From: "Halperin David,
Dr." Dept
of Musicology, Tel Aviv University ambros@post.tau.ac.il
To: Bob
Fink
Dear Bob
Your letter came today, and I immediately read the flute article. Impressive,
but I have some comments, doubts and questions; what follows is more or
less in the order in which you present your findings.
1) How do we know it's [a remnant of] a flute? Dr. Turk is, I assume,
a fine and respected paleontologist; but you know well that there are fine
and respected scholars in many areas who think that their common sense
is enough for making musicological decisions. Has Turk suggested that the
femur can only be a flute? or does he consider that it may be [I'm sticking
my neck out here, as I am not a paleontologist or an archeologist], say,
an ornamental necklace piece which had some insets in the "flute"
holes? (I know personally of a case where an archeologist with whom I'm
acquainted mistook a portrayal in a mosaic floor of the 6th-7th cy AD to
be that of a man smoking pot, where it turns out to be an oboe [or chalumeau
or aulos] player!)
2) Your assumption #2 -- a predisposition in man toward equality or
symmetry in measurements -- seems to me to be highly speculative. I have
seen constructions in archeological sites in Israel where IN-equality is
prevalent, probably due to pecunious or lazy use of at-hand building materials:
naturally occurring stones and the like. And if you're looking for biological
rationales (you cite the 5+5 finger one), how about the left-brain-right-brain
asymmetry, or the right/left-handedness of us all? The "predisposition"
you posit may in fact be just an expression of the development of esthetic
ideals, for which we have almost no documentation earlier than historical
times.
3) The inequality of our musical scale intervals is indeed a fact,
but not a universal one. Think of pelog scales; or of the Yugoslavian fipple-flutes
with holes spaced according to where the fingers fall rather than according
to a preexistent scale, a sort of conceptual symmetry (finger = finger).
4) Anne Draffkorn-Kilmer (you really should give more credit to Richard
Crocker, the musicologist with whom she collaborated) is wrong. The
Kilmer-Crocker
reading of the Ugarit tablet is far from proved, and it is rejected by
most musicologists today (see my article). The assumption of two-voice
homophony is no more than a device for producing a neat solution, but the
solution doesn't make musical sense: the tritone is highly condemning,
and the melodic motion by leaps rather than steps is even more so.
5) I would like more evidence about the distances between fingertips
and their widths in Neaderthal man: were they as in h. sapiens? and was
the opposable thumb developed to the extent that it is today? Here I'm
asking questions, not arguing: I just wonder.
6) I agree that there are extramusical reasons for our preferring anhemitonic
pentatonic or diatonic scales, and these are not only rooted in acoustic
properties of these scales but also in our psychological makeup.
7) The holes of a flute are not optimally placed at the potential nodal
locations of an air column; rather, they substitute (roughly) for foreshortenings
of the length of the column. But this will not make any substantial difference
to your arguments.
8) You tacitly assume that the flute was open-ended in its original
state. Why?
9) An overblown flute will indeed produce the octave, if it is reasonbly
cylindrical in its bore; and the octave as the most basic interval seems
to be universal. But the placement of the holes (see #7 above) for divisions
of the octave should not be at the same places where, say, a monochord
would be divided. One has to take into account he end effect and the quality
of the coupling (in physical terms) between the internal vibrating air
column and the outside atmosphere. I can't tell how much this would affect
your results without knowing the length and bore of the original flute,
and whether or not it had a reed or pair of reeds.
10) Your calculation of the original (sounding) length of the flute
assumes that the scale was diatonic. But what if it was pentatonic, with
the intervals being semitone-tone-tone. But what if they were major second-minor
third minor third (as in A-B-d-f, a segment of a hypothetical A-B-d-f-g
pentatonic scale)? I am afraid that you were caught in a sort of circular
reasoning here.
11) How do you know which was the blow-end?
