Apple Logic Express 7 - Manual (Page 470)

470 Chapter 20 Song Settings and Preferences
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.
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 taken to solve this mystery,
resulting in a range of scales, and finally arriving at the concept of “equal temperament”.
Other temperaments that have been devised throughout history maximize or
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).
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