The Bennett Scale - a partial archive
As we discover back at the start of all this – the Schema of Theobald Boehm – all modern flute scales are based on the same foundational arithmetic following acoustic principles. The first of these is to find the physical sounding length of the air column for a chosen pitch basis, defined by the fundamental frequency for A. The widely referenced web page on "Musical Mathematics" (see the About section) shows how this can be done from first principles. It confirms the reasonableness of Boehm's procedure. Corrections have to be added to both ends to arrive at the "effective sounding length". This work converges to a figure for the fundamental C1-C2 octave length, where the lower C is either the centre of the hole in a B foot-joint, or the open end of a C foot.
Upon this is computed the basic 12-tone Equal-Tempered (ET) scale, just as is used, for example, to find the fret positions on a guitar. This divides the octave (defined as a frequency ratio of 2.0) into twelve equally-spaced semitone ratios. The ET ratio of the semitone is so important a number for musicians it is well worth committing to memory:
12√2 = 1.059463094
To generate the whole scale we simply multiply this number repeatedly by itself. This is the dynamic computation provided in the first page (Basics) of the spreadsheet file included in the download. All the formal Basic scales found in the archive are derived in this way. For convenience, a range of pitch bases is also listed, also as found in the papers.By editing one field, scales to any choice of base frequency can be generated. Further simple calculations derive the distances for each note scaled to the effective sounding length. This is always twice the octave length – for any length of stretched string, pressing at the half-way point raises the pitch by one octave.
Tone Hole Sizes.
If, as Boehm did to begin with, all primary tone holes are made the same size and depth (and all keys are "plateau"-style), the only change needed is to add the required small end correction for the low open end of the tube. Boehm chose 5mm for this, but all modern scales here use 7mm. We see this everywhere in the papers. Musically, all this affects is the pitch of the lowest note, but for practical measurements we always start from the end of the foot joint.
Of course modern flutes use a range of sizes; this complicates all the calculations as it invalidates the pure ratios calculated above. We read in Cooper's book just how many variables have to be taken into account to find the best position for each tone hole (but still not as many as defined in the paper cited above!). Some of those variables arise on account of the Boehm fingering system, and the need from players to play smoothly over an extended 3-octave range - "in tune". All the concerns and compromises Cooper describes remain fully relevant. Differences, I suspect, also abound because not all flute players play in exactly the same way; nor do they always agree on what "in tune" means – which is a whole other story.
(Note: a few of the papers in the archive feature numbers that might at first glance seem to be some revision of the open end correction. Rather, they are half the measured tone hole size. For example, a 13.5mm tone hole will appear as 6.75.)
Tone Hole Displacements.
Fortunately, for making a working scale, the multitude of possible variables are here reduced to a displacement value for each size of tone hole (for which plenty of acoustic theory exists), and a further more empirical adjustment if the flute has French-style open holes. The continuous changes made to these values arise from the largely experimental way those values are decided. There is a tacit understanding that tone holes are all the same depth (bar the small C#), and that the controlling parameter is simply the diameter of the tone hole.
We will see, in particular, that early scale work assumed that the displacments were essentially linear. This can be seen, for example, in the spreadsheet by selecting the columns of hole sizes and displacements and plotting them on a line graph using the Excel command "Analyse data". Later on it was speculated that a slight stretch (see below) or curve would be needed to resolve the preception that the upper notes in the left hand still tended to be a little flat.
Towards The Scale: a case study – the "Altox" flute.
The "C# problem" will be well-known by all flute players. As Boehm explained, only one finger was available for this note, which had to serve several other functions - venting C#2,D#2, D3, G#3 and A3. Consequently it uses a very small diameter tone hole positioned much further up the tube.
In recent years a new key has entered into general use, a "C# Trill" using a full size tone hole and extra RH trill touch, and intended, as the name indicates, for facilitating not only that trill but also the G/A trill in the top octave. However, Wibb wanted something different, a way of using the full size tone hole for C#2 "automatically" - no special fingering required.
Around 2002/3 he had an initial idea for the mechanism, and commissioned the English flute maker Andrew Oxley to implement it on a custom instrument – not a new flute made from scratch, but using an Altus 907 closed hole flute transplanting the keywork onto a silver tube which Wibb provided. Hence the "Altox" name.
Fast forward about a decade, and I happened to ask Wibb if he had an open-G# flute I could borrow, for teaching purposes, while I put my own "Number 1" prototype flute (made back in 1982) on the bench for some keywork rebuilding. This was my first sight of the Altox. After Wibb's death Michie very generously suggested I keep it, and I have enjoyed playing it every other day since. Suffice it to say the prototype mechanism had been removed and the hole patched, leaving it as a standard open-G# closed hole flute with silver body (except for the head and foot sockets) and soldered tone holes.
Of course, Wibb provided Andrew with a full specification for the scale, representing his (then) current form of the "Bennett Scale". The relevant documents are included in the archive download, along with a dedicated tab in the Excel spreadsheet.
In addition to the familiar list of tone hole positions, the specification includes a solution to the "displacement problem" I have not seen elsewhere: a linear "stretch" to be added to the four principal notes above A, including the full size C#:
It also shows that this is the "Low" A=440 scale (see Altox4.jpg in the archive). This appears, for all intents and purposes, to be "The New 442".
Yet this is not quite the end of the story. Wibb revisited the flute a decade later (as he did periodically with all his flutes), writing little paper comments kept in the case, not forgetting to date them:
(side 1)
"ALTOX 4 Aug 2012 - Today B seems low (C high?) - A seems low! F should be lowered a little more."
(side 2)
"16 Nov '12 | C-nat is high | E + F might be better lower".
The year 2012 is significant. Between 2011 and 2012 Eldred Spell posted on his website two accounts of The Scale (included in the main archive). The only material difference between the versions is the figures for open-hole correction. The 2012 figures demonstrate no pattern that I can discern. For the closed hole form, the hole positions clearly agree with Wibb's – with one strange exception. Spell gives the position of the F tone hole as 169.5, while Wibb puts it at 170.5. For the flute, that is a sizeable difference. The octave length is the same, 324.1. Wibb's value is exactly correct given the stated end correction and tone hole correction. However, his comments show that he felt the F was, somehow, too high. Hence, some "plasticine" was added (shown in the section Bennett), which I have not removed. Unfortunately, Spell only lists final positions, not the underlying Basic Scale or the displacement values used, so we cannot double-check or compare the calculations.
All this work thus still falls somewhat short of a fully developed theory; that remains, one way or another, to be found. We can however note that Wibb's Altox scale dates from 2004 at the latest, and (apart from the F anomaly) nothing changed between then and 2012. So – for the closed hole flute at least, perhaps we can assert that this is as close to a definitive "Bennett Scale" as the available evidence can establish. As for the Altox, all I can say (with that little bit of 'plasticine' still in place) is that it plays as well in tune as any flute I have ever tried (including my own "Number 1", based on a vintage 1978 Wibb scale), and very smoothly indeed. Though I would like the small C# hole to be a little higher up...