Wolf interval

A wolf interval is audibly out of tune within a particular musical tuning system. A common example is the abnormally small fifth that results from the Pythagorean comma.

Definition Particularly out of tune intervals produce an unpleasant effect that remind listeners of a wolf's howl. Wolf fifths are produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone 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 temperaments. Wolf fifths were sometimes labeled imperfect or Procrustean. 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, 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 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).

Temperament

The reason for "wolf" tones in meantone tunings is the bad practice of performers pressing the key for an 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♯ when the music calls for A♭. 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") 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 but theory. In order to close the circle of fifths in 12 note scales, twelve fifths must average out to 700 cents. Each of the first eleven fifths (starting with the fifth below the 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 700 − ε cents, where ε is some small number of cents that all fifths are detuned by. In meantone 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 a diminished sixth: The interval is 700 + 11 ε cents, and is not a correct meantone fifth, which would be 700 − ε cents. The difference of 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 12 ε cents that approach 20~25 cents. 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 the key of C♮, would be

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