Re: [tuning] Digest Number 610

Seems to me that all Neil is saying is that when we talk about the 128th
harmonic (for example), in a practical sense we are not usually talking
about that pitch realtive to the root of some chord we are actually
playing. 1Hz is not audible, and so in a practical sense if we are
harmonizing with the 128th harmonic, usually it is with a pitch several
octaves down from the root which generated it. Here is the beginning of
a table of the harmonic series (through the 31st) with the non-octave
pitches divided down to their familiar form:
1/1
2/1, 3/2
4/1, 5/4, 3/2, 7/4
8/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8
16/1 17/16, 9/8, 19/16, 5/4, 21/16, 11/8, 23/16, 3/2, 25/16, 13/8, 27/16, 7/4,
29/16, 15/8, 31/16
32/1, 33/32, 17/16, etc

It is just a point about semantics or terminology. The number of
harmonic series pitches that divide down to the octave doubles each
octave, so you can actually go up quite high in the harmonic series while
staying in the range of human hearing. Low A on my piano is 55Hz, and with
88 keys I have 1+1+2+4+8+16+32+64=128 harmonics from the series
represented up to high A. For some kinds of music these may be useful
harmonized with low A, but usually we are not talking about the _actual_
pitches from the series harmonized with their generators, but an octave
equivalent.

John Starrett
"We have nothing to fear but the scary stuff."
http://www-math.cudenver.edu/~jstarret/microtone.html