I guess it's similar to the reason for Just Intonation.? Equal temperament is slightly discordant because of the interval used between notes.? Since the interval between keys or semitones (half steps) is the twelfth root of two, the only interval made up of multiples of those that gives a whole number, or a simple ratio of whole numbers, is the octave.? So, for example a "perfect" fifth is not perfect in Equal Temperament, because it's not a factor of 3/2, it's an approximation made up by multiplying seven of the twelfth root of two intervals together:
(1.0594630943592952645618252949463 ^12 = 2, so that's the ratio between adjacent keys, in Equal Temperament)
1.0594630943592952645618252949463 ^7 = 1.4983070768766814987992807320298
- which is just short of 3/2.
The reason that matters is that harmonics of the fundamental frequency beat with each other.? In a just tuned perfect fifth, some of them can be at the exact same frequency (given perfect tuning), so they don't beat - or they can be tuned so close that the beat frequency is out of our hearing range, even if the tuning isn't quite perfect.
If the lowest note of the interval (e.g. C) is f, then the higher note (e.g. G) in a Just perfect fith is 3.f/2. The harmonics of the lower note are:
f, 2f, 3f, 4f, 5f, 6f.... (we only need to care about ones we can actually hear, so the series isn't infinite)
The harmonics of the higher one are:
3f/2, 3f, 9f/2, 6f...
So, we have 3f and 6f in both, which can effectively cancel out - meaning that they don't beat.? In Equal Temperament, that doesn't happen, so some of the harmonics beat, audibly.
So, one approach would be to tune fifths on the piano to sound better together - which will mean the higher note going slightly above equal temperament (frequencies in a ratio of 1.5 instead of 1.498).? That accumulates as you go further up the keyboard, so that would be a type of stretch tuning.? I guess you could interpolate between the other notes, or try to get more intervals to sound better, by ear - adjust to get fourths sounding better, making a compromise with the way perfect fifths sound, and so on.
AndyHornBlower