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Inmusic theory, thewolf fifth (sometimes also calledProcrustean fifth,orimperfect fifth)[1][2]is a particularlydissonant musicalinterval spanning sevensemitones. Strictly, the term refers to an interval produced by a specifictuning system, widely used in the sixteenth and seventeenth centuries: thequarter-comma meantone temperament.[3] More broadly, it is also used to refer to similar intervals (of close, but variable magnitudes) produced by other tuning systems, including Pythagorean and mostmeantone temperaments.
When the twelve notes within the octave of achromatic scale aretuned using the quarter-comma meantone systems of temperament, one of the twelve intervals apparently spanning sevensemitones is actually adiminished sixth, which turns out to be much wider than the in-tune genuinefifths,[a]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.[4]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 awolf, because of a phenomenon calledbeating. Since the diminished sixth isnominallyenharmonically equivalent to a perfect fifth, but inmeantone temperament, enharmonic notes are only nearby (within about1/4 sharp or1/4 flat); 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, in12 tone equal temperament (12-TET), 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 awolf interval. For instance, in quarter comma meantone, theaugmented second,augmented third,augmented fifth,diminished fourth, anddiminished seventh may be called wolf intervals, as their frequency ratio significantly deviates from the ratio of the correspondingjustly tuned interval (seeSize of quarter-comma meantone intervals).
The reason for "wolf" tones in meantone tunings is the bad practice of performers pressing the key for anenharmonic 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 allmeantone tuning systems, sharps and flats arenot equivalent; a relic of which, that persists in modern musical practice, is to fastidiously distinguish themusical notation for two notes which are the same pitch inequal temperament ("enharmonic") and played with the same key on an equal tempered keyboard (such as C♯ and D♭, or E♯ and F♮), despite the fact that they are the same in all buttheory.
In order to close thecircle of fifths in 12 note scales, twelvefifths must average out to700cents.[b]Each of the first eleven fifths (starting with the fifthbelow thetonic, thesubdominant: F in the key of C, when each black key is tuned to a meantone sharp / no flats) has a value of700 −ε cents, whereε is some small number of cents that all fifths are detuned by.[c]Inmeantone 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 adiminished sixth: The interval is700 + 11ε cents, and is not a correct meantone fifth, which would be700 −ε cents. The difference of12ε cents between the available pitch and the intended pitch is the source of the "wolf". The "wolf" effect is particularly grating for values of12ε cents that approach20~25 cents.[d]A simplistic reaction to the problem is:"Ofcourse 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 thekey of C♮, would be
| A | [no A♯] | B♭ | B | [no B♯ and no C♭] | C | C♯ | [no D♭] | ||
| D | [no D♯] | E♭ | E | [no E♯ and no F♭] | F | F♯ | [no G♭] | G | G♯ [choose one of either G♯ or A♭] |
with this set of chosen notes in bold face, and some of the omitted notes shown in grey.[e]
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 thatBaroque period keyboard andorchestral harp performers were obliged to retune their instruments in mid-performance breaks, in order to make available all theaccidentals called for by the next piece of music.[f][g]
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 fromC major into bothA major (which needs G♯ for theseventh note) andC minor (which needs A♭ for itssixth note) 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 fifthwould be consonant, but in meantone tuning systems (whereε isn't zero) the sharp of any note isalways 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 ofε changes depending on the tuning system. In other tuning systems (such asPythagorean tuning and twelfth-comma meantone), each of the eleven fifths may have a size of700 +ε cents, thus the diminished sixth is700 − 11ε cents. If their difference,12ε , 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 offrequencyratios, in order to close thecircle of fifths, theproduct of the fifths' ratios must be128 (since the twelve fifths, if closed in a circle, span seven octaves exactly; an octave is2:1, and27 = 128), and iff is the size of a fifth,128 :f 11, orf 11 : 128, will be the size of the wolf.
We likewise find varied tunings for the thirds:Major thirds must average400 cents, and to each pair of thirds of size400 ∓ 4ε cents we have a third (or diminished fourth) of400 ± 8ε cents, leading to eight thirds4ε cents narrower or wider, and four diminished fourths8ε cents wider or narrower than average. Three of these diminished fourths form majortriads 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 thewolf major triad.
Similarly, we obtain nineminor thirds of300 ± 3ε cents and three minor thirds (or augmented seconds) of300 ∓ 9ε cents.
