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In music theory, the wolf fifth (sometimes also called Procrustean fifth, or imperfect fifth)[1][2] is a particularly dissonant musical interval spanning seven semitones. 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.[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 most meantone temperaments.
When the twelve notes within the octave of a chromatic scale are tuned using the quarter-comma meantone systems of temperament, one of the twelve intervals apparently spanning seven semitones is actually a diminished sixth, which turns out to be much wider than the in-tune genuine fifths,[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 a wolf, because of a phenomenon called beating. Since the diminished sixth is nominally enharmonically equivalent to a perfect fifth, but in meantone temperament, enharmonic notes are only nearby (within about 1/4 sharp or 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 (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 justly tuned interval (see Size of quarter-comma meantone intervals).
- ^ Silver, A.L. Leigh (1971). Musimatics, or the Nun's Fiddle (PDF) (Report). p. 354 – via lit.gfax.ch/tunings.
- ^
Paul, Oscar (1885). A Manual of Harmony for use in Music-Schools and Seminaries and for Self-Instruction. Translated by Schirmer, G. Theodore Baker. p. 165 – via Internet Archive (archive.org).
... musical interval 'pythagorean major third'.
- ^ "The wolf fifth". robertinventor.com.
- ^ Duffin, Ross W. (2007). How Equal Temperament Ruined Harmony (and Why You Should Care). New York, NY: W.W. Norton. p. 35. ISBN 978-0-393-06227-4.
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