MadSci Network: Physics

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


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

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


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

y=2        then

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

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,


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