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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
C100× 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