What does the "proportional bandwidth" version of Wien's law mean? The Wikipedia article on Wien's law explains why the photon energy corresponding to the peak of the spectral density for blackbody radiation depends on whether one is parameterizing by wavelength or by frequency, and explains how to calculate the location of the peak in both cases.
But then it says

Using the value 4 to solve the implicit equation yields the peak in the spectral radiance density function expressed in the parameter radiance per proportional bandwidth. This is perhaps a more intuitive way of presenting "wavelength of peak emission".

Can anyone clarify what exactly this statement means? What exactly is the parameter "radiance per proportional bandwidth"?
 A: When you're describing a spectral power density, it's of the form watts-per-something.  You get to pick the something. You've already seen this with watts-per-Hertz (frequency) and watts-per-nanometer (wavelength).
When you're talking about something that extends over a very large frequency band, there are a lot more Hertz in the upper frequencies than the lower ones:  Many more between 100MHz and 200MHz than there are between 100kHz and 200kHz.  But for some applications, those two ranges are very similar: They correspond to a doubling, i.e. an octave, in frequency.
Hence proportional bandwidth: it's of the form power-per-percent of bandwidth (could be some other fraction, but that one's easy to calculate with).  That's 1kHz at 100kHz, 1MHz at 100MHz, etc.
To see why it's different:  A constant-per-octave power has much less power per Hertz at high frequency that at lower frequencies. It's got the same power in 100kHZ to 200kHz as it has across all the frequencies 100MHz to 200MHz.  Each Hz in the high band has only 1/1000 as a Hz in the lower band in this example.
Proportional power is the native form for log plots.  If you plot log frequency on the horizontal, each mm of scale corresponds to the same fraction of frequency as every other.  So plotting power-at-proportional-bandwidth on the vertical is appropriate.