12) The "neutral third" seems to me to have been an attempt
to bridge the gap of a minor third in a pentatonic scale by dividing it
into two equal parts (Aristoxenus hints at this as well). If we are willing
to try to see the increasing complexity of scale constructions as an evolutionary
process (C-G-c == C-D-E-G-A-c == C-D-E-F-G-A-B-c ==
C-C#-D-Eb-E-F-?-G-Ab-A-Bb-B-c)
possibly deriving from extensions of the overtone series, this would place
the "neutral third" sqaurely in the second of these transitions.
There is quite a bit of evidence for viewing the diatonic scale as being
developed from the pentatonic (see, for example the "pien tones"
in Chinese music).
Finally: yes, the hole positions really are "consistent with 4
notes of the minor diatonic scale". My point is simply that they may
be just as consistent with some other segment of some other scale. But
I am ready to be convinced!
--David Halperin
P.S.: I have not yet studied the Archaeologia Musicalis article, but
one thing I did notice was your mention at the end of drones (bagpipes
and bi-aulos). I don't think that the sounds produced could be called
"harmonies"
in any acceptable sense.
March 6, 1997 To: David Halperin From: Bob
Fink
Dear David
Thank you for your letter. I cannot write to you properly now, but
I will later.
I had already read your article some years ago, but didn't twig to
your name when you contacted me recently. I am content-oriented rather
than name oriented, so I hope you're not offended by my forgetting that.
For reasons I will discuss in a separate discussion, I decided against
your view and in favour of Kilmer's back then. But I also confess I could
not follow your article fully at that time.
Many of your current challenges are welcome -- and I already had considered
many of them. I am always grateful for such challenges -- as truth (as
close as we can ever get to it) is more important than ego. I even spent
a month with my friend trying to debunk our own views on this flute, and
frankly, I have become a better critic of my own views than anyone at this
point. I am 99% convinced, but I could expand the remaining 1% into a formidable
case against myself.
Nothing is ever proved, as Einstein and/or Heisenberg said. Everything
is always an approximation and a degree of probability. In the meantime,
let me attach my press release. Also re-read the paper, as the end-notes
contain some answers and show awareness of a few of your objections already
(including the awareness of many other kinds of scales; equal scales; finger
or thumb-width standards, etc. Of course all this exists in history. So
do flying creatures and modern flying inventions. But do we ever doubt
that gravity still pulls all things down? It's a question of seeing the
forest -- not just the specific kinds of trees.)
And check out this website page: http://www.greenwych.ca/sherlock.htm
-- as there you'll see that I am not likely to be too impressed with
the musicology world anyway, nor with its general rejection of Kilmer's
findings. I ALMOST hold their over-specialized and narrow (non-interdisciplinary)
rejection of things to be a signal beacon that I must be on the right track.
I'm looking foward to the coming battle with them over this matter as well,
as I can assure you it will come, as quick as a knee-jerk.
Well I'm off to publicize my paper -- and I may be at risk. Instead
of "Musicologist Cracks Riddle of Ancient Flute Holes" it may
end up being "Ancient Cracked Musicologist Riddled with Holes in his
Flute." Oh, well, you only live once. Bob
Date: Sun, 16 Mar 1997 12:21:00
From: "Bonnie Blackwell,
bonn@qcvaxa.acc.qc.edu
Dept. of Geology, Queens College CUNY,
Flushing,
New York
To: Bob Fink
.
The flute fair [New York fair for flutists and
flute-makers] went really well. They were really impressed by your analysis
(if not your playing). But they were thrilled to hear the musical notes
it made, even if they smiled at the occasional missed tone. Now that I
have fully read the papers you sent me and I have thoroughly worked through
the math, I think you have hit the nail on the head. I included in the
article [for an earth-sciences journal] a note at the end, something usually
called a note added in proof, which included the following phrase (or some
such equally like it): An analysis by musicologist R. Fink (Fink, 1997,
in prep.) has suggested that the flute made the sounds neutral mi, fa,
so, minor la, on the melodic minor diatonic scale. I will fax you a copy
of the exact wording next week sometime.
Oh, one question from the crowd at the fair:
Is it a melodic minor or a "harmonic minor" scale? One guy, who
leads the NYC Reserve Regiment Band, said it was harmonic minor. --
b
.
