Bass Trombone Bell Mandrel Measurement / Comparisons

[paysonmcc]

I saw a conversation on the Bass Trombone Appreciation Facebook group earlier on the subject of bass trombone shapes and mandrels. The original topic started with an idea/ generalization that bass trombone bell shapes could be sorted into two main families- 1. Fuchs (larger throat) & 2. 70/72h (smaller throat).

I have heard many stories about bells being based off of other models, the most common being that the Bach 50 was based off of the Fuchs, and Holton copying an early Bach 50 for their first bells. I haven't heard any rumors of bells being based off of the 70/72h bell- even later George Roberts models. Let me know if you've heard of any being based on this smaller bell shape.

Now to my current situation: I own way too many basses + have too much time on my hands = I measured and took photos of each bell to compare them against each other.

I measured 4 different places that I thought were interesting.
1. The inner diameter of the top of the bell where the tuning slide enters
2. Right under the 1. Ferrule where the bell (as we see it) begins. The tallest part of the actual brass of the bell
3. 13.5" up from the bottom of the bell. This is right where most of the lower bell braces were
4. 3.5" up from the bottom of the bell. This is near where the seam to most 2-piece bells were

IMG_4068.jpeg

The basses:
1. Bach 50b
2. Edwards 454 (1341CF)
3. Edwards 502-IY
4. M&W 929 2Y
5. Holton 169
6. Thein Universal "Kruspe" cut bell

I'll follow this post up with the measurements and photos that I recorded.

[paysonmcc]

Here are the measurements. I Used digital calipers that only have one decimal place. I measured multiple times to ensure that I had as accurate of a number as possible. I'll start with the Bach as I will use that as a constant in the pictures that follow:

Bach 50 (9.5")
1. 24.5mm
2. 26.5mm
3. 34.4mm
4. 77.3mm

Edwards 454 (9.45")
1. 25.9mm
2. 25.3mm
3. 33.3mm
4. 76.4mm

Edwards 502 (9.5")
1. 25.9mm
2. 25.4mm
3. 33.3mm
4. 76.0mm

M&W 929 (9.5")
1. 24.6mm
2. 25.9mm
3. 33.7mm
4. 76.1mm

Holton 169 (9.5")
1. 24.4mm
2. 26.0mm
3. 34.0mm
4. 75.9mm

Thein Universal "Kruspe" (9.3") This bell is around 42mm shorter than the others
1. 24.8mm
2. 25.4mm
3. 32.7mm
4. 73.5mm *
*This is exactly where the cut bell ring is

[paysonmcc]

Now some photos:

I'll include a link to my Drive with higher resolution pictures. Here are some animations I made comparing the bells to each other. The Bach 50 is the control between all of these. Look at the outline, but also notice how the reflections change and show the taper rates.

https://drive.google.com/drive/folders/ ... drive_link

[paysonmcc]

Some surprises from comparing these:

Edwards is actually smaller at the top of the bell than Bach. I made sure to re-measure both multiple times because of how Edwards transition their tuning slides to their bells makes it look huge. In fact, they have the smallest diameter at the top of the bell. Who knew?

Based on how the 169 plays, I thought it was going to be larger than the Bach everywhere except the bottom of the flare. The 169 is fractionally smaller in every dimension.

169 & M&W are very close to each other. Interesting.

The Thein by far takes the most air and plays the "biggest" out of all of these, yet has the smallest bell by far. The .567 bore must play a large part.

[elmsandr]

For a complex shape like this, I would take measurements relative to some other diameter. That is, one inch above and below where the bell is exactly 3.0” in diameter. Too much funny business occurring with the bell rim and how the flares sit on the flat surface.

But even with the limitations, I would classify the variances you found as quite minimal. If you can get your hands on a 70/71/72/73H to see those measurements, I think you will see a much larger change.

Cheers,
Andy

[Burgerbob]

Very interesting!

[meine]

Interesting. I measured the Williams 10 bell once when I was looking for a copy of it. Here are the measurements I took all 5cm from the rim to the tuning slide receiver:

Rim: 254mm
5cm: 107,34mm
10cm: 72,24mm
15cm: 57,57mm
20cm: 48,40mm
25cm: 41,87mm
30cm: 37,77mm
35cm: 34,16mm
40cm: 31,13mm
45cm: 28,25mm
Inner diameter of the tuning slide receiver: 25,42mm

[paysonmcc]

Meine that seems pretty similar to the larger bells listed here.

