Physics of Music - Notes

Making Sense of Cents

Musicians often refer to intervals using "cents." One cent is the interval which is one hundredth of a half step for the equal tempered scale. That is, 100 cents is a half step. Just like other musical intervals, the "cent" is multiplicative when you are talking about note frequencies. That is, to find the frequency 40 cents above a given frequency, you multiply that frequency by a factor corresponding to 1 cent, forty times. This may be different than what you're used to -- usually one adds 1 cent 100 times to get a whole (e.g. to get a dollar).

To convert from cents to the corresponding multiplicative factor, do the following:

y = value of interval in cents
x = y/1200
z = multiplicative factor = 2x

On a calculator this last step is done using a key which is most usually labeled "xy" or "^". To make sure you know how to use this key, try 23 which should be 8. If you get 9, you computed 32 instead. Hence, a frequency 1 cent higher than a given frequency is found by multiplying the original frequency by 1.00057778951 . Middle C on a piano is about 262 Hz. If the strings are tuned 1 cent sharp, they are tuned to 1.00057778951 * 262 Hz = 262.15 Hz, or about 0.15 Hz too high. If the strings were flat, you divide instead of multiply.

To convert from a multiplicative factor to cents is a little more difficult.

The logarithm (or "log" for short) is the inverse of raising to a power. Hence, if you have the multiplicative factor above, z, and you want to find x, then you would like the log base 2 of z. However, it is a bit more difficult since our calculators don't usually have a "log base 2" on them. Instead they will have a log base 10 and/or a "natural log," which is a log base e (where e = 2.71828 is a number which shows up a lot in math, almost as much as pi = 3.14159...). The natural log is often written as "ln."

Using properties of logarithms the "log base 2" is found using:
"log base 2 of z" = "log base 10 of z" divided by "log base 10 of 2"
"log base 2 of z" = "natural log of z" divided by "natural log of 2"

The more general rule is that
"log base a of z" = "log base b of z" divided by "log base b of a"
where a and b are any two (positive) numbers.

It is usual for us to write "log base 10" simply as "log." The "base 10" is implied if no value is given. The "log" button on a scientific calculator is "log base 10."

Written as a formula the log base 2 of z is:
log2(z) = log(z) / log(2)
So, for example, the harmonic fifth has a frequency ratio of 3/2 = 1.5 to the first note of the scale. Then:
log(1.5)/log(2) = ln(1.5)/ln(2) = 0.58496
Now multiply by 1200 to convert to cents. Hence the harmonic fifth is 702 cents above the first note of the scale. Note that 1 cent is relatively small and so there is no reason to keep a lot of decimal places. The answer is usually rounded off to the nearest cent.

As another example, suppose you compare middle C from your flute with the middle C on the piano and notice that they are 4 Hz different (you hear this by listening to the "beats"). Since middle C is about 262 Hz, this is a multiplicative factor of (262+4)/262 = 1.01526717. To convert to cents, compute the log base 2 and then multiply by 1200.
log2(1.01526718) = log(1.01526718)/log(2) = 0.006580345/0.301029996 = 0.0218594 ,
1200 * 0.0218594 = 26 cents .

To compare any two notes, you compare the number of cents using addition or subtraction. (If you were using frequencies, you would compare using multiplication or division, i.e. ratios).

As an example, for the equal tempered scale the notes of the chromatic scale are (by definition) at 0, 100, 200, 300, 400, etc., cents above the first note of the scale. The fifth is then at 700 cents. The difference between the fifth of the just scale and that of the equal tempered scale would then be (702-700) cents = 2 cents.

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