Theobald Boehm and the Schema
The first thing anybody should do, to learn about the design of the Boehm flute, is to borrow, or preferably buy, a copy of the book "The Flute and Fluteplaying" by Boehm as edited by Dayton C Miller, Dover Edition (ISBN-10: 0486212599). The journey was in essence from the "simple" conical flute, with its small toneholes and dubious intonation, to a cylindrical flute (with "parabolic" tapered headjoint) with large tone holes placed according to acoustical and mathematical principles, and fully mechanised.
Central to this pioneering scientific approach was the Schema, designed to rationalise hole positions according both to Equal Temperament and to the required tuning basis, which at that time could range from A=435 to an almost "high-pitch" A=450. One may be forgiven for thinking that the latter concern no longer applies since the modern tuning standard of A=440Hz is effectively universal. The slight irony here is that, albeit over a much narrower range, modern flutes can and are made anywhere from A=440 to 444. If anything, the informal pitch standard for flutes is A=442, the idea being that this makes the flute a little easier to play in tune at a nominal A=440 in circumstances where orchestra pitch tends to rise, through a random mixture of temperature and over-enthusiasm.
It goes without saying that the flute needs to be able to produce the full 12-note chromatic scale. Previously, this required combinations of "fork" fingerings (to flatten the "natural" notes) and added auxiliary keys. As Boehm argues in the book, this was far from satisfactory. Ideally he wanted one finger per chromatic note, but as only nine fingers are available in the standard human body (the RH thumb being needed to support the instrument), some doubling up was unavoidable. The result was, of course, the Boehm flute system as we know it today.
Except in one aspect which, by all accounts, flummoxed flute players at the time
much as it seems to today – how best to make use of the little finger of the left hand?
The "Open G-sharp" Question.
As we learn from Boehm's detailed account, a driving design principle (in stark contrast to "old system" flutes full of auxiliary keys sprung closed) was that wherever possible, all keys should stand open, so that maximum tone was radiated from the instrument, and as few as possible notes depended on fork fingerings. We learn that one exception he could not avoid was the necessity of a closed key for Eb. On the other hand, there was no logical or physical reason for the G# key to be closed as on all "old system" flutes - the little finger was available for it, and moreover it need not be mechanically linked to any others. Hence the flute as designed by Boehm had an "open-G#" arrangement.
One important semantic detail is important to note here: tone holes (and the key cups which cover them) are named after the pitch they sound when open. Hence the marked G# key. Some flute players are inclined to refer, for example, to the E key as the "F# key", as being (often) the note produced when the key is pressed. Such usage naturally arises from following a fingering chart - understandable, but it is clearly important for the avoidance of ambiguity when discussing tone hole positions that the distinction is maintained between the identities of physical tone holes (plus associated key) and references based on the fingering chart.
The immediate relevance of this topic is the fact that, as we learn from the "Wibb" book, Wibb's first flute (at the age of twelve) was already an open-G# one. I like to think this gave him an unfair advantage over all other beginners, who all but inevitably began on a "standard" closed-G# instrument, and continued with it without any further questions. In my own case, I first read the Boehm book around the age of sixteen, and resolved to convert to open-G# on entry to the Royal College of Music - I think it took me about six weeks to feel the change was complete (with a lot of bad technical habits eliminated). Wibb said once that it should take a good player about a week.
The good news is that a few manufacturers have begun to offer open-G# models for the entry level player. For example, the Trevor James "10x" model is available in that form. The less good news is that, if dealer prices are anything to go by, there is no price advantage, as there should be, for a flute with so much closed-G# weight and complexity avoided. This includes the ingenious but arguably absurd "split-E" system needed to fix the Top E spoiled by the closed-G# mechanism ... so ubiquitous these days that no flute is considered "professional" without it!
In the context of Wibb's work on the flute scale it is important to remember that the size and position given for the G# tone hole are to be assumed "by default" to be for an open-G# flute, unless he expressly includes values for the closed-G# hole, as he sometimes does. The simple fact that they are different gives already a little clue to the many puzzles, enigmas and general head-scratching to come.
The numbers: Boehm's basic scale.
Boehm's book goes into comprehensive detail about the (relatively) simple mathematics involved in finding the positions of tone holes to make a practical flute on his system. Some figures, such as the "end corrections" required at both ends of the instrument, to establish the differences between theoretical and effective lengths of the air column, were clearly found by a combination of acoustic theory (something of a new science at that time) and experiment. They are in close agreement with modern analyses of flute acoustics.
We see that Boehm tabulated all his length calculations from the head end of the flute - from the stopper position (as we can see in the Schema above). This is reflected also in the key table Boehm provides relating note frequencies to hole positions (Boehm/Miller p. 35).
Modern practice from Cooper onwards is to give measurements of tone hole positions from the end of the footjoint, since any measurement from the head has to deal with the variability introduced by the tuning slide and by possible differences in headjoint lengths. The main principle to understand here is that of the "basic scale", before any end correction is added or adjustments made according to tone hole sizes. To make direct comparisons with Wibb's figures, it is sufficient to invert the right hand column above to give distances from the footjoint: simply subtract each number from the bottom one of 618.5:
0.0 37.6 73.1 106.6 138.22 168.07 196.25 222.83 247.93 271.62 293.98 315.08 335.0
This is then Boehm's basic scale (C foot) for the pitch A=435, with an octave length of 335mm.