Andy I recently sold my 72h, so I don’t have a good example to measure. I do remember it being visually tighter of a flare. I measured the top of the bell to compare it to the M&W, and it was much smaller through the tuning slide but I didn’t write down the numbers.

[hornbuilder]

Some years ago I measured the tuning slides of Bach 50 and Conn 72H. The 72 tuning slide would literally fit inside a 50 tuning slide!!

[ZacharyThornton]

One thing that messes up measurements like this: you are measuring from the outside of the bell. That leaves out the factor of what gauge the bell is.
The best way would be to measure mandrels.. but that kind of data would be much harder to collect.

[paysonmcc]

Matthew, I'd be curious how the 72h dimensions compare to a Bach 45. I remember the 72h had the shorter bell/ longer slide combo similar to the Thein bell.

It would also be cool to compare an earlier Bach 50 bell to mine. This bell is only 5 years old and I wonder how many times they've replaced the mandrel over the many years. Even small changes could really impact the shape if it has been replaced a couple of times.

[blast]

Matthew, I'd be curious how the 72h dimensions compare to a Bach 45. I remember the 72h had the shorter bell/ longer slide combo similar to the Thein bell.

It would also be cool to compare an earlier Bach 50 bell to mine. This bell is only 5 years old and I wonder how many times they've replaced the mandrel over the many years. Even small changes could really impact the shape if it has been replaced a couple of times.

Interesting point about changing the mandrel over time. With the amount of use these mandrels had , I doubt they would have needed replacement. ..... ever. The Conn Fuchs mandrel was probably cleaned up and resurfaced when 60/62H production commenced. This would mean that 60/62H bells were very very slightly smaller than original Fuchs bells. I have found this to be a trend, though 60/62H bells do vary quite a bit anyway.

[Sesquitone]

One thing that messes up measurements like this: you are measuring from the outside of the bell. That leaves out the factor of what gauge the bell is.
The best way would be to measure mandrels.. but that kind of data would be much harder to collect.

Another way is to carefully cut along the centreline of each bell and trace both sides of the metal profile onto tracing paper. Then you can overlay profiles—with slight axial shifts to see how closely (or not) they match up, independent of the gauge of the metal, and independent of bell flare size. If you're designing a new mandrel, why not take a couple of the "best" existing bells and average their profiles?

[Doug Elliott]

A cut tube tends to close due to stresses in the metal. Cutting and then measuring would be very inaccurate.

[hornbuilder]

Thank you Doug

[SamBTbrn]

Tape the top end of the bell off, (where the tuning slide pulls out) talcom powder to coat the inside of the bell and then pour "model/mould Silicon or latex into the bell and you will have an exact negative of the inside of your bell.

[Sesquitone]

A cut tube tends to close due to stresses in the metal. Cutting and then measuring would be very inaccurate.

Yes, this was also my initial reaction to the idea when I first heard about it (over thirty years ago), when, by chance, I just happened to attend a lecture/demonstration by a brass-instrument manufacturer who had recently started a new line of tenor and bass trombones—that subsequently became (and still are) very highly rated. It was a direct answer to, "How do you design the mandrels?" I recall being particularly impressed that this supposed "procedure", convincingly described in detail, seemed so straightforward. The tenor mandrel was said to be a simple average of the inside profiles of a Bach 42 and a Conn 88. I don't recall which instruments were said to have been chopped up for the bass mandrel. On the other hand, in retrospect, maybe it was just an elaborate joke: a "snippy answer" to a "stupid question"!

[Doug Elliott]

Sounds like that's what it was. Don't perpetuate it.

[peteedwards]

This subject came up years ago on TTF, around February 2015.
I cannot find the thread in the TTF archive so it might be lost.
As I recall Chris Stearn provided data on various bells, Holton 169 183 185, Bach 50, Rath R9, and a custom Kleinhammer bell. I believe the measurements were taken on the OD of the bells at evenly spaced increments.
I still have the data which I converted to CAD models and approximated the Bessel functions for each (just for fun).

