Editing Wolf interval (section)

{{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]]}}