source file: mills2.txt
Date: Mon, 23 Jun 1997 11:07:45 +0200

Subject: Octaves not exactly 2^N

From: rtomes@kcbbs.gen.nz (Ray Tomes)

While reading through the backlog of messages that built up while I was
overseas for 3 weeks, I saw one about tunings with octaves not being
exactly powers of 2.  In most cases of variation it seems that values
greater (rather than less) than 2 are found in practice.  The reasons
may relate to overtones being not exact harmonics of the fundamental.  I
would like to raise a theoretical reason for an irregular octave to see
whether anyone has practical experience of such a thing.

With just tunings, the whole idea is to have as many simple ratios
between frequencies as possible.  For a range of notes this is achieved
by having frequencies which are some fundamental times 2^A*3^B*5^C*7*D
with A, B, C and D all integers with ranges of perhaps 8, 3, 1 and 0 for
a typical western scale, or maybe 12, 5, 2 and 1 for more interesting
possibilities.

[Note: the use of 12^2 does not mean that a ratio of >4096 in frequency
must be used, as the powers of the other factors may be an offsetting
effect.  e.g. in the JI scale of 24 27 30 32 36 40 45 48, there is a
variation of 5 powers of 2 although only one octave is spanned.]

It so happens that when 8 octaves are spanned, a greater degree of
strong relationships exist when a ratio of 252 is used than when 256 is
used.  It may be argued that 8 octaves are not normally spanned, however
if the implied tuning of chord fundamentals and rhythm equivalents are
included at the low end, and the overtones of notes at the high end,
then the situation will be quite normal to span over 8 octaves.

So how do I measure this "greater degree of strong relationships"?
I use the the number of ways in which a number can be factorised, and
whereas 64 and 128 can be factorised in more unique ways than 63 and
126, the reverse applies for 256 and 252.

It is possible to make some special chords that tend to highlight this
"need" for a reduced octave.  Notes with relative frequencies of say
4:5:6:7:8:14:21:42:63 demonstrate the disparity between the 4 & 8 and
the 63.  This is a somewhat forced example to put the whole argument
into a single chord.  You can fiddle around from there to get the idea
if you want.

I am interested in whether anyone else has thought about such a note as
this reduced tonic, or felt a need for it in their music?  My reasons
are purely theoretical.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
     http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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