The circumference
of the bone at its narrowest point is (based on photographs) about 6.5cm.

For a hole to be considered out of line visually, not much of an increment
is needed to re-locate a hole in order to accomplish that visual recognition.

But the increment must, also, not assume a very unrealistic level of
precision expected of those fashioning an object like this at that time.

So I picked +/-1/4 of a hole diameter, in this case, +/-.225, which
gives us an increment of .45cm (a quarter-inch).

This is very large, and also worst for supporting the bias of my conclusions,
so it will serve to accept it here.

If we use that increment to place a hole out of line with the others,
or out of line with any other hole, then, at the narrowest point, a hole
could have appeared in 13 other places around that bone's 6.5cm circumference.
That gives 13 out of 14 ways for a hole to be out of line -- and true also
for each of the other 3 holes. The ways all 4 holes could be placed vertically
but differently, then, is:

14 ^ 4 = 38,416 (almost 40,ooo ways)

I left the number at "10" instead of 14 on the Correspondence
III page because of two reasons:

1. Other bones could be smaller, but still be capable of the properties
of a flute. A smaller bone has a smaller circumference, and will permit
possibly as few as 10, or even 9, 8 or 7 locations.

2. 10 seems like a good "average" to apply to all flute-like,
flute-sized bones, and is also easier to grasp for calculation purposes
by the average reader.

Using the figure 10, as shown on the Correspondence
III page, the odds against a line up are therefore at least10,000 to 1 for a flute-sized bone with 4 holes.

However, for this particular Neanderthal bone, the odds are closer
to 40,ooo to one.

ANOTHER
WAY TO CALCULATE

THE HORIZONTAL ODDS

Working now on the horizontal spacings of the holes, on the Correspondence
III page, we used the calculation given in the Neanderthal flute
essay Appendix, of 680 possible
spacings.

However, there are some problems with using that calculation here,
because that one was originally made based solely on the premise that it
was a flute, and to answer the question:

"What are the approximate odds that, of all possible spacings,
one will match a spacing that could play the do, re, mi and fa sequence
of notes?"

In doing this, we believed it was okay to use only the span from furthest
hole to furthest hole, rather than the full length of the bone fragment.
Also we "fixed" one hole, and just calculated the possible re-arrangements
of the remaining three. And other such things.

Also, we avoided saying that the answer was 1 match in 680, because
other matches were possible: A flip of the pattern reversing it would be
a match, but based on assumptions that the mouthpiece was at a particular
end and not the other, flipping the patterns might not allow it to be considered
viably "diatonic" any longer, hence not a match.

Also, we knew many of the holes would overlap, but we didn't calculate
that due to complexity of that issue. Thus, maybe as few as 15% (a guess)
of the 680 could be considered scale-like spacings of one kind or another,
and of these 15% -- or 102 "scales" -- only one would be "diatonic."

This caused us to realize the best that we could write at this point
was an order of magnitude of "one in hundreds," which is how
the appendix conclusion was first published, and how it remains at present.

But a better calculation can be made (although in the end it will not
result in a much different final answer) .

The new method here is based on the issue of simply how many spacings
can be achieved, without regard to whether they could be scale-like or
not, and placed somewhere within the entire length, with no holes fixed.

Now: Without choosing some kind of tolerance by how much
a hole can be moved in order to be called a "another location,"
the answer could uselessly result in "infinity."

So a tolerance has to be chosen somehow.

Since we are going to look at the final result to later ask, "How
many of these total possible spacings would match a diatonic scale-spacing
," then the only tolerance of hole movement that would give us
a noticeably brand new sounding scale that didn't still sound diatonic
(i.e., one that would be out of tune with a diatonic sequence) would be
the same .5 Pythagorean tolerance originally chosen (and by small
coincidence, the same amount as chosen above for the vertical "out-of-line
tolerance), namely +/-.22cm, which reflects the average ear's ability to
notice a tone as different from a previous tone, and happily also reflects
the toolwork ability of the hominid, if one is responsible for the event.

But first, the question at hand is not "how many scales,"
but just "how many ways can 4 holes be arranged -- including overlapping
each other -- along the whole length of the bone segment."

The whole segment is 8.5cm. Using the chosen tolerance, each hole could
move .45cm (or +/-.225cm). There are 22 segments of .45cm width available
to locate 4 holes anywhere.

The permutation formula now is:

22!/18!x4! = 7,315 different arrangements available for a 4-hole pattern.

Now we can ask: How many of those ways will match a diatonic
pattern?

Here is a list of the ways:

1. The pattern existing, and shifted left or right in 3 ways = 3

2. Another "diatonic pattern," but not of the same 4 note
sequence (which we'll only optionally consider, as each is less convincing
as a scale without the "do re or mi" there) -- of which there
could be two, = 2

3. Shrink the # 1 pattern proportionally, and it could still be a scale
(until you make it too small for fingers to play). This smaller version
of the pattern could shifted on the bone length by perhaps 3 or 4 ways.

This gives a reasonable total of 6 to 9 or more ways. And all these
ways can be "flipped" to provide a grand total of from 12 to
18 matches to a diatonic sequence (or perhaps a few more).

The calculation now is -- for 12 ways to match:

7,315/12 = 1 possibility in 609

and for 18 ways, it's

7,315/18 = 1 possibility in 406.

The odds-range for a horizontal spacing to match a diatonic pattern,
then, is very roughly, from about 1 in 400 to 1 in 600.

So using these detailed figures,
for this size bone in particular, the order of magnitude for its
holes to be both vertically lined up and also to horizontally match
4 widespread & known musical scale-notes, through randomness is, conservatively,

4oo (horizontally) x 40,ooo
(vertically) = one way in 16 million.

The range of odds is 16 million to one, for this particular Neanderthal
bone, and, as calculated on the Correspondence
III page, for also smaller but similar bones, 7 million to one.

Figures could vary further depending on the tolerances chosen and contextual
matters aurrounding the bone, but all of these cannot be calculated.

Therefore, if not an artifact, then this, to
coin a phrase, is a "one in a million" bone .