… 1.414 and 1.059…
Part 2 of this blog is about 1.059
We saw (in part 1) that 1.414 and 1.059 are approximations of irrational numbers like π, and, like π, are both quite important. If you haven’t guessed, this is about geometric means (and arithmetic means), which are important for both the audiogram and for the calculation of the frequency of musical notes.
Part 1 of this series dealt with the square root of 2, which is roughly equal to 1.414 and is important in calculating the exact inter-octave frequency- the geometric mean between 1000 and 2000 Hz is 1000 x 1.414 = 1414 Hz and not 1500 Hz.
In music, this is where it becomes really important- the calculation of the 12 notes in an octave on a piano or any other musical instrument. Short of using a table (which is actually not a bad solution), if a musician comes in and says “I have tinnitus and it’s at C# just above the center of the piano keyboard” you can quickly say “… well, that would be the twelfth root of 2 higher than C.” And everyone knows that the twelfth root of 2 is 1.059. If middle C is 262 Hz, then C# is 262 x 1.059 = 277 Hz. The next note D, is 277 Hz x 1.059 = 293 Hz, and so on.
That is, we can chop up the octave into its 12 “equal” parts. Recall that an octave is made up of not only the white notes on the piano, but the black notes as well. Adding all of the sharps and flats to the white notes in an octave, we get 12. The notes are: C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. And if you prefer to write this out with flats instead of sharps, this can be equivalently done. (Its just easier for my computer to write # instead of the flat symbol, which I am sure that I can find given an infinite amount of time).
The number 1.059 is just an estimate to three decimal places of the twelfth root of 2, just as 1.414 was just an estimate of the square root of 2, but for all intents and purposes, it’s close enough.
Following is a listing of the frequencies from 1000 Hz going up in semi-tones to the next octave higher. Recall, however, that 1000 Hz is “close” to C (1048 Hz), but our audiometers are always calibrated to be a bit flatter than the musical notes for ease of discussion.
1987 Hz (about 2000 Hz)
Obviously, 1987 Hz is not 2000 Hz (an octave above 1000 Hz) because the twelfth root of 2 is slightly larger than 1.059. The number 1.059 goes on for quite some time, but to three decimal places, it’s pretty accurate. And if you want to be really lazy, you can remember 1.06.
Given that middle C is 262 Hz, try doing this yourselves with the 1.059 multiplier to find all of the semi tone frequencies between C 262 Hz and the octave above it.