Date: Sun, 16 Mar 1997 16:57:46
From: Bob Fink
To: Bonnie Blackwell
.
He's right. The correct term is
"harmonic
minor," not "melodic minor." The 4 notes on the bone flute
will fit into either minor scale, and the only reason I used the term at
all (and wrongly) was because I left the existing natural-Ti on the flute
when I fit the 4 notes into the whole scale. But we have no idea, if they
had another hole above the minor La, whether that would have been a Flat
Ti or a natural one. So the write-up note should just say "minor"
with no qualifications. --Bob
.
Date: Mon, 10 Mar 1997 00:16:08 -0800
(PST)
From: David Halperin ambros@post.tau.ac.il
To:
Bob Fink
.
Dear Bob
-- I have some more complaints -- this time about your math and
statistics.
First about the probability calculation (17!/(14!3!) = 680). This is
correct combinatorics, but it doesn't make sense when applied to the physical
problem of how many "different" hole placements are possible.
If the length is divided into 0.45cm segments, there are indeed 17 of these
segments; but not all of the combinations can be used! I don't think that
the Neanderthal man's fingers had a width appreciably smaller than ours
(probably larger?), and holes closer than about 1.5cm (maybe even 2cm)
apart would be unusable. So the length should really be divided into 4-7
segments, and the calculation is then at most 7!/4!3! =3D 35, not 680 --
a different order of magnitude.
Better would be to determine the inter-hole distances for equal division
(7.7cm/3 =F7 2.6cm), and then work around these fictitious holes by small
(0.45cm?) increments, taking into account some sort of natural spread of
the fingers, and stopping when the spread becomes unnatural or
uncomfortable.
And now to the calculation of the intervals. I'll use your notation
for the holes and the distances, according to the figure at the top of
page 5. Your use of "percent of previous hole" is wrong. You
need to take the overall length of the flute as the basis, and calculate
the positions of the holes as fractions of that. Here's what I mean:
(This is assuming -- somewhat wrongly, but let's ignore that -- that
the pitches are directly proportional to the hole locations.) If we take
e to be, say, 5cm, then we get 26.15/33.60 =3D 0.78, quite different from
0.87 -- in fact, almost a fourth (0.75) instead of a major second (0.9
or 0.888..., depending on which second you use). The difference becomes
negligible for very large values of e, but this would need a very long
bone (by the way, has Turk determined what animal provided the femur?).
Am I making a simple matter too complicated, or is my formulation right?
Regards, --David.
March 15, 1997 To: David Halperin From:
Bob Fink
Dear David:
ORNAMENTAL NECKLACE?
You wrote: "How do we know it's [a
remnant
of] a flute?" and you suggest it could
be part of something hanging on an "ornamental
necklace."
First, let's assume that it is a "hang-on" in a necklace.
And let's assume again that there is some reason why they made the holes
unequally spaced. (We have to assume there is a reason for this, as without
a reason, necklaces (pearls, ornaments, whatever) usually tend to be strung
in roughly equal increments unless there is reason not to do so.
(See discussion of the tendency to equality below.)
It's the height of irony or coincidence, don't you think, that a Neanderthal
would wear a necklace, and without even knowing it, be wearing a
bone that if it was used as a flute (which it wouldn't be, since
it's a necklace) would by sheer accident play the musical notes:
do re mi fa? The irony of this grows even greater when you realize they
had only 1 chance in hundreds (or 1 in 35 as you claim ) to space their
holes in a manner that could play such notes. Now, to make a flute entirely
by accident, and one that -- if they ever thought of trying to play it
-- would play the notes do, re, mi and fa, also entirelyby accident,
what do you think are the odds for that??? Guess how many assumptions you'd
have to make to sustain that prognosis?