[blast]

This subject came up years ago on TTF, around February 2015.
I cannot find the thread in the TTF archive so it might be lost.
As I recall Chris Stearn provided data on various bells, Holton 169 183 185, Bach 50, Rath R9, and a custom Kleinhammer bell. I believe the measurements were taken on the OD of the bells at evenly spaced increments.
I still have the data which I converted to CAD models and approximated the Bessel functions for each (just for fun).

Yes I did. No idea what I did with the data. Just a bit of fun. The Conn 71H 73H etc are very close to my old Piering bells...traditional German. Just sayin'

[peteedwards]

Here is the raw data I got from the old TTF thread (measurements in inches, to the OD):

Untitled spreadsheet - Sheet1.pdf

here is what they look like with the curvature interpolated & overlaid:

Untitled.png

[paysonmcc]

That’s very cool! The only one missing in all of this would be a Fuchs to see which “descendant” is the closest.

[Tbarh]

Here is the raw data I got from the old TTF thread (measurements in inches, to the OD):
Untitled spreadsheet - Sheet1.pdf
here is what they look like with the curvature interpolated & overlaid:
Untitled.png

The one who says Kleinhammer, is that the Williams custom , or a prototype Holton ?

[BrettEvans]

I am glad that I kicked this discussion off. I think all of the measured bells are similar, but of course not the same.

It would be neat to see a 72H compared to this list.

[mrdeacon]

Here is the raw data I got from the old TTF thread (measurements in inches, to the OD):
Untitled spreadsheet - Sheet1.pdf
here is what they look like with the curvature interpolated & overlaid:
Untitled.png

Huh. Well that’s sure interesting! I’ll admit I didn’t expect the Bach 50B and Rath to flair out differently than the Holton and 62H bells. I figured they’d all almost be identical.

I wonder how much of the difference in design is intentional and how much is unintentional.

[boneagain]

I commented in a "Bell throat measurement thread" in 2020.
viewtopic.php?p=121984#p121984

Since that time I've gone to using log functions for the bessel curve modeling Pete Edwards mentioned above.

I've stored each bell profile as a separate CSV in a single folder on my system.

I have a python script I can run in that folder to compare any bells I want.

Why python? Well, does nicely with that log/log stuff and related regressions.
I figure if I can create a bessel model where the R-Squared value is over .99 the results are good enough.

Why good enough? Well, there's the rub.
ALL the measurement methods leave a lot to be desired.

And that's another reason for using python. By default it gives an interface where I can zoom in on any part of the plot, and pan around as desired. For these comparisons, that's HUGE. You never know WHERE on the plot things will look really poorly.

So, for measurement difficulties:
1) bell damage. Longitudinal and radial deformation WILL make usable measurements harder to get.
2) manufacturing artifacts. Spring-back is a bear in metalwork!!
3) parallax. Tough to get an EXACTLY horizontal diameter at an exact point.

Most of what I seen in inconsistencies come from the above.
As you look at the examples below, keep in mind that stupid programming mistakes on my part can EASILY account for strange results. That being said, the gist of what I'll be showing should hold true.

Here's an example of #1. I happen to have a copy of the original drawing of the Duo Gravis mandrel, complete with lengths and diameters specified to 0.001 inches. I also have camera-based measurements of my bell. Some I digitized with GIMP scripts; some with "Engauge Digitizer." The numbers agree extremely well. In fact, they agree better than the comparison of mandrel to bell.

In theory, the bell measurements should be about .022" + .022" away from the mandrel all along the length.
The plot below, based on raw measurements, is pretty far from that.

DG_Mandrel-v-Bell_measurements.png

The plot doesn't even tell the whole story. The specification on the mandrel left the points between 1/4" from the end of the bell and the large end as a smoothing exercise for the machinist. At the zero mark the taper actually stops. There is a flat area to support rolling the rim around the bell wire.

Things are crisper at the tail of the bell. In the narrowest parts the inside and outside ARE about 0.044" different from each other.

My point is that it is very possible that the large part of the flare could spring and differ from the mandrel shape. In this case the difference is somewhere between 1/8" to 1/4".

In later plots you will see that many bell shapes differ by that much. Interestingly, I have a couple examples of the same model bell measured by different people that line up almost perfectly.

Part of the point of the plot above is to come up with parameters for the Bessel model noted above. This plot visualzes the fit of the model and parameters.

