Kyle Major replies to my theory query:

>Okay, I realize I was a bit vague.  Hopefully I can clear things up a bit.
>If we start with a fundamental, say A 440, the first octave double occurs
>at 880.  (this is what Bob mentioned).  The ratio for an octave is always
>2:1, always.  So if we are working with a harmonic series where our
>fundamental is "1" the first harmonic, the octave, occurs at "2." (or at
>1/2).  Next, is "3" which creates a ratio of 3/2 with our A.  This is the
>perfect fifth, E.  Then "4" is another octave.  "5" creates a third above
>A (5/4), but not quite in tune with equal temperment.  Note that "6" will
>be another E, 6/4 or 3/2.  So if we continue through 32 we will have a
>bunch of doublings, notably the note A at 1,2,4,8,16 and 32.  We are left
>with 16 different pitches, which after ignoring register they form some
>sort of "scale."

OK, I'm getting closer, this is very helpful.  Essentially you have the
overtone sequence transposed down to the original octave above fundamental?
That makes sense.  (Hard to get away from that damn octave no matter what
system you use...)

Being a brass player and all, you have to learn all about the overtones.
There's even a funky 'partial' in the upper second octave that's not a
diatonic note; this is consistent with what you're saying.  At a certain
point, though, the human ear cannot distinguish between tones that are too
close together (i.e.  a comma), and in the third octave there are numerous
fingerings (or slide positions) for each note, most can be played open if
your lip-pitch-bending is good enough.  Oh, and your ear of course.

I wonder if all of your notes are distinct in the scale once you transpose
back down to one octave?

>Our major scale was derived by circling in perfect fifths until ending up
>at the beginning pitch.  (of course, this only almost works).

Was derived? Can be derived.  Can be derived many other ways too.  We
did a great exercise once with a monochord (basically 2 strings tuned the
same note stretched over a board, with a rod or something with which you
can shorten one of the strings) which was very helpful in understanding
relationships between pitches as mathematical ratios.  (Being at St.
John's College, of course this was all based on ancient Greek & Latin
musicological texts, Pythagoras, etc.) Interesting to find that all of our
diatonic intervals are rational numbers except the tritone, which works out
to 1:square root of 2.  (Where's the radical key on my keyboard????) Which
is of course not rational at all.

>And to say this only works for one octave is correct!  I am glad Bob
>pointed this out.

Whew.  Thought I was losing it for a sec.  And keep you at it, and us
posted.

Bob K.