All the discussion on temperament, if peripheral, has certainly been
interesting and, if I may, I would like to add my two cents.
It is my understanding that it is only the extreme octaves (top and bottom)
of a piano that are 'stretched' to accommodate the non-harmonic overtones
of the (inflexible) strings.
There is an implied assumption in the prior discussion that 'just'
intonation, with small number ratios of the frequencies, is in tune, while
other ratios, such as equal temperament, are slightly out of tune. That is
not necessarily so. If two notes, about a major third apart, are compared,
and one adjusted until it is 'in tune', the resulting frequency ratio
depends very much on the nature of the comparison. In particular, it
matters whether the two notes are sounded together, as a chord, or
sequentially and, if the latter, on how long an interval exists between
hearing the two notes. It also depends on whether the two notes are rich
in harmonics or not. If rich in harmonics and sounded together, then the
Pythogorean, or just, major third will sound in tune. But if a solo cello,
say, plays one note and then the other a little later, we (anyone with a
good ear) tend to settle on a smaller interval for the major third. [In
another life I taught a course on the Physics of Music and, in the class
lab, conducted a few experiments on this topic.]
It is interesting that many diatonic African instruments, such as the
Balanje (bush piano) in Sierra Leone, increase the interval between both
semitones of the equivalent Western major scale, so to our ears the thirds
and sevenths sound a bit flat, and the fourths and tonic sharp. I've
noticed that some kids, learning the violin, tend to do the same thing!
Derek Lane-Smith
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