Inquarter-comma meantone, the frequency ratio for the fifth is 4√ 5 , which is about3.42157 cents flatter than an equal tempered700 cents, (or exactly one twelfth of adiesis) and so the wolf is about737.637 cents, or35.682 cents sharper than aperfect fifth of ratio exactly3:2, and this is the original "howling" wolf fifth.
The flat minor thirds are only about2.335 cents sharper than asubminor third of ratio7:6, and the sharp major thirds, of ratio exactly32:25, are about7.712 cents flatter than thesupermajor third of9:7 . 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-limitjust interval49:32, which has a size of737.652 cents.
In12-tone Pythagorean temperament, there are elevenjustly tuned fifths sharper than700 cents by about1.955 cents (or exactly one twelfth of aPythagorean comma), and hence one fifth will be flatter by twelve times that, which is23.460 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 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 toPythagorean tuning, 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 important to note that the two fifths, three minor thirds, and three major sixths marked in orange in the tables (ratio40:27,32:27, and27:16; 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, i.e.,81:80, or about21.5 cents) large enough to be clearly perceived asdissonant.
Five-limit tuning determines one diminished sixth of size1024:675 (about722 cents, i.e.20 cents sharper than the3:2 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 twoimpure perfect fifths of size40:27.Five-limit fifths are about680 cents; lesspure than the3:2 Pythagorean and/orjust701.95500 cent perfect fifth .They are not diminished sixths, but relative to the Pythagorean perfect fifth they are less consonant (about20 cents flatter) and hence, they might be considered to be wolf fifths. The correspondinginversion is animpure perfect fourth(also called Acute Fourth[5]) of size27:20 (about520 cents). For instance, in theC majordiatonic scale, an impure perfect fifth arises between D and A, and its inversion arises between A and D.
Since in this context the termperfect is interpreted to mean 'perfectly consonant',[6] the impure perfect fourth and perfect fifth are sometimes simply calledthe imperfect fourth and fifth.[2] However, the widely adopted standard naming convention formusical intervals classifies them asperfect intervals, together with theoctave andunison. This is also true for any perfect fourth or perfect fifth which slightly deviates from the perfectly consonant4:3 or3:2 ratios. For instance, those tuned using12 tone equal orquarter-comma meantone temperament. Conversely, the expressionsimperfect fourth andimperfect fifth do not conflict with the standard naming convention when they refer to a dissonantaugmented third ordiminished sixth, e.g. the wolf fourth and fifth in Pythagorean tuning.
Wolf intervals are a consequence of mapping a two-dimensional temperament to a one-dimensional keyboard.[7]The only solution is to make the number of dimensions match. That is, either:
When the perfect fifth is tempered to be exactly700cents wide (that is, tempered by about1/11 of asyntonic comma, or precisely1/12 of aPythagorean comma) then the tuning is identical to the now-standard 12 toneequal temperament.
Because of the compromises (and wolf intervals) 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 the55 equal temperament fifth of 698.182 cents, will have a wolf of720 cents:18.045 cents sharper than a justly tuned fifth. This howls far less acutely, but is still noticeable.
The wolf can be "tamed" by adoptingequal temperament or awell temperament. The very intrepid may simply want to treat it as axenharmonic music interval; depending on the size of the meantone fifth, the wolf fifth can be tuned to more complexjust ratios 20:13, 26:17, 17:11, 32:21, or 49:32.
With a more extreme meantone temperament, like19 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. (In 19 equal temperament, the diminished sixth is enharmonically equivalent to theaugmented fifth.)
A lesser-known alternative method that allows the use of multi-dimensional temperaments without wolf intervals is to use a two-dimensional keyboard that is "isomorphic" 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 thesyntonic temperament—that is, by theoctave and temperedperfect fifth—so they are isomorphic.
On anisomorphic keyboard, 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, 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. In19-TET, the interval from E♯ to C♭ would be (enharmonic to) a perfect fifth.
Such edge conditions produce wolf intervals only if the isomorphic keyboard has fewer buttons per octave than the tuning hasenharmonically distinct notes.[7] For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge condition, from E♯ to C, isnot a wolf interval in12-TET,17-TET, or19-TET; however, itis a wolf interval in 26-TET,31-TET, and53-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.[9]
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.[9] Plamondon, Milne & Sethares (2009),[9] Figure 2, shows the valid tuning range of the syntonic temperament.
... musical interval 'pythagorean major third'.
perfect concord.
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