140
value with itself 12 times, or take a pocket calculator with the “power” (^) function and tap in
1.059463 ^ 12. The result is exactly 2 (or 1.9999 recurring). A perfect octave. Sounds complicated,
but it seems to work – so where’s the problem?
Let’s get back to our C-major chord and work out the frequency ratios using this “magic” number.
The interval C-E is four semitones, which means we should multiply the magic number with itself
four times (1.059463 ^ 4). Feel free to try it – your Virus is doing this kind of thing all the time :-). The
result is 1.2599. Fairly close to 1.25 (see above), but not near enough: It is detuned all of 14 cents
(14% of a semitone). A singer with perfect intonation would sing the E exactly 14 cents lower than
this!
The interval from C to G is seven semitones: 1.059463 to the power of 7 is 1.4983. Not bad, but not
quite the 1.5 me might have expected. This detuning is equivalent to only 2 cents, but still...
So that’s the big disadvantage of the tempered scale, the standard for all modern (western) key-
board instruments. Why can’t the notes simply be tuned “correctly” instead? Just intonation
depends upon the current situation, upon the context: An A-minor triad (A, C, E) also includes the
notes E and C like in our C-major chord, but their functions are now Fifth and minor Third (instead of
major Third and Root). So the frequency ratio should certainly not be 5:4 (1.25) for them to be per-
fectly in tune!
Is it actually possible to hear the disadvantages of Equal Temperament? Not immediately – we have
become too accustomed to it throughout our lives. However, we can certainly react to its more
unpleasant side-effects: Electric guitarrists avoid playing full chords when using very distorted
sounds. Instead, they often limit their playing to the so-called “power chords” which leave out the
Third (so they are neither major nor minor). Attempting to add the Third to a power chord only
results in a very rough and muddy sound.
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