justly tuned fifths sharper than 700 cents by about 1.955 cents (or exactly one twelfth of a
Pythagorean comma
In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C ...
), and hence one fifth will be flatter by twelve 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
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note ...
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
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...
, which can be considered a 3-limit just intonation tuning. Namely, while Pythagorean tuning determines only 2 wolf intervals (a fifth and a fourth), the 5-limit symmetric scales produce 12 of them, and the asymmetric scale 14. It is also important to note that the two fifths, three minor thirds, and three major sixths marked in orange in the tables (ratio 40:27, 32:27, and 27:16 (or G−, E−, and A+), even though they do not completely meet the conditions to be wolf intervals, deviate from the corresponding pure ratio by an amount (1
syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...
, i.e., 81:80, or about 21.5 cents) large enough to be clearly perceived as
dissonant
In music, consonance and dissonance are categorizations of simultaneous or successive Sound, sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness ...
.
Five-limit tuning
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note ...
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). These are not diminished sixths, but relative to the Pythagorean perfect fifth they are less consonant (about 20 cents flatter) and hence, they might be considered to be wolf fifths. The corresponding
inversion
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
is an ''impure'' perfect fourth of size 27:20 (about 520 cents). For instance, in the
C major
C major (or the key of C) is a major scale based on C, consisting of the pitches C, D, E, F, G, A, and B. C major is one of the most common keys used in music. Its key signature has no flats or sharps. Its relative minor is A minor and ...
diatonic scale
In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps, ...
, an impure perfect fifth arises between D and A, and its inversion arises between A and D.
Since the term ''perfect'' means, in this context, perfectly consonant,
[ Definition of ''Perfect consonance'']
in Godfrey Weber's General music teacher, by Godfrey Weber, 1841. the impure perfect fourth and perfect fifth are sometimes simply called ''imperfect'' fourth and fifth.
However, the widely adopted standard naming convention for
musical intervals classifies them as ''perfect'' intervals, together with the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
and
unison
In music, unison is two or more musical parts that sound either the same pitch or pitches separated by intervals of one or more octaves, usually at the same time. ''Rhythmic unison'' is another term for homorhythm.
Definition
Unison or per ...
. This is also true for any perfect fourth or perfect fifth which slightly deviates from the perfectly consonant 4:3 or 3:2 ratios (for instance, those tuned using
12-tone equal or
quarter-comma meantone
Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80 ...
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
In classical music from Western culture, an augmented third () is an interval of five semitones. It may be produced by widening a major third by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . For ...
or
diminished sixth
In classical music from Western culture, a diminished sixth () is an interval produced by narrowing a minor sixth by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . Specific example of an d6 not ...
(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 (one-dimensional) piano keyboard, and shift to a one-dimensional temperament (e.g.,
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
), or
* Keep the two-dimensional temperament, and shift to a two-dimensional keyboard.
Keep the piano keyboard
When the perfect fifth is tempered to be exactly 700
cents wide (that is, tempered by approximately of a syntonic comma, or exactly of a Pythagorean comma) then the tuning is identical to the familiar 12-tone
equal temperament
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
.
Because of the compromises (and wolf intervals) forced on meantone tunings by the one-dimensional piano-style keyboard,
well temperament
Well temperament (also good temperament, circular or circulating temperament) is a type of tempered tuning described in 20th-century music theory. The term is modeled on the German word ''wohltemperiert''. This word also appears in the title of ...
s and eventually equal temperament became more popular.
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
An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...
or a
well temperament
Well temperament (also good temperament, circular or circulating temperament) is a type of tempered tuning described in 20th-century music theory. The term is modeled on the German word ''wohltemperiert''. This word also appears in the title of ...
. The very intrepid may simply want to treat it as a
xenharmonic music
Xenharmonic music is music that uses a tuning system that is unlike the 12-tone equal temperament scale. It was named by Ivor Darreg, from Xenia (Greek ξενία), ''hospitable,'' and Xenos (Greek ξένος) ''foreign.'' He stated that it was ...
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.
With a more extreme meantone temperament, like
19 equal temperament
In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...
, 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.
Keep the two-dimensional tuning system
To use a two-dimensional temperament without wolf intervals, one needs a two-dimensional keyboard that is "
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
" with that temperament. A keyboard and temperament are isomorphic if they are generated by the same intervals. For example, the Wicki keyboard shown in Figure 1 is generated by the same musical intervals as the
syntonic temperament
A regular diatonic tuning is any musical scale consisting of " tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same s ...
—that is, by the
octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
and tempered
perfect fifth
In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so.
In classical music from Western culture, a fifth is the interval fro ...
—so they are isomorphic.
On an
isomorphic keyboard An isomorphic keyboard is a musical input device consisting of a two-dimensional grid of note-controlling elements (such as buttons or keys) on which any given sequence and/or combination of musical intervals has the "same shape" on the keyboard wh ...
, any given musical interval has the same shape wherever it appears—in any octave, key, and tuning—except at the edges. For example, on Wicki's keyboard, from any given note, the note that is a tempered perfect fifth higher is always up-and-rightwardly adjacent to the given note. There are no wolf intervals within the note-span of this keyboard. The only problem is at the edge, on the note E. The note that is a tempered perfect fifth higher than E is B, which is not included on the keyboard shown (although it could be included in a larger keyboard, placed just to the right of A, hence maintaining the keyboard's consistent note-pattern). Because there is no B button, when playing an E
power chord
A power chord (also fifth chord) is a colloquial name for a chord in guitar music, especially electric guitar, that consists of the root note and the fifth, as well as possibly octaves of those notes. Power chords are commonly played on am ...
, one must choose some other note that is close in pitch to B, such as C, to play instead of the missing B. That is, the interval from E to C would be a "wolf interval" on this keyboard. In
19-TET, the interval from E to C would be (enharmonic to) a perfect fifth.
However, such edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning has
enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written n ...
ally distinct notes.
For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge condition, from E to C, is ''not'' a wolf interval in
12-TET,
17-TET, or
19-TET; however, it ''is'' a wolf interval in 26-TET,
31-TET, and
53-TET. In these latter tunings, using electronic transposition could keep the current key's notes centered on the isomorphic keyboard, in which case these wolf intervals would very rarely be encountered in tonal music, despite modulation to exotic keys.
A keyboard that is isomorphic with the syntonic temperament, such as Wicki's keyboard above, retains its isomorphism in any tuning within the tuning continuum of the syntonic temperament, even when changing tuning dynamically among such tunings.
Figure 2 shows the valid tuning range of the syntonic temperament.
Listen to it
Listen to the two fifths here:
References
{{DEFAULTSORT:Wolf Interval
Intervals (music)