>>However did the twelfth root of two (or its early approximations) become
>>part of man's psyche?
>>(I speak of the ratio of frequencies represented
>>by the semitone interval - the indivisible step in music ranging from
>>Bach to boogie, and beyond).
Allow me to suggest that the fundamental feature of pitch that allows us
to have music is not the semitone but the octave. That is what, as the
song says, "will bring us back to 'doe'." I have read that we perceive
pitch
as somehow returning to its initial value when frequency
is doubled, because of the spiraling shape of the cochlea in the inner
ear. A tantalizing hint from the Scientific American "Sound and Music"
book, but that is all they said about it. I would love to learn more.
More broadly, the octave phenomenon (and I wish there was language
to speak of it without the bias towards "eight") puts me in mind of
Nietzsche's doctrine of the "Eternal Return," or even more fantastically,
of the "Spindle of Necessity" from Plato's Republic.
The semi-tone is by no means constant, as Vernon suggests. Any group of
classically-trained string players will, more or less in unison, adjust the
distance between tones to bring it closer to "natural" or "just"
intonation,
if the music allows. (This is particularly evident in major 3rds, which in
equal temperment (12th root of 2) is noticeably sharp---you can hear
"beats" when that interval is played in equal temperment, but not if played
as the ratio 5/4, which is the value of a 3rd in just intonation (it may be
6/5, I can't remember).)
For literally 1000s of years, theorists in China have been devising
different
ways to divide octaves, with one ancient method stipulating 120 divisions!
Most of these assays are based on a phenomenon called the "Pythagorean
comma", a curious anomaly, susceptible to algebraic proof, that a
succession of perfect fifths will never add up perfectly to an exact octave
of the starting point. The remaining tonal space is smeared on the rest
of the notes in various ways, equally in equal temperament, which I believe
came about circa 1600.
Generating 12 perfect fifths in a row (the Pythagorean
method of scale generation) brings you quite close to an octave of your
starting point, and that is one main reason why we have 12 notes in the
dominant equal-tempered scale. But closer passes are to be had at 43
notes and 120 notes. 17th Century Dutch physicist Christiaan Huygens
concentrated on developing a scale based on 43 notes, with a smaller
"Pythagorean comma" spread over more notes, hence more perfectly
sonorous notes, while at the same time more vigorous transposition
possibilities, the real reason for the whole line of inquiry. The drawback,
of course, is managing 43 notes per octave!
>> Is it possible that an anthropoid
<<possessing such a faculty had some survival advantage over a brother who
>>hadn't? If so, what might that advantage have been?
Maybe not getting the hook at the Apollo? : )
seriously,...
This is a really interesting question, because it delves into that
period of human development (which lasted around 100,000 years,
according to Frederick Turner) where human culture actually
influenced human evolution. But I think that the best answer
would probably be found not in the realm of musical instruments,
but in the prehistory of singing.
Brendan Frost