Apple Logic Pro 8 User Manual
Page 946
946
Chapter 40
Project Settings and Preferences
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 was 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.
 Perfectly tuned octaves with the final fifth (F to C) out of tune.
It goes without saying that detuned octaves are more noticeable to the ears.
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).
Other historical temperaments that have been devised emphasize different aspects of
harmonic quality. Each compromises in some way or another. Some maximize pure
thirds (Mean Tone) while others emphasize pure fifths, at the expense of the thirds
(Kirnberger III, for example).
Every temperament has its own character, and a given piece of music may sound fine
in one key, but awful in another. Transposing a piece to a new key can completely
change its character.
Careful attention must be paid to the selection of temperaments for authentic
performances of historic keyboard music. The wrong choice could result in an
unsatisfactory and historically inaccurate musical experience.
About Equal Temperament
Equal temperament takes the tuning error (the comma), and spreads it equally
between each step of a chromatic scale. The result is actually a scale of equally mis-
tuned intervals, with no interval grossly out of tune, but none in perfect tune. Equal
temperament has become the de-facto standard for two main reasons:
A
168.75
A#
180.2032
B
189.8438
C
202.7287
Note
Frequency (Hz)
Notes