Note you are adding unlikely assumption after assumption so far:
Assumption 1) It would be almost as newsworthy that Neanderthals had
an "ornamental" necklace as that they had a flute. This assumption
is remote, as nothing like a necklace or ornaments have been found (to
my knowledge -- but like you, I'm not a paleontologist). At least we have
a bone with holes for the flute's evidence. In this regard you contradict
another assumption you make, when you wrote: "The 'predisposition'
(to equal spacings/symmetry)... may in fact be just an expression of the
development of esthetic ideals, for which we have almost no documentation
earlier than historical times." In reply, I ask you: If you say such
esthetic ideals or ideas are not shown to be a tendency among Neanderthals,
then why do you now allow yourself to think they could have the esthetic
concept of an "ornamental" necklace or ornaments of ANY kind?
Of course, I believe there is a very wide use of equal spacings --
and, that since it exists in recorded history, therefore, it likely existed
before recorded history as well -- just as I believe that music did, or
for that matter, gravity. Yours is a strange method (to me, anyway), to
make assumptions that what exists (especially nearly universally) on the
known historic record, somehow ceases to exist before records were kept
or found, simply because of the lack of such records. I don't see the logic
in that method of making assumptions.
Your assumption 2):
Ivan Turk -- and a whole team of archaeologists and paleontologists
-- are out to lunch? (Surely you must know that Turk doesn't operate in
a solitary manner?) Of course, this is possible, just as it is possible
that most musicologists are out to lunch, but again, without extensive
evidence, the assumption is improbable, rather than likely. (Unless you're
like one of those "biblical creationists" who have "tons"
of evidence invented against the evolutionists and paleontologists.) Anyway,
the archaeologists point out all other flutes (whistles) found in pre-historic
times are very similar. The others have little mouthpiece slits (no extra
parts like inserted reeds), but no more than 2 holes because all these
relics were previously broken off where additional holes could have been
revealed to archaeologists. Some of the relics may have been just mouthpieces
themselves, slipped into larger, longer bones that were too large to be
comfortable in the mouth. This similarity of Neanderthal bone flute to
other known pre-history flutes, plus the line-up of the holes, is cited
as evidence the Slovenian bone (bear cub femur segment) is evidence of
a flute.
But on all this you should argue with Ivan Turk and Bonnie Blackwell
in New York as they can explain their point of view better than I. But
please bear in mind -- isn't it excessive or gratuitous to argue against
their "proof" it's a flute, when no one has claimed to have proof?
That's just a "straw man" method, I think.
EQUALITY OF SPACINGS
You wrote: "a predisposition in man
toward
equality or symmetry in measurements -- seems to me to be highly
speculative."
I cannot agree at all with your view regarding the tendency to equal
spacings being less than near-ubiquitous. You are not allowing your own
experience to tabulate the true quantities involved in this tendency. Without
keeping count exactly, surely you can tell the overwhelming majority of
cases in life -- maybe even 80%, if we actually counted instances -- would
be a case of people only going for non-equality when there is a specific
reason to do so. Without reason, the mental or psychological "default
mode" on this point would be to do equal spacing (if not in actual
workmanship, then clearly in intent-- if workmanship skills were
not up to snuff). If you don't believe this, then spend part of a day counting
every instance of repetition that you see in every detail of pictures,
or on the street, etc., (wristband holes; belt holes; plaid shirts; ancient
architecture, ancient calendars dividing the day into hours, the incessant
attempts to find equal numbers of months in a year; and the like) and seeing
how many of them (by modern people or by ancient ones) use UNequal
spacings and how many use equal spacings. If you are rigorously methodical,
and aren't prejudicially selective, that is, skip nothing (except
those elements that are not intended to be repetitive, like a wall with
only one door), and do this for several hours, then the comparative
percentages will be as I say.
For example, your reference to "at-hand" stones used in
construction
as being an example of "un-equality" is not relevant, since the
elements that are repeatedly used (various shaped stones in construction)
are not identical elements, therefore, they are not repeatable in the first
place. Think in terms of pearls, or pre-made bricks, or beans, or any other
elements that are virtually or naturally identical. When these are spatially
"arranged" by people, what prevails? Equality or non-equality?
Look around.
