"David Webber" <dave at musical.demon.co.uk> wrote:
>This is an oversimplification. You can find other rational numbers x and y
>such that 2^x and 2^y are close to 3/2 and 4/3 or whatever other intervals
>you want. You have to work harder to come up with 12. The concept of how
>close it has to be to be "nearly indistinguishable" is one of the
>considerations - the maximum number of different notes you can cope with in
>an octave is another.
David,
no, you haven't quite hit the point. Let me try to explain in
more detail.
When you try to construct a tone ladder with equidistant tones
whose steps multiply up to exactly 2 and where some intermediate
tones happen to hit 3/2 and 4/3, there is no other solution with
small numbers except if you use 12 steps. 3/2 and 4/3 are
important, of course, because they are naturally occurring as
harmonics when you ignore the additional factors of 2, so our
ear was already used to them before our forebears invented
music, and they occur on musical instruments (flageolettes,
resonance points of trumpets, etc.).
You cannot find any other ladder that has these same properties
with any small number other than 12.
The discrepancy is small enough to allow tempered tuning (such
that you can transpose a piece of music by any multiple of the
twelfth root of 2, play it on the same piano, and it sounds the
same, only higher or lower. This proves that the human ear
doesn't care much about the small discrepancy. Almost all pianos
today use tempered tuning.
I hope I use the right words. English is not my mother language,
so I may not always hit the proper word.
>But music does seem to require its own sort of braininess which has
>something in common with mathematical braininess but is not the same. :-)
Yes, I agree fully.
Hans-Georg
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