# Wolf fifth

Diatonic intervals
Prime
second
third
fourth
fifth
sixth
seventh
octave
none
decime
undezime
duodecime
tredezime
semitone / whole tone
Special intervals
Microinterval
Comma
Diësis
Limma
Apotome
Ditone Tritone
Wolf
fifth
Natural septime
units
Cent
Millioctave
Octave
Savart
The two wolf fifths fill the missing pitch between 7 octaves and 11 mean-tone or pure fifths

A wolf fifth is understood to be a fifth that is very “out of tune” compared to the perfect fifth ( frequency ratio  ; listening ? / I ) and that occurs in different historical moods . ${\ displaystyle {\ frac {3} {2}} \ {\ widehat {\ approx}} \ 701 {,} 955 \; \ mathrm {Cent}}$

The fifth with equal tuning (700  cents ) deviates only slightly from the perfect fifth and is therefore not considered a wolf fifth .

(In order to be able to better compare intervals in terms of their size, their size is often given in the logarithmic unit of cents . 1  octave  = 1200 cents.)

## Pythagorean wolf fifth

The circle of fifths in equal mood. Twelve fifths are a little too big in Pythagorean tuning, a little too small in mid-tone tuning .

When using the Pythagorean tuning , the tones of the scale (scale) are defined with the help of superimposed perfect fifths. Actually, the keynote should be reached again after 12 steps . In fact, however, this tone is higher than the corresponding octave of the fundamental tone by the Pythagorean comma .

In order to be able to “play” all keys and to close the fifth row to the circle of fifths, the twelfth fifth is reduced accordingly. The tone series consists of 11 perfect fifths and one that is reduced by a Pythagorean comma. This fifth strongly contradicts the classical idea of euphony . It used to be said to howl like a wolf , which is why it is called the Pythagorean wolf fifth .

Frequency ratio of the Pythagorean comma: .${\ displaystyle \ left ({\ frac {3} {2}} \ right) ^ {12} \ cdot \ left ({\ frac {1} {2}} \ right) ^ {7} \ {\ widehat { \ approx}} \ 23 {,} 460 \, \ mathrm {Cent}}$
${\ displaystyle \ Rightarrow}$Frequency ratio of the Pythagorean wolf fifth: .${\ displaystyle {\ frac {3} {2}} \ cdot \ left ({\ frac {2} {3}} \ right) ^ {12} \ cdot 2 ^ {7} = {\ frac {262144} { 177147}} \ {\ widehat {\ approx}} \ 678 {,} 495 \, \ mathrm {Cent}}$

Alternative calculation: Since the 11 perfect fifths are each 1.955 cents larger than the equal fifth with 700 cents, the 12th fifth is correspondingly smaller: (700 - 11 · 1.955) cents = 678.495 cents.

## Mid-tone wolf fifth

In the mid-tone tuning , an even less useful fifth emerges, which is also called a wolf fifth . However, it is not a fifth, but a diminished sixth (usually G sharp – E flat). This arises as the remaining interval between 11 layered mean-tone fifths and the seventh octave of the fundamental. The mean-tone wolf fifth is therefore calculated as the difference between 7 octaves and 11 mean-tone fifths. It is larger than a perfect fifth and therefore also larger than a mean-tone fifth.

The 1/4 decimal point mean fifth is 1/4 of the syntonic comma smaller than the perfect fifth:
Frequency ratio of syntonic commas .${\ displaystyle {\ frac {81} {80}} \ {\ widehat {\ approx}} \ 21 {,} 506 \, \ mathrm {Cent}}$
${\ displaystyle \ Rightarrow}$Frequency ratio 1/4-comma mean-Quinte: .${\ displaystyle {\ frac {3} {2}} \ cdot {\ sqrt [{4}] {\ frac {80} {81}}} = {\ sqrt [{4}] {5}} \ {\ widehat {\ approx}} \ 696 {,} 578 \, \ mathrm {Cent}}$
Frequency ratio 1/4-comma mean-wolf fifth: .${\ displaystyle 2 ^ {7}: \ left ({\ sqrt [{4}] {5}} \ right) ^ {11} \ {\ widehat {\ approx}} \ 737 {,} 637 \, \ mathrm {Cents}}$

Alternative calculation: Since the 11 mean-tone fifths are each 3.422 cents (more precisely 3.42157 cents) smaller than the equal fifth with 700 cents, the 12th fifth is correspondingly larger: (700 + 11 · 3.42157) cents ≈ 737.637 cents .