Calculate flute pitch from sounding length

[Andro]

Jem Hammond has a youtube video on how to measure the sounding length of flutes:

https://youtu.be/2sz-ixD34ho

When I measure my flutes, and then calculate A from the flute lowest note (assuming ET as an approximation) my results are nowhere near what I would expect. I have asked this question here before - how do you establish the pitch or pitch range for a flute from the basic sounding length measure? The answer to my question previously was to look here:

http://www.mcgee-flutes.com/flutelengths.htm

This is a good page but its a catalog of measure lengths, with no indication of the pitches.

Do I have to make complex calculations from theory and incorporate end corrections, hole effects, and so on? Or is there a rough relationship? Jem in his video says that information about that is readily available elsewhere and he does not discuss it. But I am unable to find anything.

For keyless conical bore flutes in D with open holes for C sharp and C, is the sounding length measured to the middle of the C sharp hole?

I use 343.2 m/s for the speed of sound in air at 20C.

Andrew

[paddler]

There are several complications in trying to determine a flute's target pitch from it's sounding length, and when you
combine them and try to generalize across a wide range of flutes with different designs, that renders the measurement
almost useless. Within a family of flutes that share common design characteristics, the measurement can tell you quite
a lot, though.

The first major problem is how to deal with the tuning slide. Some flutes were designed to play in tune with themselves
with the tuning slide almost closed, while others were designed to play in tune with themselves with it open almost 20 mm.
If you measure the sounding length of one of the latter flutes with the slide closed then you are going to be 20 mm off,
which represents a massive change in your estimated target pitch. For the measurement to make any sense you need to adjust
the tuning slide so that the flute is in tune with itself (not with an A=440 hz tuner) across the full range, and then do your
measurements.

The second major problem is that flutes with identical pitch targets can vary a lot in the length of the foot section below the
bottom vent hole. On a keyless, long foot, flute with two vent holes, the section below the bottom hole is acoustically irrelevant,
at least as far as pitch target is concerned. Even among flutes from modern makers targeting A=440hz, this measurement can
vary by over 10 mm.

Scale length measurements, such as Terry's C#-D# measurement make a lot more sense, but even these have a very large
variation when applied to flutes with different bore shapes and tone hole sizes. To see this, take a look at some of the
American flutes in my here.

My Antique Flute Collection

On this page I've started to document some of the relevant details of flutes in my collection. You will see that the information
is not complete for the flutes lower down the page, and, in fact, I still have a lot of flutes to add to the site. But take a look at
the following flutes. These flutes are fully restored and playable, and I have been able to accurately determine the target pitch
by playing them in controlled conditons.

Geib: target pitch 430 hz, C#-D# length 263 mm
Firth Son & Co: target pitch 440 hz, C#-D# 260 mm
William Hall & Son: target pitch 440 hz, C#-D# 263
Cloos: target pitch 425 hz, C#-D# 257mm
Pfaff: target pitch 440 hz, C#-D# 256 mm

There is a large tuning difference between an A=440 hz flute and an A=425 hz flute, but the C# to D# measurements can
be almost identical, or in the case of the Cloos, it has the lowest target pitch and the shortest C#-D# measurement. Crazy,
eh?

The tuning difference is attributable to differences in the bore size and shape, and the tone and embouchure hole sizes.
This is why I think these measurement are really only useful when applied to flutes that are members of a narrower family
that share common design characteristics. It is not that the physics is wrong. Its just that accurately modeling real world
flutes is more complicated than most models, or simple metrics derived from them, acknowledge.

[Andro]

Hello @paddler,

This is a great reply. I was beginning to suspect that sounding length was not a good indicator of pitch, and your excellent reply confirms this.

I was looking to buy a Rudall Carte from Gumtree, but wanted to know the expected pitch, which the seller does not know. So I had hoped sounding length might give it to us, but I see now that really only knowledge of RC instruments and experience give it to you. Nothing is simple. [That's why flutes are wonderful!]

And temperature affects the pitch. It's going to be 44 degrees centigrade (111.2 F) here today in Melbourne. I'm not playing my flutes outside!

Andrew

[paddler]

There is good collective knowledge on this forum about sounding lengths, C#-D# measurements,
and how they match to pitch for Rudall & Rose and later Rudall, Carte & Co flutes, so if you or Gumtree
posts the relevant measurements for the flute in question I suspect somebody here would be able to
give you a fairly reliable estimate of the target pitch. I don't have enough samples in my collection,
but I know others here have an immense amount of experience and data on those flutes.

