View source for Wolf interval

{{short description|Dissonant musical interval}}
{{Refimprove|date=June 2008}}
[[Image:Wolf fifth on C.png|thumb|Wolf fifth on C {{audio|Wolf fifth on C.mid|Play}}]]
[[Image:Pythagorean wolf fifth.png|thumb|Pythagorean wolf fifth as eleven just perfect fifths]]

In [[music theory]], the '''wolf fifth''' (sometimes also called '''Procrustean fifth''',
or '''imperfect fifth''')<ref>
{{cite report |first=A.L. Leigh |last=Silver |year=1971 |title=Musimatics, or the Nun's Fiddle |page=354 |url=http://lit.gfax.ch/tunings/Musimatics_or_the_Nun's_Fiddle.pdf |via=lit.gfax.ch/tunings}}
</ref><ref name=Baker>
{{cite book |last=Paul |first=Oscar |year=1885 |title=A Manual of Harmony for use in Music-Schools and Seminaries and for Self-Instruction |page=165 |publisher=Theodore Baker |translator=Schirmer, G.  |url=https://archive.org/details/bub_gb_4WEJAQAAMAAJ |via=Internet Archive (archive.org) |quote=...&nbsp;musical interval 'pythagorean major third'.}}
</ref>
is a particularly [[Consonance and dissonance|dissonant]] musical [[interval (music)|interval]] spanning seven [[semitone]]s. Strictly, the term refers to an interval produced by a specific [[musical tuning#Tuning systems|tuning system]], widely used in the sixteenth and seventeenth centuries: the [[quarter-comma meantone]] temperament.<ref>
{{cite web |title=The wolf fifth |website=robertinventor.com |url=http://robertinventor.com/wiki/index.php?title=Wolf_fifth&oldid=22807}}
</ref> 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]]s.

When the twelve notes within the octave of a [[chromatic scale]] are [[Musical tuning|tuned]] using the quarter-comma meantone systems of temperament, one of the twelve intervals apparently spanning seven [[semitone]]s is actually a [[diminished sixth]], which turns out to be much wider than the in-tune genuine [[perfect fifth|fifths]],{{efn|
Specifically, the actual note present on the keyboard where the desired next fifth ''would'' be, is ''not'' a fifth, but rather a [[diminished sixth]].
}}
In mean-tone systems, this interval is usually from C{{music|#}} to A{{music|b}} or from G{{music|#}} to E{{music|b}} but can be moved in either direction to favor certain groups of keys.<ref name=Duffin2007>
{{cite book |last=Duffin |first=Ross W. |date=2007 |title=How Equal Temperament Ruined Harmony (and Why You Should Care) |page=35 |location=New York, NY |publisher=W.W. Norton |isbn=978-0-393-06227-4}}
</ref>
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 [[beat (acoustics)|beating]]. Since the diminished sixth is ''nominally'' [[enharmonically equivalent]] to a perfect fifth, but in [[meantone temperament]], enharmonic notes are only nearby (within about {{sfrac|1|4}} sharp or {{sfrac|1|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, in [[12-tone equal temperament#Twelve-tone equal temperament|12&nbsp;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 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 [[Just intonation|justly tuned]] interval (see [[Quarter comma meantone#Size of intervals|Size of quarter-comma meantone intervals]]).

{{Listen|filename=Mean5th Wolf 5th.ogg|title=Meantone and wolf fifths|description=A mean fifth followed by a wolf fifth in [[quarter-comma meantone]] temperament|format=[[Ogg]]}}

== Temperament and the wolf ==
The reason for "wolf" tones in meantone tunings is the bad practice of performers pressing the key for an [[enharmonic equivalence|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{{music|#}} when the music calls for A{{music|b}}. 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]] ([[enharmonic equivalence|"enharmonic"]]) and played with the same key on an equal tempered keyboard (such as C{{music|#}} and D{{music|b}},  or  E{{music|#}} and F{{music|n}}), despite the fact that they are the same in all but [[musical theory|theory]].

In order to close the [[circle of fifths]] in 12&nbsp;note scales, twelve [[perfect fifth|fifths]] must average out to {{nobr|{{math|700}} [[cent (music)|cents]]}}.{{efn|
No such {{nobr|{{math|700}} [[cent (music)|cents]]}} exact average for fifth inervals exists meantone systems: Their fifths – and all repeated intervals – form a [[helix]], not a circle.
}}
Each of the first eleven fifths (starting with the fifth ''below'' the [[tonic (music)|tonic]], the [[subdominant]]: F in the key of C, when each black key is tuned to a meantone sharp / no flats) has a value of {{nobr|{{math|700 − ''ε''}} cents}}, where {{mvar|ε}} is some small number of cents that all fifths are detuned by.{{efn|
The size of {{mvar|ε}} is around 1–4&nbsp;cents, and is different for each particular meantone system used. As a technicality, [[equal temperament]] happens to be a [[meantone temperament]] for which the value of {{mvar|ε}} is zero.
}}
In [[meantone temperament]] tuning systems, the twelfth and last fifth does not exist in the 12&nbsp;note octave on the keyboard. 

