Let’s start with a different chain of justly tuned perfect fifths (see my previous blog entry):

Eb-Bb-F-C-G-[D]-A-E-B-F#-C#-G#

This chain has [D] as the starting point and fifths are calculated moving both clockwise and counterclockwise from it.

Why [D]? Because doing so we have 3 white notes on each side of the chain before encountering black notes (I refer to the black and white notes of a piano keyboard)

So what? Traditionally the idea has always been to give tuning priority to keys based on white notes so, starting on [D] we make sure that those fifths close to the starting point will be justly tuned and if we have to make some adjustments, those will appear on seldom used ones (on black keys).

We have already seen that if we keep stacking justly tuned perfect fifths and we want to fit them inside one octave, after 11 of them, the 12th will not match the starting point (by one Pythagorean comma): a tempered fifth measures 700 cents, a just one 701.955 cents. If we multiply the difference between the 2 fifths times 12 (as the notes of a chromatic scale) we get: 1.955*12=23.46 (the size of a Pythagorean comma).

So the 12th fifth (in order to fit within an octave) will have to be:
701.955 - 23.46 = 678.495 cents

This flat fifth is often called a “wolf fifth” (because it reminds of the howling of a wolf). Have you ever heard such an interval?

EXAMPLE #1: Single notes


(click on the image to watch the movie)

We move along the above mentioned chain of justly tuned perfect fifths (and relative inversions, just perfect fourths, 498.045 cents).
Do you hear the wolf interval? It is between G# and Eb.
Does it sound like a howling wolf?
This interval should be called augmented third that in equal temperament equals a perfect fourth (500 cents).
Our wolf fourth measures 521.505 cents (a justly perfect fourth + a Pythagorean comma!) because it is the inversion of the wolf fifth (521.505 + 678.495 = 1200 cents = 1 octave!)

EXAMPLE #2: Double notes


(click on the image to watch the movie)

Same progression but with double notes (simultaneous justly tuned perfect fourths and fifths except for the wolf fifth G#-Eb, that should be called diminished sixth)
Is it easier to hear the wolf interval?

EXAMPLE #3: Double notes again


(click on the image to watch the movie)

Same progression but lower inversion, with double notes (simultaneous justly tuned perfect fourths and fifths except for the wolf fourth Eb-G#)
Does the lower register make a difference perceiving the wolf interval?
The sequence ends with 2 notes a Pythagorean comma apart.