From time to time, we hear of Boehm's Schema, a semi-mystical artefact in some way instrumental in the perfection of flutes. Let's see what we can find out about it.
So what's a Schema anyway?
One web resource tells us it's:
And what is Boehm's Schema?
Precisely that - a diagram, designed to help a flute maker set out the correct positions of flute holes. But not just at one pitch - a list of numbers or a the distances marked on a rule would do that. Boehm's Schema permits the user to determine the locations of the tone holes for any pitch. To use today's terms, it's "interactive".
Scale vs Schema
Confusion often sets in between the use of the terms scale and schema. The scale of a flute, or any other instrument, is the series of notes we get when we play it.
Because the notes have a physical relationship to the structure of the instrument, it's also valid to call the actual measurements of tone-hole spacing, or fret spacing on a guitar, or string lengths on a harpsichord, "the scale" of that instrument. So we might refer to a flute "scaled at English High Pitch", or "having Cooper's Scale". These are actual pitches, or the dimensions that will yield these pitches.
The Schema is not just a scale, although it includes Boehm's scale. It is a device for translating Boehm's scale into other useful pitches. So it is not accurate or helpful to refer to, say, Cooper's Schema (unless he happened to have designed a pitch translator too!) when we mean the scale he used for his flutes. That is Cooper's Scale.
Why was the Schema needed?
A fair question. Surely any flute maker simply took the measurements Boehm used on his own flutes? Aha, not so easy. In Boehm's time, pitch was not standardised around the world. While Germany and France had settled on A 435Hz, England was having a veritable civil war - the domestic piano was probably still tuned to A 430, the Philharmonic Orchestra played up around 452-455 Hz, and the church organ was probably where the last well-intentioned meddler had left it.
Now that diversity of pitch centre seems not to have bothered most English flutemakers, who seemed to take the attitude: give them a tuning slide long enough and a good player will make the best of it - but it clearly bothered Boehm. He had invested years of his life making the world's first properly tuned flute, he wasn't going to let any hick manufacturer tarnish his reputation by making the wrong flute for the region, or perhaps worse, by guessing what changes to the dimensions were needed to make it right.
And we can imagine what Boehm was up against. While the precision of figures in his treatise and essay suggests he was comfortable with the use of 7-figure logarithm tables, we can't safely accuse the average London flutemaker of being so. Count the holes in an 8-key flute, sure, but none of your "12th root of 2" mumbo-jumbo, thank you very much.
What did it look like?
Like a wide rule, essentially, about 700 long and 60 mm wide (28" x 2.5"). The line along the middle (at B) shows the position of the stopper, embouchure hole and the tone holes on Boehm's flute. The line above (at A) shows the same for a flute pitched 1 semitone higher, the lower line (at C) a flute pitched a semitone lower. Diagonal lines at each of the key holes permit locations of holes at intermediate pitches to be interpolated.
And how is it used?
The close-up below reveals how it is constructed and how it is used. You'll notice that the middle line assumes A = 435Hz, the pitch used on the Continent at the time. To demonstrate its use, Boehm has drawn lines for two additional pitches, A 445 and A430.
An aside on the illustrated pitches
This is an interesting choice of pitches - 430 Hz is what had generally prevailed in England up to about this time, while the Philharmonic movement were soaring up around 453 Hz, though this was not illustrated by Boehm. The other pitch he chooses to show is 445Hz - the proposed compromise pitch being promulgated by the Society of Arts in England. Was Boehm attempting to lend some support for this more moderate pitch, or were his two choices purely random? We certainly know he was aware of the Society - he mention in one of his letters something about "your friend at the Society of Arts". So his choice of pitches seems to indicate the Schema was developed with England in mind. Seems odd not then to have illustrated the pitch most likely to be demanded by English professional musicians.
Boehm gives two approaches for using the Schema - by tuning and by numbers. We'll look at them separately.
Supposing you need to devise a new scaling for domestic use in 19th century England - i.e. flatten Boehm's flute from A435 to A430. Pull out the tuning slide on the A435 flute until the A is in tune with the pitch in question, 430Hz. The notes above and below A won't be in tune, but that doesn't matter at this point. Measure the change in tuning slide position that has achieved this - it will be found to approximate 4.63mm (Boehm obviously calculated this, as you'll see below!). Mark a point 4.63mm further along the centre line and drop a vertical to intersect the diagonal line. Run a long horizontal line through this intersection. This line now intersects all of the diagonal lines at the spacing needed to make the flute play accurately at 430Hz on all the notes (and not just the A).
