The wolf fifth

You can listen to the wolf fifth yourself.

Click the New Scale button.

Choose Pythagorean temperament from the list of Scales drop list, choose Follow Scale from the Arpeggios or modes drop list, show a New Seed window, and play notes F sharp and D flat.

See Playing along to find out how to play them.

Compare with the interval C to G, which is pure in both these temperaments. (C is white key to left of the pair of black notes, and G is the white key between the 1st and 2nd of the group of 3 black notes.)

The ratio of the flattened fifth to the exact one is -23.46 cents (2^19/3^12), or a little under a quarter of a semitone.

Try also the just temperament scale. This has two flattened fifths - from B flat to F and D to A, and one sharpened fifth, from F sharp to D flat.

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The real wolf fifth

Though the sour fifths in Pythagorean, just temperament, and other temperaments are called wolf fifths in a general way, the original wolf fifth is in the quarter comma mean-tone system.

This is the system in general use for organs at the time of Bach's youth. The wolf fifth is between E flat and G sharp (right-most of the pair, and middle of the triple of black keys).

All the other fifths are flattened more than they need to be, to give as many pure major thirds as possible. The idea is that in the sequence of fifths C G D A E the E should be a pure major third (plus two octaves) above the C. So the fifths have to be tempered by a quarter of the difference between a pure major third and the Pythagorean major third - an interval known as the syntonic comma. The tempering is by a quarter of a comma, which is the reason for the name. The meantone part of the name refers to the whole tones - you get them by a division of the pure major third into equal parts - all the whole tone intervals except two of them which are a wider. In just intonation the whole tones are 9/8 and 10/9, and are unequal.

So in this system the upper note of the wolf fifth is sharp, rather than flat as in the other systems. The wolf fifth is 737.64 cents instead of the pure fifth of 701.96 cents, or +35.68 cents.

This Tune Smithy file plays all the wolf and tempered fifths mentioned, alternating with pure fifths:

pure_tempered_and_wolf_fifths.ts

Order played:

3/2 , 2^18/3^11, 3/2 , 696.578 cents, 3/2 , 737.638 cents, 3/2 , 700 cents, 3/2

I.e.

pure fifth, pythagorean wolf, pure, quarter comma meantone tempered fifth (not a wolf), pure, quarter comma meantone wolf, pure, equal temperament tempered fifth, pure.

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