The actual note available is really a [[diminished sixth]]: The interval is {{nobr|{{math|700 + 11 ''ε''}} cents}}, and is not a correct meantone fifth, which would be {{nobr|{{math|700 − ''ε''}} cents.}} The difference of {{nobr|{{math|12 ''ε''}} cents}} between the available pitch and the intended pitch is the source of the "wolf". The "wolf" effect is particularly grating for values of {{nobr|{{math|12 ''ε''}} cents}} that approach {{nobr|{{math|20~25}} cents}}.{{efn|
{{nobr|{{math|20~25}} cents}}, or a quarter-sharp / quarter flat, is the typical size of the several discrepant musical intervals called [[comma (music)|"commas"]]. Note that a quarter-comma is a different interval than a quarter-sharp.
}}
A simplistic reaction to the problem is: ''"Of {{underline|course}} it sounds awful: You're playing the wrong note!"''

With only 12&nbsp;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]]{{music|n}}, would be
:{| style="text-align:center;"
|-"
| '''A''' &emsp;|| {{grey|[no A{{music|#}}]}} &emsp;|| '''B{{music|b}}''' &emsp;|| '''B''' &emsp;|| {{grey|[no B{{music|#}} and no C{{music|b}}]}} &emsp;|| '''C''' &emsp;|| '''C{{music|#}}''' &emsp;|| {{grey|[no D{{music|b}}]}} &emsp;|| &nbsp; &emsp;|| &nbsp;
|-
| '''D''' &emsp;|| {{grey|[no D{{music|#}}]}} &emsp;|| '''E{{music|b}}''' &emsp;|| '''E''' &emsp;|| {{grey|[no E{{music|#}} and no F{{music|b}}]}} &emsp;|| '''F''' &emsp;|| '''F{{music|#}}''' &emsp;|| {{grey|[no G{{music|b}}]}} &emsp;|| '''G''' &emsp;|| '''G{{music|#}}''' &thinsp; {{grey|[choose one of either G{{music|#}} or A{{music|b}}]}}
|}
with this set of chosen notes in bold face, and some of the omitted notes shown in grey.{{efn|
Of course, [[double sharp]]s and [[double flat]]s are infeasible for the key of C&nbsp;major / A&nbsp;minor.
}}

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 (music)|Baroque period]] keyboard and [[orchestral harp]] performers were obliged to retune their instruments in mid-performance breaks, in order to make available all the [[accidental (music)|accidentals]] called for by the next piece of music.{{efn|
If a performer could get the use of an extra instrument, an alternative to retuning is to switch seats to a spare instrument already tuned for the upcoming piece.
}}{{efn|
Note that [[wind instrument]]s, [[bowed stringed instrument]]s, and [[singing|singers]] have no such need for a retuning session, since players always microtune every note they produce "on the fly". On the other hand, players of stringed instruments with movable [[fret]]s, such as the [[oud]] face a similar problem; performers on fixed-fret instruments likewise are limited to the keys which are compatible with the positions of the frets, although it is possible to microtune by tugging on a string using the finger that presses it down.
}}

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]] (which needs G{{music|#}} for the [[major seventh|seventh note]]) and [[C minor]] (which needs A{{music|b}} for its [[minor sixth|sixth note]]) is not possible, since each of the two meantone notes, G{{music|#}} and A{{music|b}}, 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 (where {{big|{{mvar|ε}}}} isn't zero) 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&nbsp;notes in an octave. 

The value of {{mvar|ε}} changes depending on the tuning system. In other tuning systems (such as [[Pythagorean tuning]] and twelfth-comma meantone), each of the eleven fifths may have a size of {{nobr|{{math|700 + ''ε''}} cents,}} thus the diminished sixth is {{nobr|{{math|700 − 11 ''ε''}} cents.}} If their difference, {{nobr| {{math|12 ''ε''}} ,}} 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]] [[ratio]]s, in order to close the [[circle of fifths]], the [[product (mathematics)|product]] of the fifths' ratios must be {{math|128}} (since the twelve fifths, if closed in a circle, span seven octaves exactly; an octave is {{nobr|{{math|2:1}},}} and {{nobr|{{math|2{{sup|7}} {{=}} 128}}),}} and if {{mvar|f}} is the size of a fifth, {{nobr|{{math|128 : ''f''{{sup|&nbsp;11}}}},}} or {{nobr|{{math|''f''{{sup|&nbsp;11}} : 128}},}} will be the size of the wolf.

