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logarithms

The cent

In music the Well Temperament divides the octave neatly in 12 equal parts. However, one should realize: in order to calculate the frequency of a tone, multiplying is needed, not adding. Example: getting a fifth higher (from any tone) you need to multiply with 3/2. So for each semitone one needs to multiply with the same number ($2^{\frac{1}{12}}$) (=$\sqrt[12]{2}$). This calls for a multiplication unit, and yes, there is one: the cent.
1 cent = $2^{\frac{1}{1200}}$. So raising with a 100 cents means multiply a 100 times with $2^{\frac{1}{1200}}$. In formula:

$b=a\times 2^{\frac{n}{1200}}$ where $b$ and $a$ are the frequencies of the tones of an interval and n the number of cents.

So, in order to calculate the number of cents of an interval this applies: $n=1200\times{ }^{2}\log \frac{b}{a}$

This way it is possible to think linear at intervals rather than exponential. And independently of the frequency one can speak about difference of pitch.

The guitar neck

The exponential function of frequencies is also visible in the construction of instruments. The distances between the frets of the gitar neck are also logarithmic. In this graph I started with string length 100. String length 50 means one oktave up. In between there is a logarithmic/exponential curve.

 

The nautilus shell NautilusCutawayLogarithmicSpiral

The spiral of the Nautilus shell is not based on Fibonacci numbers but it is logarithmic/exponential. The principle of growth being proportional and uniform with the existing structure causes this kind of growth and can be seen in many biological systems.
The beautiful clip made by Christóbal Vila (see Fibonacci and the Golden Ratio) also contains this shell, however that is not justified (which Vila admits).

 

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