Now if you need to go sharper than the original 435, you'll run into a practical problem - you can't push the slide in far enough on Boehm's original flute to get more than a Hz or two higher. Confounded? No! Just tune the Bb to the A in question, eg A 445. You'll need to pull out about 13.4mm to do this, now measure back 13.4 from the Bb point (Boehm illustrates that at 8.96 to the left of A, surely an unnecessary calculation, but again see below), raise a vertical (because we're going sharper), and then run the horizontal through the intersection with the diagonal. A flute laid out using the lengths along that line will play accurately the Society of Arts scale, A445Hz.
Supposing though we don't have a note to tune to, or indeed, we don't have Boehm's A435 flute at hand to do it with. Easy, we can calculate our way out of trouble. The length of flute needed to give A430 for example, L430 is given by:
Subtract L435, and you get the 4.63mm offset we pulled out in the Tuning example above. Proceed as before. The same will work for A445, except we don't need to do the Bb thing, as we don't run into the practical problem of pushing a tuning slide in too far.
The Secret of the Schema
So how does it work - how can this simple graphical approach correct the lengths above and below A in the right proportions? Look again at the full picture of the Schema above. Note how the diagonal lines look to be approximately parallel until you compare the right hand end ones with the ones in the middle of the image. The right hand end lines are tilted more and more, so a horizontal line below the centre line (eg our 430Hz line) will cut off increasingly longer spacings as it comes to the end of the flute. The slide isn't just being pulled out, making one note correct, the flute is being stretched, keeping them all in tune.
When did the Schema come about?
The earliest reference to it I'm aware of is an account given by William Pole, a reporter at the London Exhibition of 1862:
One would imagine however that Boehm would have realised the need much earlier, as soon as he decided to license a manufacturer in England to make his instruments. We know he provided an "Essay on the Construction of Flutes" to Rudall & Rose in 1847 which they failed to publish until Broadwood dug it out from under them 35 years later. There does not appear to have been a Schema associated with that; indeed Broadwood seems unaware of the 1862 London version and cites the 1867 French version as the earliest. It is possible then that Boehm simply did all the work for Rudalls and provided them with an example of the flute they were to build, conveniently cut to the right lengths. A letter dated 2 Sept 1847 from Rudall to Boehm requesting such a model seems to confirm the fact.
The Schema then was perhaps intended for the next batch of manufacturers who came on line once the original patent period was up in 1861. That would be consistent with the 1862 timing, but raises the question, who were all these manufacturers clamouring for new models? We know Clinton made at least one Boehm flute and a number of flutes based on Boehm's bore, but who else? Pratten, maybe, over at Boosey & Co, getting more complicated with his Perfected flute? Doesn't seem to amount to "many".
Further, what pitches had the Boehm flute yet to be transposed to? Boehm worked at 435, so that would satisfy the needs of Germany and France. He provided a model to Rudall for England that was presumably high pitch. The same would presumably have satisfied the Americans. What was left?
Which came first, the flute or the Schema?
Definitely the flute - Boehm makes that very clear. Although he also uses maths to lay out the Schema, he went to a lot of trouble making special experimental flutes with moveable holes to get his basic scale to his satisfaction. Again, the Schema is a device to translate a known good design to other pitches, not as the starting point for his design.
How did he devise the Schema?
I don't want to go into mind-numbing detail here - more can be found by reading Boehm's own account (see Acknowledgements below). But here's the thumbnail picture.
Anyone attempting to scale flutes by simply applying a scaling factor (eg to convert a flute into a piccolo, simply divide everything by 2) will become quickly aware it doesn't work quite like that. Firstly, parts of the flute that are tapered react differently to cylindrical parts. And secondly, there are "end corrections", that are best treated like a constant, and not scaled at all. Boehm's Schema has to somehow deal with these.
