Myth: Octaves are Stretched on Pianos because of Equal Temperament

Yes, octaves are "stretched" on pianos but it has nothing to do with Equal Temperament.  Twelve notes DO have to be distributed within a slightly stretched octave but stretching and temperament are different phenomena altogether. Be sure to read my previous post about Equal Temperament.
OCTAVE STRETCHING
   What does this even mean?  Theoretical OCTAVES are pitches that have fundamental frequencies that progress by a factor of "2"  as shown here starting with C below middle C:

  C3       130.8 hz ( vibrations per second)
  C4       261.6  hz
  C5       523    hz
  C6      1046.5  hz    etc.
          ( multiply any frequency by 2 to get the pitch ONE octave higher)

Octaves on a freshly tuned piano actually end up a bit different than what you see above. This measurement comes from my own just-tuned Kawai KG6C a 7'4" grand piano ( every piano will end up a bit different, that's why tuning them is so much fun ).  Using C3 as a Zero reference, the other notes end up as shown below,  measured in cents (1/100 of a half-step):

  C3        0
  C4      4.4 cents sharper than the theoretical 2:1 octave
  C5     5.9 cents #
  C6     8.2 cents #
  C7     15.4 cents #
   C8     35 cents #

      THAT is octave stretching.   Relative to the 2:1 theoretical octave, the octaves are tuned wide, hence the word "stretch".  They were not stretched this way arbitrarily, though there is some latitude when stretching octaves, particularly in the very high registers. They are stretched this way so that the octaves will sound pure and smooth when played together.  On any other instrument, 2:1 octaves would sound pure, but not on acoustic pianos!  The stretching is done to accommodate the fact that, on the piano, the partials of any given note, which sound together with the fundamental are sharper than expected --- but only on pianos.  When playing octaves, the fundamental of the upper note primarily interacts with the 2nd partial of the lower note ( among other interactions!) and the tuning must accommodate the sharpness of the partial in preference to the theoretical 2:1 octave or the octave will sound out of tune. Tuners that tune aurally do this without even thinking about it - a pure octave on the piano is a stretched octave.  Notice that C8 in the above example is a full 35 cents sharp!  It's no wonder that other instruments often have real problems playing in tune with pianos.
    Why does this happen on pianos. Physics.  It is a quirky condition, which also is totally unrelated to Equal Temperament, called...

INHARMONICITY
 There is something to do with the physical nature of high-tension stringing ( 200 lbs. per string, more or less) and a "stiffness" factor near the string termination points that cause the effective speaking length of the string, for the high partials, to be shorter. The high partials are just sharp. Period.  There's no way to fix it but I'm not sure there's a need to.  Here is a measure of one string on my piano showing the octave partials that are generated by that one note ( C3):

       C3  0
       C4   3.4 cents #     2nd partial
       C5   4.1 cents #    4th partial
       C6    6.4  cents #     8th partial
       C7    125   cents #    16th partial  ( over 1/2 step sharp!!)

Interestingly, the first few octaves line up somewhat with the stretched octaves in the previous chart, but the the upper partials go pretty wild -- fortunately they are pretty much inaudible. Tuners use direct measurements of those out-of-tune partials when tuning octaves, so it's not really a surprise that the actual tuning mirrors the inharmonicity.
   There is a bit of serendipity in all of this because it turns out, that our ears distort our perception of pitch relative to the mathematics, and as luck would have it,  we humans prefer that the upper pitches be tuned sharp, otherwise we perceive them as flat.  This is easily and clearly demonstrated by playing a series of octaves on an electric piano - the upper notes sound disturbingly flat!   Because of this bit of PSYCHOPHYSICS ( physical science relative to perception), I believe that one of the big reasons that the piano has stuck around for so long is that it just coincidentally deals with the "pitch issue" naturally - if you tune the octaves for pureness, the pitch alterations pretty much compensate for the higher frequency pitch-compression that takes place in the ear and brain. 

 To make life in the piano-tuning world even more fun, every time there's a change in steel string diameter, which happens about every 6 notes,  the inharmonicity characteristics change significantly - tuners work hard to achieve EQUAL temperament, but sometimes it's not so EQUAL as we would like.