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The average value of the twelve fifths must equal the 700 cents of equal temperament. If eleven of them have a flattened meantone value of 700-ε cents, the wolf will equal 700+11ε cents. In terms of frequency ratios, the product of the fifths must be 128, and if f is the size of the meantone fifths, 128/f11 will be the size of the wolf. In 1/4-comma meantone, the meantone fifth is of size 51/4, 3.422 cents flatter than 700 cents, and so the wolf is 37.637 cents sharper than 700 cents, which is 35.683 cents sharper than a just fifth of size exactly 3/2, and this is the original howling wolf fifth. A fifth of the size Mozart favored, at or near the 55-equal fifth of 698.1818 cents, will have a wolf of 720 cents, 18.045 cents sharper than a justly tuned fifth. This howls far less acutely, but still very noticeably.
We likewise find varied tunings for the thirds. Major thirdIntervals The musical interval of a Major third is the relationship between the first note (the root or tonic) and the third note in a major scale. It is the inversion of the minor sixth. It is abbreviated as M3 . It can be produced by starting on a highs must average 400 cents, and to each pair of meantone thirds of size 400-4ε cents we have a sharp third (or diminished fourth) of 400+8ε cents, leading to eight thirds of size 400-4ε cents and four of size 400+8ε cents. Three of these form major triadIn music or music theory, a triad is a tonal or diatonic tertian trichord. In other words, a chord with three notes, built from thirds, and a diatonic function. The three notes are called the root, third, and fifth (a third above the third). The functions with meantone fifths, and one triad is the wolf major triad, with a wolf fifth and a sharp major third. Similarly, we obtain nine minor thirdThe musical interval of a minor third is the relationship between the first note (the root or tonic) and the third note in a minor scale. It is the inversion of the Major sixth. It can be produced by starting on a high note and playing the third below ors of 300+3ε cents and three flat minor thirds (or augmented seconds) of 300-9ε cents.
In 1/4-comma meantone, the flat minor thirds are only 2.335 cents sharper than a subminor third of size 7/6, and the sharp major thirds, of size exactly 32/25, are 7.712 cents flatter than the supermajor third of 9/7. Meantone tunings with slightly flatter fifths produce even closer approximations to the subminor and supermajor thirds and corresponding triads. These thirds therefore hardly deserve the appellation of wolf, and in fact historically have not been given that name.
In Pythagorean tuningPythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. Its discovery is generally credited to Pythagoras. It is the oldest way of tuning the 12-note chromatic scale and, as such,, we have eleven justly tuned fifths sharper than 700 cents by 1.955 cents, and hence one fifth will be flatter by eleven times that, which is a Pythagorean commaWhen you ascend by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, you eventually reach a note around seven octaves above the note you started on, which, when lowered to the same octave as your starting point, is 23. 46 cents hi flatter than a just fifth. A fifth this flat can also be regarded as howling like a wolf. We also now have eight sharp major thirds, and four major thirds only 1.954 cents flat.
The wolf can be tamed by adopting equal temperament or a well temperamentWell temperaments are a form of musical tuning, also known as circular temperaments or irregular temperaments . Any tuning system which is not equal temperament but which tunes the chromatic scale in a way suitable for a keyboard instrument and in such a. The very intrepid may simply want to treat it as a xenharmonicXenharmonic is fairly recent broad musical term used to refer to tuning systems or music using those systems not using the common twelve-tone equal temperament. It is also used more generally to refer to all microtonal music. The term was invented by Ivor interval; depending on the size of the meantone fifth it can be made to be exactly 20/13 or 17/11.