> I would have to do additional math to understand this thoroughly, but > I can now see how 4ths and 5ths might be tuned so that they beat at > the same rate across the keyboard (or close enough to appear that way) > if the octave stretch is large enough. I think "close enough to appear that way" is the key phrase here, like virtually everything else we attempt to do. No two pianos tune precisely *absolutely* the same, nor does any single piano in successive tunings, even if done by the same tuner, let alone different ones. The musically acceptable target range is also quite broad and incompletely defined - hence these discussions. Ron N
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