How the program turns your numbers into a tune
Scales of interest to musicians
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How the program turns your numbers into a tune
Scales of interest to musicians
In the Pythagorean temperament (if constructed in the way described), the wolf fifth is between the notes F sharp and D flat
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). One could place the mean-tone wolf fifth anywhere, but this is the normal choice.
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, i.e. a quarter of a comma.
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 midi clip plays all the wolf and tempered fifths mentioned, alternating with pure fifths:
pure_tempered_and_wolf_fifths.mid
Order of play:
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, pure, quarter comma meantone wolf, pure, equal temperament tempered fifth, pure.