By the way, you seem to be hitting all around the subject of how our musical scale was originally designed. You were asking about the twelveth root of 2 thing, and now you are talking about harmonics. You are probably aware of this, but just in case you aren't, I'll mention it. Bach fudged the scale when he designed the well-tempered clavier, and lots of contemporaries hated him for it, saying that his instruments were out of tune. (Which is technically true, but usually no one but a musician can ever hear it.) The original scale was built using harmonics.
If you take a string and make it vibrate in thirds of its length, you will get a different note, one that just happens to be a fifth of an octave from the original. (Actually, an octave and a fifth, but you just lower the note by an octave to put it into the octave you are building.) Imagine that you are building an ancient lyre where you have one string for each note of the octave. No sharps or flats here. We are talking primitive. If your first note is a C, you will get a G. So you tune another string to that note. If you make a string vibrate in fifths of its length, you get another note, a third of an octave. It's funny that the 3 and 5 numbers are inversely related like that, but that is just how the numbers work out. From a C, 1/5 of the length of the string gives you an E. So you tune another string to that note. You continue the process, sometimes taking thirds or fifths of your new strings to make more notes.
The whole pattern is this: note length relative
frequency
I C 1 1
II D 1/3 * 1/3 9/8
III E 1/5 5/4
IV F (3/4 of string)
4/3
V G 1/3 3/2
VI A 1/3*1/3*1/3 27/16
VII B 1/5*1/3 15/8
VIII C 1/2 2
The terms like "1/3*1/3" mean "one third of one third". There, you will be taking one third of the length of the string of a new note that was made by using 1/3 of the length of the original string. The one funny exception to the rule is that F. They couldn't find any harmonic ratio that would make a note that would fit in there, so they broke the pattern and resorted to using 3/4 of the length of the string to get a note to fit in there. So that note is only sub-harmonically related to the rest of the octave. But some of its harmonics will be identical to some of the harmonics of some of the other notes, so it still sounds harmonic.
So what's the matter with this scale? Just one thing: If you start the scale on some other note, you will find that some of the notes that you get will be a little different, and some of them will really sound out of tune relative to the first scale. Bach decided to split the differences, to build a musical instrument with sharps and flats that could be played in any key without retuning. Hence the modern keyboard. And hence the newer convention of each note being 1.0526 times the previous note, including all sharps or flats. Still, I hear that some jazz musicians deliberately tune their guitars to the old scale, so that certain chords will sound more harmonic.
You can get away with that if you aren't changing keys a lot.
* Terrance Hodgins *