However -- again, this is really off the topic, as this is, for me,
a supportive point (I would've taken it to be common knowledge, like the
knowledge of gravity (despite flying things to the contrary). Acceptance
of this tendency is not required to make the conclusions I've drawn.
MODEL OR IDEAL FLUTE
CALCULATIONS
As to the the "percentages of previous hole" dimensions as
shown in the illustration called "Model Flue," we checked them
many, many times, as have others, and I'm sure they're dead right.
I don't know which draft you have. I did find an error in one calculation,
perhaps you got the draft copy before we corrected it? That error was in
one of the minor-key holes in the "Model" illustration. (Also,
though irrelevant to this issue, the pictures of the holes should have
been placed on the right side of the vertical dimension lines, not
the left side. I've mailed off a current version.)
You seem to be suggesting the use of "percentages of previous
holes" are "wrong" because we shouldn't be using that kind
of procedure?
In all this, I apologize, as it is now clear to me (from looking at
your complaints about the math) that my lack of clarity in this has led
you to totally mis-interpret the Model illustration and the purpose for
it. It was made in order to become a measuring tool. The ratios are standard;
the proportions (percentages of previous holes) are simple arithmetic comparing
the distances between holes. Aside from possible arithmetic errors, it
can't be as you said, "wrong." What's wrong with finding the
proportional differences between holes in a standard flute? It serves,
as written in the paper already, as a way to show the unique pattern existing
in any set of 4 notes in the standard scale. The patterns then become a
ruler or guide or template for measuring to see if the bone's holes mirror
the same pattern of spacings. This is wrong to learn or find out?? I'm
sure you can't possibly mean that.
After we're all done finding these percentages for all the holes (and
the bone flute still does not exist for us during this exercise
in reconstructing a model or ideal flute), then we have attained the proportions
found between holes on a theoretically perfect and standard "model
flute." (If you want to dispute these results then you need to fight
with the tuning-fork manufacturers and the acousticians in the physics
departments, not me. The ratios all come from comparisons of vibrations
of notes -- standard stuff etched in physical law).
Now, and only now, we can go to the bone flute, armed with our fixed
proportions "ruler" (although, quite frankly, except for the
need to be rigorous in academia, visual observation would almost suffice
to see a match on this bone!!). On the bone flute, using these percentages
taken from the model flute, we calculated and wrote down in the tables
the dimensions we'd "expect" from any reasonable match (and we
found two matches in one direction on the bone) to see if the match is
arithmetically as good as it looks compared to "actual" dimensions.
Namely, is it within the tolerances of what the ear can tell is in tune?
These tolerance ranges are already established in musicology, as well as
empirically (by using my own and others' ears). So if you might be considering
disputing these, take it up with specialists of the ear or physiology.
Based on "actual" dimensions in each match, we can now
roughly
calculate the possible original length of the bear cub bone required to
play these notes relatively in tune. The maximum length of bone possible
for the 2cm (posterior minimum diameter) of bear cub femur has been checked
out by myself with zoologists and archaeologists, and we're told the length
we needed is possible to exist. The bone could have been long enough for
it to have been a flute able to play 4 diatonic notes.
.
YOUR MATH
Your math was done based on an earlier length estimate. I apologize
for not sooner sending you an up-to-date draft to use. The new figure for
the length of the bone is 37cm (plus one cm and minus 5 cm). I believe
these new figures were in the summary I sent to you. You probably should
have read that before doing your calculations. That 37(+1/-5)cm is as accurate
as we can get.
Your results will be completely thrown off by the 37cm length -- but
as I explained in that original draft (or didn't I?) the 41.6cm length
is the effective "AIR-column" length, not actual bone length
needed. Actual flute material (any flute) need not be as long as the air
column because part of the air-column extends out into the air at the open
end of a hole or at flute's end. Thus a shorter flute length sustains an
operating air-column that is longer than the flute. (Check Helmholtz; Sir
James Jeans; many flute-makers whom I have consulted).
