CORRESPONDENCE I -- NEANDERTHAL FLUTE
 
 
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Date: Thu, 6 Mar 1997 00:37:50
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 entirely by 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 idea that 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 the odds 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."
Why not?
-- Bob

Updated Mar 2003-- Two New Books on Music Origins & Music Archaeology

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