MandrelModelFit.png

The upper plot shows two things: the original data and, to the right a bit, the Bessel model. The model is shifted right because the function goes to infinity as distance from big end of bell approaches zero. Part of the regression process is to find out how far the function shifts "left" to put the arbitrary cut-off of the function at the physical bell 0 point.

The lower plot shows how well the measured data follows a theoretically perfect line. The blue line is most telling. The straighter it is, the closer to a good Bessel the data is. That goodness is reflected in the "r-sq" number in the title of the plot. I suspect it is .9969 instead of something likek .9999 because of the flat spot at the large end of the mandrel. The deviation in the blue line supports that guess.

The window title also has the Bessel parameters that gave the best line up of measured points with the function. I know from speaking to George McCracken that he was aiming for something around m=.71, and this is mighty close.

Interestingly, the bell outside measurement has m=.7088, with r-sq of .9999, so a better fit to its Bessel parameters.

Having these parameters makes it possible to synthesize full curves and compare them.

The last picture I'll put in this post is such a comparison. It points out possibilities for other measurement errors (from the list above).

50B_Comparisons.png

You'll note this goes quite a bit to the right. It actually goes to the point where the Bessel model predicts that the taper will decrease to the bore size plus arbitrary wall thickness of .022".

The Sandhagen and Stearn 50B measurements lie directly on top of each other. They extend around 40" from the big end to bore size. That happens to be just about the point where the slide joins the bell.

The PaysonMc measurements have a MUCH narrower throat, and go about 60" to reach bore size... so a lot of straight slide tube gets in there.

It does NOT take much in terms of parallax to get measurements snarled. Taking very consistent measurement points and scupulously horizontal caliper readings averaged at several spots around the bell (dent removal ovalizes more or less, depending on the skill of the de-denter) make HUGE differences.

I'll try to post a couple more comparisons from the data Pete Edwards put up (which I finally found in my archives) and that from Meine and paysonmcc.

I think what we will see, though, is that the more careful the measurements, the more we see VERY little variation in bell shapes.

It would be fun to discuss the reason for that

Dave Adams

[Pieter]

Are you comparing actual bell measurements against a model of just a single Bessel function?

There is quite some research out there of the flaws of a pure Bessel bell flare. You would need small modifications in order for the horn to play in tune across the partials. Some have tried a model with sine/cosine functions as modifications. Others have expressed bells as several Bessel functions linked together as segments. They usually checked for the partials to be in tune with some kind of acoustic modelling. That is for a trombone perhaps not the most important thing, since we have this slide to correct things.

It is how I made the Printbone. A 3D printed trombone, can be found on thingiverse: the bell is modelled as several segments, each represented by a Bessel function plus the begin and end of each segment. You could even use that to print a couple of bells and compare the results...

[hornbuilder]

There is no flat spot on a bell mandrel to spin the wire into the rim. Case in point, Bach 36 and 42 are spun on the exact same mandrel. Bach 50B 9.5" and 50BL 10.5" are both spun on the exact same mandrel. I have bells spun at 9", 9.5", 10" and 10.5" , all on the same mandrel. You couldn't do that if there were flat spots for the bell wire to be spun on.

[boneagain]

There is no flat spot on a bell mandrel to spin the wire into the rim. Case in point, Bach 36 and 42 are spun on the exact same mandrel. Bach 50B 9.5" and 50BL 10.5" are both spun on the exact same mandrel. I have bells spun at 9", 9.5", 10" and 10.5" , all on the same mandrel. You couldn't do that if there were flat spots for the bell wire to be spun on.

That's helpful Matt! Thanks!
Here's what that corner of the mandrel looks like:

Bell_BassTrb_Duo-Gravis_mandrel_corner.jpg

The rightmost vertical line is marked with diameter of 9.625".
The vertical line left of it is marked with diameter of 8.856".
I measured my old SilverSonic DG at very close to 9.5".
So, I'm not sure what went on there.

I can see on the Eroica bell drawing that the taper goes past the end of the final bell size.

I can see (faintly) in THIS DG drawing that George DID have his curve continuing in his working lines.

Got any ideas on what's going on there with the Duo Gravis?

Dave

[hornbuilder]

No clue, sorry.

[boneagain]

Are you comparing actual bell measurements against a model of just a single Bessel function?