Your mistake is going to the bone-flute illustration first and using
it for calculating rather than understanding why I first built a "model
flute." To do calculations of ratios from the bone flute's supposed
length (L dimension) is impossible, because we don't have, nor will we
ever know, the length of that original bone-flute. That is, no "L"
dimension exists for the bone flute!! That's the "unknown"
we seek to solve or estimate. That's why we constructed a
"model"
illus. of a flute. Once we got the "percentage of previous holes"
calculations from that model, we then looked for a match on the bone flute's
hole-distance percentages or proportional distances among the holes. Note
again, that each set of 4 holes on the model flute would have a unique
arrangement of these percentages, as if they were like a fingerprint.
After finding a match, we took the best bone-hole dimension as a given
and then used it, reciprocally, to try to calculate from it the missing
length of the bone flute.
In other words: Rather than calculate from the "L"
to arrive at "something" (which you didn't seem to clearly name),
we calculated from what we already found in order to come up with an
"L" -- a length -- just the reverse. (I say 'something' above
because it's not clear to me what you're aiming to find in your math. You
haven't laid out the conceptual procedure or labelled what figures and
results represent -- i.e., are they "expected" or "actual"
dimensions or what?; and how do you determine your assumed dimensions?
No algorhythm nor english explanation is given of the point of the math.
Also, you seem to have used match #1 rather [you don't say which, actually]
than the one we chose as best, namely, match #2.)
Where did you get 5cm for dimension "e"? This is not a
dimension
that suppositions can be made about. In the model flute illus, dimension
e is a totally fixed entity arrived at by using the Helmholtz ratios. It's
the difference between L and the hole located at 8/9ths of L.
E.g., if L=100cm, then e=L minus 8/9ths of 100, or 11.11cm, and that
is 11.2% of L. Right?
Again: If L=50cm then e=L minus 8/9th of 50 and that's 5.56cm, and
that's, again, the SAME percentage: 11.2% of L.
Pick any dimension for L you like. The percentages (or proportional
comparisons to previous holes) are fixed and always the same. Again: they
are a "ruler." The only issue remaining is whether the corresponding
dimensions on the bone flute (which are labelled dimension P, Q and R)
exhibit the same percentages of the preceding holes, or enough of the same,
to constitute a match. (Nothing really new is being created here. We're
just making a simple numerical comparison: Let's say we have holes A, B,
and C, and another set: D, E and F. We note that A is twice as far from
B than it is from C. Using that knowledge as a "ruler" we ask:
Is D also twice as far from E than it is from F? Let's check! If so, then
D to E to F is a relative "match" of the distances-pattern of
A to B to C.)
Of course, there IS no preceding hole available for dimension P, because
the bone is broken off, so the table says "not applicable."
So we do it again, this time for the next hole to the right (again
using only the MODEL illus). Now this next hole involved is a minor 3rd,
or 5/6ths of L, and the percentage of the distance apart of that hole from
the previous hole is 49% of e (or .49e), using the row representing minor
intervals of the scale. If you claim some mistake here, what is it?
And we keep doing it for each hole. If this isn't clear yet, then I
don't know how to make it any clearer, except to say, in one sentence,
the following: It's simple: All we've done is to check to see if the pattern
of spacings on the bone flute closely matches any of the spacing-patterns
on a standard flute (or not). That's it. Period. End of story.
Maybe it would have been better, for reader clarity, to have dropped
the math and tables altogether, and to have instead simply produced pictures
true-to-scale or at visually comparative sizes and then suggested to readers
that they simply use a pair of dividers (or proportional dividers) to check
out each of the two matches for themselves, or overlay one above the other
to see the match.
You have indeed immensely complicated a simple matter, but that's my
fault. But as I said, if we made any mistakes in our adding/dividing, etc.,
in the model flute calculations, we need to have them corrected and would
appreciate it if you find any there. We can't find any.
PROBABILITY
CALCULATIONS
You wrote: "First about the probability
calculation (17!/(14!3!) = 680). This is correct combinatorics, but it
doesn't make sense when applied to the physical problem of how many
"different"
hole placements are possible."
