How do we calculate the cents that separate two notes?

We have found two formulas to calculate the cents knowing the frequencies of two notes:

\[ cents = ( log_2(f_2)-log_2(f_1) ) * 1200 \]

\[ cents = log_2 \left( \frac{f_2}{f_1} \right) * 1200 \]

Let's calculate the cents of the major thirds that we discussed earlier.

Pitagorean major third:

\[ cents = ( log_2(329.06)-log_2(260) ) * 1200 \]

Base 2 logarithms of 329.06 and 260 are:

\[ log_2(329.06) = 8.36220685522536 \]

\[ log_2(260) = 8.02236781302845 \]

We substitute the values and calculate:

\[ cents = ( 8.36220685522536 - 8.02236781302845 ) * 1200 \]

\[ cents = 0.33983904219691 * 1200 = 407.8 \]

Just intonation major third:

\[ cents = ( log_2(325)-log_2(260) ) * 1200 \]

Base 2 logarithms of 325 and 260 are:

\[ log_2(325) = 8.34429590791582 \]

\[ log_2(260) = 8.02236781302845 \]

We substitute the values and calculate:

\[ cents = ( 8.34429590791582 - 8.02236781302845 ) * 1200 \]

\[ cents = 0.32192809488737 * 1200 = 386.3 \]

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