[Terry McGee]

Confirming the discussion so far, in determining the playing pitch:

- the "Overall Length" of a flute is of absolutely no practical value
- the "Sounding Length" ditto. Both include too many red herrings.
- the C#-D# length is the best I've been able to come up with so far, but we can't push it far enough to yield a pitch.
- it can lump instruments of like type together, and that might be helpful
- any better approximation would require the kinds of measurements most people aren't in a position to take with sufficient accuracy. At least some finger-hole diameters and bore diameters at 2 or 3 points along the flute would need to be included, as well as the location of those points.
- it's possible that computer modelling such as WID could be used to identify which minimum number of parameters would be needed to get within say 5Hz.

My $0.02 worth....

[Andro]

Thanks Terry, I wonder how sounding length ever gained any currency to begin with. I see it referred to often.

Andrew

[Terry McGee]

I wonder if it's a relatively modern thing, e.g. post-Boehm flute? An interesting change seems to have kicked in back there somewhere. In the era of our conical flutes, pitch was all over the place, and we see that reflected in really long tuning slides. But suddenly tuning slides shortened drastically, and it remains that way today. Sounding Length becomes a bit more meaningful if the tuning slide can only introduce a few mm of variance.

One of the most bizarre outcomes of the suddenly-short tuning slides seemed to impact largely on Boehm-bore wooden simple-system flutes, eg this flute by Moon in my research collection:



In order for it to be in tune, the head has to be withdrawn to the point of falling off. And it's not an isolated case - I've struck many a flute from the late 19th century with that syndrome. Indeed here's an image of an original head and a new head made by me to reduce the pitch of the flute from 455 to 440. I'd moved the embouchure hole to the left by 21mm (53/64"). Weirdly, the flute was more in tune at 440 than 455!



What were they doing back then?

[paddler]

Terry, I've noticed a similar phenomenon with conical bore antique flutes, particularly the low-end
models that come without a tuning slide at all. Several such flutes from American makes, such as
E. Baack, Firth, Hall, Pond etc, have the same body tone hole configurations and bore shape as their
more expensive models that have a tuning slide, but the head length for the non-tuning slide flutes
tends to be set to match what would be a closed tuning slide position for a flute with a slide. This is
unfortunate because the flutes with slides tend to play in tune with themselves with the slide open
about 10 mm or so. So the inexpensive, slideless flutes tend not to play well in tune with themselves
unless the tenon is pulled out to the point where there is a large cavity in the bore and some instability
in the joint.

I have no idea why they did this rather than simply making the head a bit longer on the inexpensive flutes.
It would have been a trivial fix. So yes, what were they thinking?

[jemtheflute]

Sounding length is pretty reliably useful with antique Böhm cylinder concert flutes which haven’t (as often with French metal ones) had their heads cut shorter. They pretty consistently have a 19mm body bore, so variation of that is rarely a factor. The Bb-D key measurement (centres of key platters, or better, to the centres of the B and E tone-holes below them) can be a good guide to their design pitch scaling, too. Generally speaking, 600mm +/-3mm SL and no evidence of cut head = built for A440. 580mm = built for A452.4. 605-613mm = built for A435. High pitch flutes are found at other SLs such as 575mm, 585mm, 590mm. Bear in mind that makers usually allow 2-3mm for sharpening capacity and the fact that no two players blow exactly the same, so the closed up SL is not the actual working SL for the associated playing pitch.

Conical bore flutes are more difficult and variable, but I still find the SL to be a helpful buyer's guide as to the likely core pitch of a flute. Some years back I did try (quite hard! I liked Terry's argument) using Terry's C#-Eb measurement, but gave up with it as I found it useless as a predictor of even approximate design pitch or usable pitch. I suspect many makers worked out their mechanism design first and then sized the Eb tone-hole to suit the key position rather than siting the hole acoustically optimally and making the key to fit! Even on one class of flute (eg later C19th English "nach Metzlers") that dimension proved very inconsistent and no sort of predictor of at what pitch a given flute would play well.

I pretty much agree with paddler's posts above.

I have collected SL data of various styles and sizes of conical-bodied flutes and piccolos and, whilst I wouldn't state that a given measurement will give a certain pitch (waaay too many variables for that!), I do pretty well at predicting whether a given flute is likely to be usable at modern pitch A440 somewhere within its tuning capacity. The C#-Eb measurement is IME completely useless for that function.