I use a form of Bessel recommended by Benade in "Fundamentals of Musical Acoustics":
D=B/(y + y0)^m
D is the diameter of the bell at a given distance from the large end.
B settles the small end of the taper.
y0 settles the large end of the taper.
m settles the rate of the curve.
I say "settles" rather than defines on purpose.
In practice this seems to work a lot like a parabolic equation:
We get one actual curve. Additional parameters change the "zoom-in" on the curve and position in the x-y axes.
Benade notes "their shapes correspond closely enough to be musically serviceable..."
He makes no claim that these well-known forms are EXACT.

In the particular case of the Duo Gravis trombone and, even more so, the Eroica horn, the r-square values are, respectively, .9969 and 1.0, which I would certainly call "close enough." This closeness should not be surprising, though. McCracken calculated his mandrel specifications out with the model Benade cited in mind.

Erik Jansson of the Royal Institute of Technology, Stockholm, worked with Benade on this stuff in the 1960's and 70's. Jansson put together an Algol program to print horn dimensions, given parameters, and predict nodal points in the flare. I have a run they did for George on a couple instruments, including the Duo Gravis. The notes were a life-saver (but some of the notes were lost between pages, not a complete life-saver.) Between Jansson and Benade I understood a few key factors:
1) exploring the data to find "m" by using log/log relationships
2) finding B and y0 in sequence after finding "m".
3) visualizing the differences between:
3.1 the end point of the function, which approaches infinity as "bell end" of the FUNCTION approaches zero
3.2 the actual end of the bell, which is the only real physical hook can hang onto
3.2 the ACOUSTICAL end of the bell, which can be modeled as a point on a spherical wave front emmanating from the bell.

Of course, all that being said, I would expect that a Conn, for example, would not get as high an r-square as a King. Earl Kent had some very productive ideas about joining sequences of catenary curves to achieve desired intonation. And the rest of your post (copied below) bring up other likely-to-be more accurate modeling for general purposes.

But it IS interesting to see the r-square values for these derived curves. All of them, given good measurements, are well over .99, which startles me.

When I say good measurements I do not mean those that give r-square better than .99; I mean measurements without obvious horrible mistakes in them. For instance, I had one series where the measurer had transposed some numbers. The result was about a half inch bulge in the bell. Not going to see that in the real world!

But now you have me CURIOUS. I'll look up your Printbone and see if I can pull dimensions from the STL file. Would like to see what the bessel segments do the the r-square on the single one, and what the "average" resulting "m" would be, and how it compares to these others.

This will be a pretty unique regression. No possibility of the errors I noted above. No springback in 3D printing... but maybe that is replaced by settling during cooling? No parallax or leveling or photographic barrel distortion errors... hmmm.... dang it, this curiosity gets control WAY too easily...

Dave

There is quite some research out there of the flaws of a pure Bessel bell flare. You would need small modifications in order for the horn to play in tune across the partials. Some have tried a model with sine/cosine functions as modifications. Others have expressed bells as several Bessel functions linked together as segments. They usually checked for the partials to be in tune with some kind of acoustic modelling. That is for a trombone perhaps not the most important thing, since we have this slide to correct things.

It is how I made the Printbone. A 3D printed trombone, can be found on thingiverse: the bell is modelled as several segments, each represented by a Bessel function plus the begin and end of each segment. You could even use that to print a couple of bells and compare the results...

[Pieter]

The printbone bell is based on measurements someone made of a conn 88H.
The printbone small is not - roughly based on photos of smaller horns.

You can absolutely make it print dimensions, using the openscad source files, I have done so in the past.

[HawaiiTromboneGuy]

My non-scientific contribution . The Wick mute inserts just a hair more in the Williams compared to my early NY 50. Both with 9” bells.

Also, as a side, the NY 50 tuning slide gets swallowed in the Williams, while the Williams is too large for the NY 50.

[Tbarh]

My non-scientific contribution . The Wick mute inserts just a hair more in the Williams compared to my early NY 50. Both with 9” bells.

Also, as a side, the NY 50 tuning slide gets swallowed in the Williams, while the Williams is too large for the NY 50.

As for the tuning slide,the length of the bell section and where the slides are placed also come into play !

