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:
- A diagrammatic representation of the structure or framework of
something. It's the logical and physical definition of data elements,
physical characteristics and interrelationships.
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
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
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
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
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
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:
L430 = L435 x
= 398.38 x 435/430
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:
Boehm has "sent for exhibition a
geometrical diagram, with explanations, by which makers of tubular
instruments can, with the greatest readiness and accuracy, construct
their instruments according to any of the recognised pitches.
Having been applied to by many factors [manufacturers] for new models,
M. Boehm desired to give his diagram and explanation the greatest
publicity and usefulness by sending them to this exhibition."
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
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:
allow low C to be a cut-off, rather than a side hole in
a longer tube.
incorporate any allowance to tune sharper
allow for holes of different sizes, either all over, in
steps or graduated
calculate those funny little holes, upper c# and the
allow for the shading of F# by R2 or R3 being closed
consider variations in hole depth
Fortunately Boehm deals with the first two of these
elsewhere in his treatise:
Cut the tube off 5mm below the calculated position of
low C for a flute ending at C.
Boehm claims to make his tuning slides 2mm shorter than
the Schema suggests to permit the flute to play a tad sharper when
needed. (DCM comments he has found some with 3mm shortening.)
Our kind translator, the scientist Dayton C Miller, took
on the duty of answering more of them, largely by examining rather a lot
of Boehm's flutes:
the shading of F# by R2 or R3 is ameliorated by moving
the F# hole 1.2mm above the theoretical,
the D# trill is typically 7.6mm diameter at 216.3mm from
the D natural trill is also 7.6mm diameter at 233.4mm
from the cork,
the c# key cannot be determined by the Schema because it
also forms the middle d vent. It's usually 6.6mm at 253.5mm from
(Keep in mind that all these figures relate to Boehm's
flutes at 435Hz, and will need to be adjusted appropriately at other
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:
"But the surest proof of the authenticity of my
invention may be a statement of my motives for constructing a new flute,
and the explanation of the acoustical and mechanical principles I made
use of; for he alone is capable of producing a rational work who is able
to account for every detail, from its conception to its completion."
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 ....)