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## Pythagorean Tuning

In the Pythagorean Tuning 12 tones are built by stacking 12 perfect fifth (frequency ratio 3/2) above each other. This gives the following table:

Note Cents
(oct. reduced)
Compare: Cents
in Equ. Temp.
Ab 0 0
Eb 701.955 700
Bb 203.910 200
F 905.865 900
C 407.820 400
G 1109.775 1100
D 611.730 600
A 113.685 100
E 815.640 800
B 317.595 300
F# 1019.550 1000
C# 521.505 500
G# 23.460 0

A cent is the 100th part of a halftone in the Equal Temperament on a logarithmic scale, so the number 701.955 in the table comes from:

$\text{Pythagorean fifth in cents} = \log(\frac{3}{2}) \cdot \frac{100}{\log(\sqrt[12]{2})} = 701.95500$

The Bb is twice such a fifth, hence 1403.910, in the table cropped to the octave which is 1200, giving 1403.910 - 1200 (same for the other numbers in the second column, possibly with multiples of 1200).

In the table you can easily see that a G# and an Ab do not equalize! They are 23.460 cents apart which is called the Pythagorean Comma.

Because of this oddness and the fact that our ears tend to favor better sounding thirds over perfect fifth, the Pythagorean Tuning plays no role nowadays, and hasn't been doing so for centuries now.