We likewise find varied tunings for the thirds: [[Major third]]s must average {{nobr|{{math|400}} cents,}} and to each pair of thirds of size {{nobr|{{math|400 ∓ 4 ''ε''}} cents}} we have a third (or diminished fourth) of {{nobr|{{math|400 ± 8 ''ε''}} cents,}} leading to eight thirds {{nobr|{{math|4 ''ε''}} cents}} narrower or wider, and four diminished fourths {{nobr|{{math|8 ''ε''}} cents}} wider or narrower than average. Three of these diminished fourths form major [[triad (music)|triad]]s 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 third]]s of {{nobr|{{math|300 ± 3 ''ε''}} cents}} and three minor thirds (or augmented seconds) of {{nobr|{{math|300 ∓ 9 ''ε''}} cents.}}

=== Quarter comma meantone ===
In [[quarter-comma meantone]], the frequency ratio for the fifth is {{math|{{radic| 5 | 4 }} }}, which is about {{nobr|{{math|3.42157}} cents}} flatter than an equal tempered {{nobr|{{math|700}} cents,}} (or exactly one twelfth of a [[diesis]]) and so the wolf is about {{nobr|{{math|737.637}} cents,}} or {{nobr|{{math|35.682}} cents}} sharper than a [[perfect fifth]] of ratio exactly {{nobr|{{math|3:2}},}} and this is the original "howling" wolf fifth.

The flat minor thirds are only about {{nobr|{{math|2.335}} cents}} sharper than a [[Septimal minor third|subminor third]] of ratio {{math|7:6}}, and the sharp major thirds, of ratio exactly {{nobr|{{math|32:25}},}} are about {{nobr|{{math|7.712}} cents}} flatter than the [[Septimal major third|supermajor third]] of {{nobr|{{math|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 intonation|just]] interval {{math|49:32}}, which has a size of {{nobr|{{math|737.652}} cents.}}

=== Pythagorean tuning ===
In [[Pythagorean tuning]], there are eleven [[just intonation|justly tuned]] fifths sharper than {{nobr|{{math|700}} cents}} by about {{nobr|{{math|1.955}} cents}} (or exactly one twelfth of a [[Pythagorean comma]]), and hence one fifth will be flatter by twelve times that, which is {{nobr|{{math|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]] 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&nbsp;wolf intervals (a fifth and a fourth), the 5-limit symmetric scales produce 12 of them, and the asymmetric scale&nbsp;14. 

It is important to note that the two fifths, three minor thirds, and three major sixths marked in orange in the tables (ratio {{nobr|{{math|40:27}},}} {{nobr|{{math|32:27}},}} and {{math|27:16}}; or G↓, E{{music|flat}}↓, 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&nbsp;[[syntonic comma]], i.e., {{nobr|{{math|81:80}},}} or about {{nobr|{{math|21.5}} cents}}) large enough to be clearly perceived as [[consonance and dissonance|dissonant]].

[[Five-limit tuning]] determines one diminished sixth of size {{math|1024:675}} (about {{nobr|{{math|722}} cents,}} i.e. {{nobr|{{math|20}} cents}} sharper than the {{math|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 {{math|40:27}}.
Five-limit fifths are about {{nobr|{{math|680}} cents;}} less ''pure'' than the {{math|3:2}} Pythagorean and/or [[just intonation|just]] {{nobr|{{math|701.95500 cent}} perfect fifth .
}}
They are not diminished sixths, but relative to the Pythagorean perfect fifth they are less consonant (about {{nobr|{{math|20}} cents}} flatter) and hence, they might be considered to be wolf fifths. The corresponding [[Inversion (interval)|inversion]] is an ''impure'' perfect fourth(also called Acute Fourth<ref>{{cite web | url=https://www.huygens-fokker.org/docs/intervals.html | title=Stichting Huygens-Fokker: List of intervals }}</ref>) of size {{math|27:20}} (about {{nobr|{{math|520}} cents}}). For instance, in the [[major scale|C&nbsp;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',<ref>{{cite book |first=Godfrey |last=Weber |year=1841 |section=Definition of ''perfect consonance'' |title=General Music Teacher |publisher=Boston, J. H. Wilkins & R. B. Carter |quote=perfect concord. |via=Internet Archive (archive.org)
|url=https://archive.org/details/bub_gb_20oBAAAAYAAJ}}</ref> the impure perfect fourth and perfect fifth are sometimes simply called ''the imperfect'' fourth and fifth.<ref name=Baker/> However, the widely adopted standard naming convention for [[Interval (music)|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 {{math|4:3}} or {{math|3:2}} ratios. For instance, those tuned using [[12-tone equal temperament|12&nbsp;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.<ref name=Milne2007>
{{cite journal
 | first1 = Andrew  | last1 = Milne
 | first2 = William | last2 = Sethares   | author-link2 = William Sethares
 | first3 = James   | last3 = Plamondon
 | date = December 2007
 | title = Invariant fingerings across a tuning continuum
 | journal = [[Computer Music Journal]]
 | volume = 31  | issue = 4  | pages = 15–32
 | doi =  10.1162/comj.2007.31.4.15 | doi-access = free  | s2cid = 27906745
 }}
</ref>
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]]), 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 {{nobr|{{math|700}} [[cent (music)|cents]]}} wide (that is, tempered by about {{sfrac|1|11}} of a [[syntonic comma]], or precisely {{sfrac|1|12}} of a [[Pythagorean comma]]) then the tuning is identical to the now-standard 12&nbsp;tone [[equal temperament]].

