> Except for one little thing--the twelve-tones of the chromatic scale
> reflect the basic physics of sound. The interval of the fifth is the
> second overtone of the fundamental vibration and the twelve-tone
> series is nothing but the succession of fifths:C-G-D-A-E-B-F#-C#-G#-D#-
> A#-E#.
Not so. It appears to be so if you run through the fifths on say a modern piano, because the fifths are "tempered" -- that is, they are not acoustically perfect fifths.
If you construct a circle of acoustically perfect fifths (and divide back down where necessary into the original octave) you don't end up with an octave divided into twelve equal parts. In fact you don't end up with an octave at all (google "pythagorean comma").
In other words twelve fifths don't equal seven octaves. You can check this with a calculator:
(3/2)^12 != 2^7
The equal-spaced twelve-tone octave is something we have created by *departing* from "the basic physics of sound" and the "laws of nature".
--
Michael Smith mjs at smithbowen.net http://stopmebeforeivoteagain.org