You must be missing something here. Of course the formula
certainly
does show how many total placements of any kind of spacings are possible,
if you assume a different placement exists even when only one
hole is more than +/-.225cm off from current locations. Even a reverse
placement could be a mis-match to the model flute if no bone-matter extends
in the right direction to confirm the match. In any event, unless we assume
this is a flute for sure, then there are 680 placements within those
tolerances of +/-.225cm. (However, I didn't do these figures. Mike Finley,
a mathematician, did, and we agreed that since some of the 680 instances
could not be considered usable IF this was assumed to be a flute, that
we would stick only to an order of magnitude rather than try to excruciatingly
quantify the matter further.)
You write "...but not all of the
combinations
can be used! I don't think that the Neanderthal man's fingers had a width
appreciably smaller than ours (probably larger?), and holes closer than
about 1.5cm (maybe even 2cm) apart would be unusable."
I have trouble again with the lack of any consistent premise(s) involved
in your critique. Why would they be unusable? How do you define
"usable"?
Are you now ADMITTING this is a flute? If this bone and its holes are a
product of chance, and not even a flute, then all or some of the
holes can even touch each other, right? Indeed, they don't even have to
be in line, nor be four of them. One hole with no others would also
count as a "miss" of a match if only chance was at work. (But
we didn't even count these possibilities.) And the number of misses due
to chance or other Neanderthal motivations, if this is not a flute, are
certainly in the hundreds.
If it is a flute, then still, holes as close as 1.5cm would be
usable by many Neanderthals, or by many people, as
we have no idea of the sex nor age of the player. Here again
is an assumption on your part without which the critique fails. However,
granting your assumption, for a larger person, it could be uncomfortable,
but I think still useable, especially if you covered 3 holes with two fingers
and then "rolled" the fingers off of any hole. I am 6-foot 2
inches tall, very large-boned and overweight as well. I could still use
such close holes.
But not even counting or assuming techniques like these, the low number
you propose of 35 is wrong just on experimental grounds (as I manually
counted the mismatches). Remember, the holes do not have to be equally
spaced to be counted as a miss of our model's pattern. Any variety of spacings
can be a miss. Any one hole that is inexplicably far off the
"expected"
location can create a mis-match of all 4 holes even without moving any
of the other three. The holes can all be nearly touching, and if you
shift them, as a group, by .45cm, then that's yet another countable mismatch
even though the distances between all holes remain the same. Just count
these possibilities yourself, without using math.
But of course, this whole debate is completely off the point, I think.
We will never know certain things. We can only look at the existing holes
and say that they "are consistent with" notes in a diatonic flute
(or not). The conclusion goes no further than that, and I fail to see the
relevance of most of your objections, unless you insist that we've somehow
claimed that we've "proven" this to be a diatonic flute.
And finally, I can only repeat -- even if you're right, One chance
in 35 still makes a powerful case for the ideathat this bone is
a flute, and has holes intentionally consistent with those on a diatonic
flute. We are certainly on the right side of the odds.
I'd be willing to add your "worst-case" calculation into
the final draft, if you like, citing you as the source.
OTHER
SCALES
You yourself already wrote: "I agree that
there are extramusical reasons for our preferring anhemitonic pentatonic
or diatonic scales, and these are not only rooted in acoustic properties
of these scales but also in our psychological makeup.... Finally: yes,
the hole positions really are 'consistent with 4 notes of the minor diatonic
scale'. My point is simply that they may be just as consistent with
some other segment of some other scale." (emph.added)
On that point you wrote: "Your
calculation
of the original (sounding) length of the flute assumes that the scale was
diatonic. But what if it was pentatonic, with the intervals being semitone-tone-tone.
But what if they were major second-minor third minor third (as in A-B-d-f,
a segment of a hypothetical A-B-d-f-g pentatonic scale)?"