Because of the compromises (and wolf intervals) forced on meantone tunings by the one-dimensional piano-style keyboard, [[well temperament]]s 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&nbsp;cents, will have a wolf of {{nobr|{{math|720}} cents:}} {{nobr|{{math|18.045}} cents}} 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 intonation|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.

=== Keep the two-dimensional tuning system ===
[[Image:Isomorphic Note Layout.jpg|thumb|top|550px|Figure 1: The Wicki isomorphic keyboard, invented by Kaspar Wicki in 1896.<ref name=Gaskins2003>
{{cite web
 | first1 = Robert  | last1 = Gaskins
 | date = September 2003
 | title = The Wicki system – an 1896 precursor of the Hayden system
 | website = Concertina Library: Digital Reference Collection for Concertinas
 | url = http://www.concertina.com/gaskins/wicki/
 | access-date = 2013-07-11
}}
</ref>]]
[[Image:Rank-2 temperaments with the generator close to a fifth and period an octave.jpg|right|250px|thumb|Figure 2: The [[syntonic temperament]]’s tuning continuum.<ref name=Milne2007 />]]
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 the [[syntonic temperament]]—that is, by the [[octave]] and tempered [[perfect fifth]]—so they are isomorphic.

On an [[isomorphic 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{{music|sharp}}. 

The note that is a tempered perfect fifth higher than E{{music|sharp}} is B{{music|sharp}}, which is not included on the keyboard shown, although it could be included in a larger keyboard, placed just to the right of A{{music|sharp}}, hence maintaining the keyboard's consistent note-pattern. Because there is no B{{music|sharp}} button, when playing an E{{music|sharp}} [[power chord]], one must choose some other note that is close in pitch to B{{music|sharp}}, such as C, to play instead of the missing B{{music|sharp}}. That is, the interval from E{{music|sharp}} to C would be a "wolf interval" on this keyboard. In [[19 equal temperament|19-TET]], the interval from E{{music|sharp}} to C{{music|flat}} 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 has [[enharmonic]]ally distinct notes.<ref name=Milne2007/> For example, the isomorphic keyboard in Figure 2 has 19 buttons per octave, so the above-cited edge condition, from E{{music|sharp}} to C, is ''not'' a wolf interval in [[Equal temperament|12-TET]], [[17 equal temperament|17-TET]], or [[19 equal temperament|19-TET]]; however, it ''is'' a wolf interval in 26-TET, [[31 equal temperament|31-TET]], and [[53 equal temperament|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.<ref name=Plamondon2009>
{{cite conference
 | first1 = J.   | last1 = Plamondon
 | first2 = A.   | last2 = Milne
 | first3 = W.A. | last3 = Sethares 
 | year = 2009
 | title =  Dynamic tonality: Extending the framework of tonality into the 21st&nbsp;century
 | conference = Annual Conference of the South Central Chapter of the College Music Society
 | book-title = Proceedings of the Annual Conference of the South Central Chapter of the College Music Society
 | url = http://sethares.engr.wisc.edu/paperspdf/CMS2009.pdf
 | via = sethares.engr.wisc.edu
}}</ref>

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.<ref name=Plamondon2009/> Plamondon, Milne & Sethares (2009),<ref name=Plamondon2009/> Figure&nbsp;2, shows the valid tuning range of the syntonic temperament.

== Footnotes ==
{{notelist}}

== References ==
{{reflist|25em}}

{{Intervals|state=expanded}}

{{DEFAULTSORT:Wolf Interval}}
[[Category:Intervals (music)]]