Ratios and Octaves

To determine the number of octaves represented by a given ratio, take the base-2 logarithm of that ratio. Since your pocket calculator probably only has base-10 (common log) and base-e (natural log or ln), note that:

log2(X)  =  log10(X) / log10(2)  =  ln(X) / ln(2)

Given two frequencies, such as upper and lower limits for a band of noise, you can find the number of octaves between them by first finding their ratio, then taking the log of it, then dividing that by the log of 2. (It doesn't matter whether these are base-10 logs or natural logs.)

For example, if the frequencies are 800 and 1200 Hz, their ratio is 1.500. Then log10(1.500) / log10(2) = 0.585 octaves.

Conversely, to go from an octave value to a ratio, just raise 2 to that power.

One common requirement is for an octave band of noise, centered at a given frequency. How do you determine the upper and lower limits of the band? The simplest way is to note that the ratio between the given center frequency and either limit is half the specified octave width.

If the total width is one octave, then find 2^(0.5) = 1.414. So the upper limit is 1.414 times the center, and the lower limit is the center divided by 1.414. If the center is 16000, then the upper limit is 16000 * 1.414 = 22627.4 Hz and the lower limit is 16000 / 1.414 = 11313.7 Hz.

You can use the same method for any fractional octave. If the total width is 1/3 octave, just compute 2^(1/6) and use that to multiply and divide the center frequency.

Note that the Frequency Entry Step Mode of the Daqarta Generator can take care of this for you. Start out in Direct mode and enter the center frequency for both the upper and lower values. These might be start and end limits for a frequency sweep or rise and fall band edges for a band of noise, for example.

Then go to the step control dialog and select Octaves, and set the size to 0.500 octaves, or half of whatever octave width you desire. Now exit that dialog and scroll the center frequency up one step to set the upper limit, or down one step to set the lower limit.

See also Octave Frequency Entry Mode, Cents Frequency Entry Step Mode, Standard Musical Note Frequencies, Band-Limited Noise, Frequency Sweeps