Boehm's approach to all these pesky inconveniencies is breathtakingly simple - shove them all up to one end of the diagram to get them out of the way. He found by experiment that the distance from cork to hole is 618.5mm for low C (C4 in the modern system) and 335mm for middle C (C5 in the modern system) (this all of course being at his A=435Hz). Double 335mm gives you 670mm, and this should equal the first measurement, 618.5mm. Ooops, it doesn't? Oh well, there's all your end corrections in a bucket - slap the leftover 51.5mm up past the stopper and use that as the datum point for all other calculations.
Once the end correction problem is out of the way the rest falls into place. Each subsequent semitone is placed further along by the ratio of the 12th root of 2, which calculates as 1.0594631. Do that 12 times and you get a factor of 2 - the octave.
Limitations of the Schema
Now, if you're thinking, whoopee, I'll just rush off and design myself a flute using the Schema, you'll find it has a few quite significant limitations. Boehm's Schema doesn't attempt to:
Holes of a different size?
The remaining fly in the ointment is how to deal with holes of a different size. Strangely, Boehm doesn't attempt to deal with the issue at all. He says earlier in the treatise that he found the holes should be at least 3/4 the diameter of the tube (14.25mm) to give the best results, but that 13.5mm was about the biggest easily practical in silver, and 13mm in wood.
Miller counters that most of the Boehm wooden flutes are closer to 12.8mm - we might conclude that the 1.5% difference could be combination of undersize drilling due to the elasticity of the wood and subsequent shrinkage in America's drier climate. Miller continues though that the largest hole found on the body of a Boehm silver flute are 13.4, while most are 13.2, on body and foot. He notes some exceptions - 14.5mm on the foot-joint of the "Macauley" flute, and the "Heindl" flute having holes graduated in 0.2mm steps all the way from the thumb-key hole at 11.4mm to the C# at 13.6mm.
So despite all these variations in hole diameters, no allowance is made in the Schema or its instructions for dealing with different sized holes. Miller makes the observation that the calculations hold good for holes of 13.2mm, with a rise of the pad of about 3mm. He doesn't make clear the basis of the observation.
Why publish it?
It does make you wonder why Boehm was so keen to have it published that he sent it to the London Exhibition in 1862, the Paris Exposition of 1867, and the journal of the Bavarian Polytechnic Society in 1868. By 1862, Boehm was 68 years of age, and of a mind to put all his matters in order, as we see by the writing and publication of his major work which he finished 6 years later. That work also included a less detailed account of the Schema.
It's hard to imagine that flute players at large were much motivated by it, especially as the explanation that accompanies it takes some working through. After 15 years, his flute was making steady gains in popularity - the diagram was not likely to alter this. And those who needed to calculate different lengths for different pitches had presumably already done it - the next major shift in pitch not until the high pitch fraternity capitulated in 1895.
A possible explanation?
We've seen that Boehm was keen to publish the Schema, but we cannot see much in the way of practical demand for it. Is there a sub-plot operating here?
Remember that Boehm had been accused of stealing his design for his original ring-key conical flute from Gordon. This accusation had a habit of re-appearing whenever it suited anybody to raise it again, and with the death of Gordon there seemed little hope of it being proved or disproved convincingly. Boehm mounted this elegant argument in his favour:
So perhaps the publication of the Schema is not so much to assist unspecified and unlicensed makers transpose his design to pitches we appear not to be aware of, but more to assist in clearing Boehm's name of the charge of plagiarism? That would be a worthy goal, even if it is sad to think the great man needed to continue that fight. And of course he did. Rockstro, probably Boehm's harshest critic ever, was yet to enter the fray.
So, what's this Schema good for?
Probably most useful for analysing flutes of the past, to see how they match or otherwise. Just as we can use the Schema to design a flute of a certain pitch, we can reverse it to ascertain what pitch the maker was attempting to design a flute for. Simple as marking out the scale of the flute in question on a strip of paper, and moving it up and down on the Schema to find the best match. Or perhaps there's an even easier way ....
(To be Continued ....)
I've taken the images of the Schema, both full and detail, from Dayton
C Miller's 1922 translation of "The Flute and Flute Playing", by Theobald
Boehm. This is an excellent place to read more on the Schema - it
should be readily available in libraries, second-hand bookstores and in
facsimile. A copy of the letter from Rudall to Boehm was kindly
supplied by Ludwig Boehm.
First draft, Dec 2005.