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/128)11, 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.
Read more about this topic: Wolf Interval
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