|
In music theory, the wolf fifth (sometimes also called Procrustean fifth, or imperfect fifth)〔 (A.L. Leigh Silver (1971) ), p.354〕〔 Paul, Oscar (1885). ''(A manual of harmony for use in music-schools and seminaries and for self-instruction )'', p.165. Theodore Baker, trans. G. Schirmer.〕 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.〔(The wolf fifth )〕 More broadly, it is also used to refer to similar intervals produced by other tuning systems, including most meantone temperaments. When the twelve notes within the octave of a chromatic scale are tuned using the quarter-comma meantone tuning system, one of the twelve intervals spanning seven semitones (classified as a diminished sixth) turns out to be much wider than the others (classified as perfect fifths). Typically, this interval is from G to E. The eleven perfect fifths sound almost perfectly consonant. Conversely, the diminished sixth is severely dissonant and seems to howl like a wolf, because of a phenomenon called beating. Since the diminished sixth is meant to be enharmonically equivalent to a perfect fifth, this anomalous interval has come to be 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, 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 may be regarded as howling like a wolf may be called a wolf interval. For instance, in quarter comma meantone, the augmented second, augmented third, augmented fifth, diminished fourth and diminished seventh may be considered wolf intervals, as their size significantly deviates from the size of the corresponding justly tuned interval (see Size of 1/4-comma meantone intervals). == Temperament and the wolf == In 12-tone scales, the average value of the twelve fifths must equal the 700 cents of equal temperament. If eleven of them have a value of 700−ε cents, as in quarter-comma meantone and most other meantone temperament tuning systems, the other fifth (more properly called a diminished sixth) will equal 700+11ε cents. The value of ε changes depending on the tuning system. In other tuning systems (such as Pythagorean tuning and 1/12-comma meantone), eleven fifths may have a size of 700+ε cents, thus the diminished sixth is 700−11ε cents. If 11ε is very large, as in the quarter-comma meantone tuning system, the diminished sixth is regarded as a wolf fifth. In terms of frequency ratios, the product of the fifths must be 128, and if ''f'' is the size of eleven fifths, 128/''f''11, or ''f''11/128, will be the size of the wolf. We likewise find varied tunings for the thirds. Major thirds must average 400 cents, and to each pair of thirds of size 400−4ε (or +4ε) cents we have a third (or diminished fourth) of 400+8ε (or −8ε) cents, leading to eight thirds 4ε cents narrower or wider, and four diminished fourths 8ε cents wider or narrower than average. Three of these diminished fourths form major triads with perfect fifths, but one of them forms a major triad with the diminished sixth. If the diminished sixth is a wolf interval, this triad is called the wolf major triad. Similarly, we obtain nine minor thirds of 300+3ε (or −3ε) cents and three minor thirds (or augmented seconds) of 300−9ε (or +9ε) cents. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wolf interval」の詳細全文を読む スポンサード リンク
|