Octaves vs Decades Fractions At this USGS very good site about rms noise in sensors, a phrase regarding third octave bands (the original topic) appeared.


  The figure displays the minimum vertical seismic noise directly as rms amplitudes in a bandwidth of one-sixth decade. By coincidence, these amplitudes may also be interpreted as average peak amplitudes in a bandwidth of one-third octave (a standard bandwidth in acoustics)


One sixth decade factor is $10^{1/6}=1.467799$; the 10th factor $10/6=1.666666$, and one third octave factor $2^{1/3}=1.259921$, none of them being comparable.

It happens actually to be comparable with one tenth decade factor $10^{1/10}=1.258925$.

Note the rms and peak distictions... Why the author state as coincident both log factors?



In addition, the same paragraph continues with:


  An example: the minimum vertical ground noise between the periods of 10 and 20 sec is at -180 dB relative to  ${\rm 1 m/s^2}$, thus  ${\rm 10^{-180/20}\;m/s^2 = 1\;nm/s^2}$ average peak in one-third octave. 


Which is ok, but then: 


  Since the bandwidth considered is a full octave, the total average peak amplitude in this band is  ${\rm\sqrt{3}\;nm/s^2}$.


In here i honestly dont know why a $\sqrt 3$ factor appeared...
I think they made a mistake. $2^{1/3} \approx 1.259921$ is not close enough to $10^{1/6} \approx 1.467799$ to say that there is any coincidence. Even the half-octave factor $2^{1/2} \approx 1.414214$ is closer.