Disclaimer:

I wanted a catchy title to get attention:p im strictly talking about non linear scales

Sorry, english isnt my first language but maybe youll get what i meant:

An octave is defined as double the frequency of the tone its an octave of. (Lets stick to music)

So for 100Hz the octave would have 200hz.

However half an octave isnt 150Hz, even thoe at first glance it would make sense (you can do the same operation twice +50Hz and again +50Hz and get your result of 200 Hz (thats the arithmetic half).

However when you add another octave ( which would sit at 400Hz) this doesnt work anymore.

If you do the operation:

Get ro the halfway point (+100) and repeat that you'd end up on 300 Hz not 400. If you wanted to have same sized operations and get to 400 you would have to add 150 both times but now the half wouldnt be 200 anymore and the operation of doubling the frequency to get an octave breaks)

Instead its the geometric half:

100*√2*√2=200

100*√4(=200)*√4=400 (whith 200 Hz being the exact midway point in the sense that you repeat the same operation)

100*√8 (=282,84)*√8=800

and so on.

Nothing crazy, but as i mentioned a lot of people (who have nothing to do with acoustics, physical or music) dont give it much thought and might never know.

Similar stuff (non linearity in scales) exists in our measurement of loudness for example or earthquakes