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The cent is defined as one hundredth of an equal tempered semitone, which is equivalent to one twelve-hundredth of an octave since there are 12 semitones to the octave. Thus one cent can be expressed as:

1200 \sqrt{2} o r 2^{[\frac{1}{1200}]}

$1200 \sqrt{2} o r 2^{[\frac{1}{1200}]}$

The frequency ratio of any interval (F1/F2) can therefore be calculated from that interval in cents (c) as follows:

\frac{F 1}{F 2} = 2^{| \frac{c}{1200} |}

$\frac{F 1}{F 2} = 2^{| \frac{c}{1200} |}$

and the number of cents can be calculated from the frequency ratio by rearranging to give:

\log_{2} [\frac{F 1}{F 2}] = [\frac{c}{1200}]

$\log_{2} [\frac{F 1}{F 2}] = [\frac{c}{1200}]$

Therefore:

c = 1200 \log_{2} [\frac{F 1}{F 2}]

$c = 1200 \log_{2} [\frac{F 1}{F 2}]$

(A3.1)

For calculation convenience, a logarithm to base 2 can be expressed as a logarithm to base 10. Suppose:

log₂ [X] = y

(A3.2)

Then by definition:

X = 2^y

Taking logarithms to base 10:

\log_{10} [X] = \log_{10} [2^{y}] = y \log_{10} [2]

$\log_{10} [X] = \log_{10} [2^{y}] = y \log_{10} [2]$

Substituting in Equation A3.2 for y:

\log_{10} [X] = \log_{2} [X] \log_{10} [2]

$\log_{10} [X] = \log_{2} [X] \log_{10} [2]$

Rearranging:

\log_{2} [X] = [\frac{\log_{10} [X]}{\log_{10} [2]}]

$\log_{2} [X] = [\frac{\log_{10} [X]}{\log_{10} [2]}]$

(A3.3)

Substituting Equation A3.3 into Equation A3.1:

c = 1200 {\frac{\log_{10} [\frac{F 1}{F 2}]}{\log_{10} [2]}} = [\frac{1200}{\log_{10} [2]}] \log_{10} [\frac{F 1}{F 2}]

$c = 1200 {\frac{\log_{10} [\frac{F 1}{F 2}]}{\log_{10} [2]}} = [\frac{1200}{\log_{10} [2]}] \log_{10} [\frac{F 1}{F 2}]$

Evaluating the constants to give the equation for calculating the cents value of a frequency ratio:

c = 3986.3137 \log_{10} [\frac{F 1}{F 2}]

$c = 3986.3137 \log_{10} [\frac{F 1}{F 2}]$

(A3.4)

In semitones (s), this is equivalent to:

s = [\frac{c}{100}] = 39.863137 \log_{10} [\frac{F 1}{F 2}]

$s = [\frac{c}{100}] = 39.863137 \log_{10} [\frac{F 1}{F 2}]$

(A3.5)

Rearranging Equation A3.4 to give the equation for calculating the frequency ratio from a cent value:

[\frac{F 1}{F 2}] = 10^{[\frac{c}{3986.3137}]}

$[\frac{F 1}{F 2}] = 10^{[\frac{c}{3986.3137}]}$

(A3.6)

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Table of Contents for
Appendix 3 Converting between Frequency Ratios and Cents

Appendix 3: Converting between Frequency Ratios and Cents

Table of Contents for Appendix 3 Converting between Frequency Ratios and Cents

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Appendix 3: Converting between Frequency Ratios and Cents

Table of Contents for
Appendix 3 Converting between Frequency Ratios and Cents