The pentatonic I refer to as historically significant is the
"whole-tone"
(anhemitonic) one. There of course is a smaller (underline smaller)
chance the arrangement of the 4 holes was intended to match a non-diatonic
and non-whole-tone pentatonic scale. But these scales are not nearly as
ubiquitous as the others. Indeed, when they are found in actual use, it
makes ethnomusicological news. If I am to make assumptions, I have to assume
not that it was intended to match a pentatonic that has half-tones, but
rather a diatonic scale. I always go with the likelier assumptions, and
try to avoid making any at all, whenever possible.
However, if you think it will match other scales, which ones exactly
do you think it will match, how common (across cultures) have such
scales been (and where), and how good a match would it be? What
"standard"
exists (and where) to define the tuning of these scales that you have in
mind?
And finally - most damning of all: We have a bone whose holes (you
surmise) may match some other scale besides the diatonic. Fine. What
are theodds that we'd find a bone whose holes just happen to
match your scale, and at the same time happen to be just those very
4 holes, which, from your scale, will also match a diatonic scale? This
is another assumption of yours that requires slim odds to be at work in
order to sustain it, I think.
Also, while this additional connotation is not in the Harvard Dictionary
of Music (yet), the term diatonic is often used to include any similar
scale that has no half-tones as well as denoting the usual 7-note (5 whole-tones
+ 2 half-tones) scales. This could include the whole-tone pentatonic and
a scale with neutral 3rds and 7ths. But this is not relevant to the matter
at hand except as a possible nomenclature problem. If so, I can look at
clarifying my conclusion -- i.e., "we have a neutral third -- or we
have a diatonic minor third that is out of tune but still within the recorded
historic tolerances found in tuning thirds." Because of this historic
record, I have arbitrarily granted these tolerances to myself as a necessary
and reasonable assumption to use in my analysis.
MISCELLANY
You wrote: "The inequality of our
musical
scale intervals is indeed a fact, but not a universal one. Think of pelog
scales; or of the Yugoslavian fipple-flutes with holes spaced according
to where the fingers fall rather than according to a preexistent
scale...."
This is already answered in the notes, where there is recognition of
the existence of finger widths used for hole spacings. But as in any evolutionary
process, I have sought out the overall cross-cultural tendencies rather
than being blinded by the local cultural exceptions.
You write: "The holes of a flute are not
optimally placed at the potential nodal locations of an air column; rather,
they substitute (roughly) for foreshortenings of the length of the
column."
Of course, you're right, and this misnomer will be corrected in the
course of future editing and publishing.
You wrote: "You tacitly assume that the
flute was open-ended in its original state. Why?"
The answer to this was given in the paper. (I'm not qualified to decide
how a bone fragment might fit into the whole length of the bone. But it
just didn't look possible in the "other" direction -- although
our best possible match would have been in that direction. It may be we'll
never find out out which end of this bone is up or how close it was to
a knobby end.) However, as stated in the paper, match # one "tacitly"
assumes a closed-end flute. You missed the discussion of this issue of
"open or closed" in the paper.
You wrote: "There is quite a bit of
evidence
for viewing the diatonic scale as being developed from the pentatonic (see,
for example the "pien tones" in Chinese music)."
I agree -- it's one of the fundamental theses in my writings. There
are many more examples across many cultures and periods than just the "pien
tones" you mention. I go into this issue at length, especially in
the full version of the Origin of Music, a copy of which is at the Jewish
Nat'l University Library in Jerusalem. If you have any literature on the
pien tones, I'd love to look at it or know about it.
Regarding the Kilmer matter, you can duke it out with her. My reasons
for accepting her view are based in her (and Crocker's) evidence relating
to syllable counts and the numbers of words, among other things. She makes
(as far I can understand it) fewer assumptions than you do, and ones which
I think are likelier or more justified than some of yours. But I hesitate
to argue this matter on behalf of Kilmer, nor until I can figure out the
math you use in your article -- which I cannot follow at all. I think it
would even take me hours just to write out the questions I have about it.
This will have to wait.
You wrote: "I have not yet studied the
Archaeologia
Musicalis article, but one thing I did notice was your mention at the end
of drones (bagpipes and bi-aulos). I don't think that the sounds produced
could be called "harmonies" in any acceptable
sense."
