Decibels and Octaves

Happy New Year everyone. It has been a while since I blogged last. Sometime back I started learning Piano. That was my first introduction to world of music - theory and composition. As you may know already, a Piano has several white keys with black keys spread between them. The arrangement has a repetitive symmetry to it, even a casual observer would notice. Little did I know that there were fundamentally just 12 keys - 7 white and 5 black - collectively called as an 'octave' and this set is 'kinda' repeated over and over throughout the piano. Not quite repeated, but a key in one octave will emit a sound of frequency exactly twice as that of the corresponding key in the previous octave. And as for the relationship between keys within the same octave - it is quite simple - since you do not want an abrupt transition when you go from one octave to another, the 2nd key in an octave has a frequency (f2), that is the frequency of the 1st key (f1) multiplied by 12th root of 2. i.e. f2 = f1*(2^-12). This is so that automatically when we reach the first key of the next octave, which can be referred to as (f13), we have f13 = f1*(2^-12)^12 = f1*2 - Thus 1st key of next octave has frequency twice as much as that of the 1st key of this octave - therefore no abrupt transitions in frequency and at the same time an octave is doubled in frequency as it is repeated over.

Since the intention of this blog is not to give you a lesson in music, let me get to the point. When I first heard the term 'octave' while learning Piano, I didn't think about it too much - but it did ring bells in my subconscious mind. One day I was working at my office dealing with a measurement which used log2. When I was comparing one value with another one, I was writing down in my record book, "2nd measurement is one octave higher than the 1st one" - oops, 'octave'!! Now I remembered - in engineering like we use 'decibels' as units when we refer to a log10 value, we use 'octaves' when we use log2. The etymology of 'decibel' is known - it is from Graham Bell, an unit often used to measure sound as Graham Bell invented the sound device of telephone. But in college I always wondered where 'octave' came from - because what has log'2' in common with number 8 ('oct'ave)? Well it is all clear new - every '8'th white key in a piano will belong to the next 'oct'ave, that will have a frequency 'twice' as much as the 1st white key. In other words, if we take two white keys of frequencies fx and fy and compared them logarithmically, and if log2(fy/fx) = 4 then one white key is 4 'oct'aves away from the other. Well, people were measuring sound long before they even heard of electricity - I guess they used the same units they used for sound, which they were familiar with, when they started measuring electricity first.

After all that being said, I still have no clue why an octave contains 12 keys (why not, say, 14?). It might hit me someday!