# Simple mathematical process to calculate octaves and decades

#### Article By : John Dunn

Calculating the number of octaves and/or decades for numerical value ratios is actually a simple mathematical process.

The so-called frantic 50s and swinging 60s are decades meaning time spans of 10 years, but when we use that term decades in engineering, we are referring to numerical values in 10-to-1 ratios. “Ten” is one decade above “one,” “six-hundred” is one decade above “sixty” and so forth. Similarly, we have the term “octave” where “two” is one octave above “one,” “nine-hundred” is one octave above “four-hundred-fifty” and so forth.

However, when we have number pairs that are not in a convenient two-to-one ratio or a ten-to-one ratio, we need to do a little math to find their relationship in terms of octaves and/or decades.

Figure 1 Here is the math for finding octaves and decades.

In the method of calculation from Figure 1, “x” is either octaves or decades.

For example, if we arbitrarily let “Value 1” be 394 Hz and we let “Value 2” be 17831 Hz, we find these two frequencies to be [ log(17831/394) / log(2) ] = 5.5 octaves apart and also [ log(17831/394) / log(10) ] = 1.656 decades apart from each other.

Please feel free to check these numbers on a calculator or in Excel. You will find that 2^5.5 and 10^1.656 are equal to each other and come to 45.256. You will also find that 394 × 45.256 = 17831. Although there are some rounding errors due to the numbers of significant digits we’re using, that’s not germane to this thesis.

Please also note that it does not matter which base of logarithms you use as long as you are consistent in your choice. You can use common logarithms or you can use natural logarithms, but the ratios of the logarithms will be the same.

Figure 2 You can use common logarithms or you can use natural logarithms, but the ratios of the logarithms will be the same.

We now look at a few examples of calculating octaves and decades.

Figure 3 Here are a few examples of calculating octaves.

Figure 4 Here are examples for calculating decades.

Looking at the impacts of decades or octaves can sometimes be a bit startling; see this 2015 post on galactic oscillation.

Back in 2011, astronomers disclosed that the disk of the Milky Way is vibrating or oscillating at a frequency of 64 octaves below middle-C. You can play middle-C on the piano and it’s easy to hear, but going that many octaves down yields the impressive result that the period of the Milky Way oscillation at sixty-four octaves down from middle-C comes to more than 2 billion years.

Figure 5 The disk of the Milky Way is oscillating at a frequency of 64 octaves below middle-C.

Even Paul Robeson couldn’t have matched this bass note.