Apple Logic Pro 7 User Manual
Page 631
Chapter 21
Song Settings and Preferences
631
To simplify this example, we’ll start tuning at a frequency of 100 Hz and we’ll call it ‘C’ (a
real ‘C’ would be closer to 130 Hz). The first fifth would be tuned by adjusting the pitch
until a completely clear tone is produced, with no beats (beats are cyclic modulations
in the tone). This will result in a ‘G’ at exactly 150 Hz. This is derived from this
calculation:
•
the fundamental (100 Hz)
×
3 (=300 Hz for the second harmonic)
•
divided by 2 (to drop it back into the same octave as your starting pitch).
This relationship is frequently expressed in terms of the ratio 3:2.
For the rest of the scale:
Tune the next fifth up: 150
×
3 = 450/2 = 225 (which is more than an octave above the
starting pitch, so you need to drop it another octave to 112.5.
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.
There is, as you can see, a choice. Either:
•
each fifth is perfectly tuned, with octaves out of tune, or
•
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.
Note
Frequency (Hz)
Notes
C
100
×
1.5/2
C#
106.7871
divide by 2 to stay in octave
D
112.5
divide by 2 to stay in octave
D#
120.1355
divide by 2 to stay in octave
E
126.5625
divide by 2 to stay in octave
F (E#)
135.1524
F#
142.3828
divide by 2 to stay in octave
G
150
(
×
1.5) divided by two
G#
160.1807
A
168.75
A#
180.2032
B
189.8438
C
202.7287