From: William Djaja Tjokroaminata <billtj@...>
Date: 2003-02-11T14:57:41+09:00
Subject: OT: Twelve Tone System

Hal E. Fulton <hal9000@hypermetrics.com> wrote:
> I asked my college band director (assistant, actually)
> why we used a twelve-toned scale. He was the first
> person ever to answer that for me. It turns out, yes,
> it is arbitrary (like our base-ten system)... and yet
> not arbitrary at all. It has to do with things "working
> out" mathematically in a "nice" way. (Sorry, I've
> forgotten the details.) There are other scales -- in fact
> some cultures get along with a pentatonic scale -- but
> for most humans 12 is just about right.

Hi,

I never researched the actual answer, but this is my guess.  When a string
of length L which is fixed at both its end points is vibrated, then from
the undergraduate physics the following wavelengths are possible: L, 1/2
L, 1/3 L, 1/4 L, etc.  Mathematics call these the harmonic numbers.

The wavelength L corresponds to the fundamental tone.  The wavelength 1/2
L corresponds to a tone which is twice the frequency of the
fundamental.  If we call the fundamental tone as C1, then the first
overtone is one octave higher, which is C2.  The wavelength 1/3 L
corresponds to a tone which is three times the frequency of the
fundamental.  In our twelve tone system, this will corresponds to G2.  So
when a string with fundamental tone C1 is played, actually we get all
these overtones:

    C1 C2 G2 C3 E3 G3 B&3 C4 ...

Therefore, although it is true we can create any arbitrary scale, my guess
is that they will not sound as natural... (probably I should some day
experiment this in my Roland synthesizer...)  Regarding the pentatonic
scales, usually they are just subsets of the twelve-tone scale, such as [C
E F G B] and [C D E G A].

Regards,

Bill