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The false piano (1)

The following remark sounds strange for most people:

a well-tuned piano is a little bit false.

The way musical instruments are tuned nowadays is the result of centuries of searching in order to make music sound pleasant. So, can't it be done in a simple way? Do you need special tuning tricks?

Well, yes.
Let's have a closer look.

A tone is a vibration, expressed in Hz (Herz = cycles per second).
Looking at the keyboard of the piano you will see that all the tones are lying neatly next to one another.
Distances in between seem to be equal, but in the case of Hz they are not:
one octave higher doubles the amount of Hz.
For example: from a tone of a 100 Hz, the tone one octave higher counts 200 Hz, the next 400Hz, then 800Hz etc.

So 7 octaves higher means multiplying the initial number of Hz by 27 :

When one multiplies the initial number of Hz by 3 instead of doubling, then the octave plus fifth is reached (in theory, see 'Pythagoras and strings') e.g. from C to G:

One octave higher means: number of Hz x2, so one octave lower means: number of Hz divided by 2:

so, a fifth higher means multiplying the number of Hz by 1,5:

A fifth equals 7 semitones (white and black keys), an octave counts 12. So after 12 fifths one reaches 7 octaves:

However that would mean that 1,512 equals 27 ! This is not so. Between the two numbers is a small difference called 'the Pythagorean comma'.
Well, by reducing the fifth slightly the system can be made perfect. Precisely that, is the task of the piano tuner: to reduce the fifth so slightly that 12 fifths exactly equal 7 octaves. In other words: make the factor x (instead of 3/2) that belongs to the fifth with the result that x12 equals 27. Results in: 1,4983 instead of 1,5.
(Exact: x = $$2^{\frac{7}{12}}$$.)

How does the piano tuner do that? Read more in The false piano (2).

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