Apple Logic Express 9 - Manual (Page 1183)

For the rest of the scale: Tune the next fifth up: 150 x 3 = 450. Divide this by 2 to get 225
(which is more than an octave above the starting pitch, so you need to drop it another
octave to 112.5).
The following table provides a summary of the various calculations.
Notes Frequency (Hz) Note
x 1.5 divided by 2. 100 C
Divide by 2 to stay in octave. 106.7871 C#
Divide by 2 to stay in octave. 112.5 D
Divide by 2 to stay in octave. 120.1355 D#
Divide by 2 to stay in octave. 126.5625 E
135.1524 F (E#)
Divide by 2 to stay in octave. 142.3828 F#
(x 1.5) divided by 2. 150 G
160.1807 G#
168.75 A
180.2032 A#
189.8438 B
202.7287 C
As you can see from the table above, there’s a problem.
Although the laws of physics dictate that the octave above C (100 Hz) is C (at 200 Hz),
the practical exercise of a (C to C) circle of perfectly tuned fifths results in a C at
202.7287 Hz. This is not a mathematical error. If this were a real instrument, the results
would be clear.
To work around the problem, you need to choose between the following options:
• Each fifth is perfectly tuned, with octaves out of tune.
• Each octave is perfectly tuned, with the final fifth (F to C) out of tune.
Detuned octaves are more noticeable to the ears, so your choice should be obvious.
The Comma
The difference between a perfectly tuned octave and the octave resulting from a tuned
circle of fifths is known as the comma.
Over the centuries, numerous approaches have been tried to solve this mystery, resulting
in a range of scales (before arriving at equal temperament—the 12 tone scale).
1183 Chapter 42 Project Settings in Logic Express