### Re: What is the operation to convert Hertz to cents deviation in music pitch?

Date: Wed Mar 14 11:53:28 2001
Posted By: Bruno Putzeys, Staff, Electroacoustics and Analog Electronics, Philips ITCL
Area of science: Physics
ID: 984035112.Ph
Message:
```
Hi,

An octave equals multiplying frequency by 2, two octaves equals
multiplying frequency by 2x2, three octaves equals multiplying frequency
by 2x2x2, it means that moving up x octaves equals multiplying by

x
2         (two to the power of x).

X does not need to be an integer. One can raise to a real number, or to a
fraction. For example, one cent is 1/1200th octave, so 1 cent up from a
certain frequency equals multiplying with

(1/1200)
2

To find the exponent if you know the base (in our case 2) is expressed as
a logarithm. If

x
y=2        then

x=log y    (The base-2-logarithm of y)
2

Now, the two normally used logarithm bases are e (approximately 2.718) and
10 (for reasons of our using a 10-digit numeric system).
Converting from, for example the base-e-logarithm (called the "natural"
logarithm, written ln) simply is done as follows:

ln y     ln y
log y = ---- = --------
2    ln 2   0.693147

Kind regards,

Bruno

```

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