[boneagain]

Also, as a side, the NY 50 tuning slide gets swallowed in the Williams, while the Williams is too large for the NY 50.

As noted by Pieter above, using different taper methods for different segments is a likelihood.
The main tuning slide area is a case in point.
I've seen a number of references, but my favorite remains Benade's "Fundamentals of Musical Acoustics." He tended to explain things so even my limited brain can absorb them.
The specific reference here is in Chapter 20, section 20.5 "Musically Useful Shapes..." in a "Digression on the Effect of Bends and Loops in the Tubing of Brass Instruments" on page 409 of the Dover reprint.

"Even a pipe of uniform cross section acts somewhat like a flaring duct if it is given a curved shape."

So, for example, if we took the equation from the same chapter and tried to make a one-piece bell smoothly back to the slide receiver, the J portion of the horn would not work right.

[HawaiiTromboneGuy]

My non-scientific contribution . The Wick mute inserts just a hair more in the Williams compared to my early NY 50. Both with 9” bells.

Also, as a side, the NY 50 tuning slide gets swallowed in the Williams, while the Williams is too large for the NY 50.

As for the tuning slide,the length of the bell section and where the slides are placed also come into play !

Very true! Hadn’t thought about that!

[Pieter]

now you have me CURIOUS. I'll look up your Printbone and see if I can pull dimensions from the STL file. Would like to see what the bessel segments do the the r-square on the single one, and what the "average" resulting "m" would be, and how it compares to these others.

This will be a pretty unique regression. No possibility of the errors I noted above. No springback in 3D printing... but maybe that is replaced by settling during cooling? No parallax or leveling or photographic barrel distortion errors... hmmm.... dang it, this curiosity gets control WAY too easily...

Oh i forgot to mention, the printbone is not a single trombone design: it is a parametric trombone 3d model generator. You input parameters such as the lengths and bore diameters of the various parts, including neck pipe, tuning slide and connection to the slide or you choice. And the bell, as several Bessel or conical segments. It will then generate the STLs. No valves. Bell sections are easier to print than slides, although you need a big printer for bass trombone bells.

Note that there is a feel and sound difference between plastic and brass belle. But for comparing different bell flares... You could also learn to make brass bells of course

Generally if printed in PLA the prints are relatively accurate. The largest problems is removing the support material for the bell, and warping potentially is an issue.

For mouthpieces however, FDM printing is not ideal. You need to sand them to get rid of the layer lines, and it is very hard to do so consistently. Trombone bells luckily are much bigger than that.

[boneagain]

now you have me CURIOUS. I'll look up your Printbone and see if I can pull dimensions from the STL file. Would like to see what the bessel segments do the the r-square on the single one, and what the "average" resulting "m" would be, and how it compares to these others.

This will be a pretty unique regression. No possibility of the errors I noted above. No springback in 3D printing... but maybe that is replaced by settling during cooling? No parallax or leveling or photographic barrel distortion errors... hmmm.... dang it, this curiosity gets control WAY too easily...

Oh i forgot to mention, the printbone is not a single trombone design: it is a parametric trombone 3d model generator. You input parameters such as the lengths and bore diameters of the various parts, including neck pipe, tuning slide and connection to the slide or you choice. And the bell, as several Bessel or conical segments. It will then generate the STLs. No valves. Bell sections are easier to print than slides, although you need a big printer for bass trombone bells.

Note that there is a feel and sound difference between plastic and brass belle. But for comparing different bell flares... You could also learn to make brass bells of course

Generally if printed in PLA the prints are relatively accurate. The largest problems is removing the support material for the bell, and warping potentially is an issue.

For mouthpieces however, FDM printing is not ideal. You need to sand them to get rid of the layer lines, and it is very hard to do so consistently. Trombone bells luckily are much bigger than that.

I got far enough into the openscad to get an inkling of the parameterization and chunk approach to the build. Kill a few birds with one stone: better printability; "fire-walls" to limit costs of reprinting to test a different curve in, say, the tail; convenient break points to minimize how many parameters you need in a single module, for example. Nice!

BTW: I looked up the references in your github (nice job on including references too!) That got me refering back to my hardcopy of Alistair Braden's PhD thesis. Great stuff!

Now I just need to find time to characterize from flair to tail and compare that to the actual curves of the flare, throat, and tail. Time, yep...