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

To answer the question "why 31"--why not.  It seems natural to me to end at
an octave of the fundamental, first of all.  There is no reason why I could
not go up to 64 or 128 and have more pitches, or stop at 16 and have fewer.
But ending at 32 gives me what I feel is a workable number of pitches, 16.

And about pitches getting closer together.  In any equal temperment the
ratio of the frequencies of adjacent pitches is constant.  In our 12 tone
system, the ratio from A to A sharp or C sharp to D is something like 1.06.
(Try 440 * 1.06^12 and see if it is at least close to 880.) Now I believe
I said that in the 16 tone scale the relationship between pitches is
arithmetic, not geometric.  So starting on A440, the next pitch is 27.5 Hz
higher or 467.5, then 495, etc.  Our perfect fifth occurs at 660 (try it
out, it works).  And to say this only works for one octave is correct!  I
am glad Bob pointed this out.  From A880 to A1760 the difference becomes
55, twice 27.5.  All the pitches are the same in this octave as the one
below it.  (if you don't believe me, try it out.)

So the intervals between pitches decrease as you ascend the octave.  There
are many pitches between A and E as there are between E and A.  But this,
I believe, will happen on any finite harmonic series ending on an octave of
the fundamental.  If we go from 1-4 we have pitches in the ratio, when put
in a scale, of 1, 1.5, 2.  From 1-8 we have 1, 1.25, 1.5, 1.75, 2.  And
that same crazy additive business always happens.  I didn't plan any of
this, it simply works mathematically.

Well, I hope I explained it fairly well.  But the more questions, the
merrier.

Kyle Major
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