In Search of the Optimum Bore

[Tunborough]

Interesting. I hadn't noticed the correspondence.

From Moloney and Hatten, with a bit of arithmetic, friction losses at the wall give Qwall=0.303*a*sqrt(f) and radiation losses give Qrad=L*c/(pi*a^2*f) where a is pipe radius, L is length, c is speed of sound, f id frequency, and a^2 is a-squared; overall Q is given by 1/Q = 1/Qwall + 1/Qrad. I then assumed L=c/(2*f). This is questionable in two respects: in practice, L is a couple of cm short of a half wavelength, and in the second octave there is a full wavelength in the tube. I tried L=c/f (full wavelength) in the second octave, but the results didn't agree with anything. I got the numbers in the table by solving Q(a,f) = Q(a,4*f) for a at the different frequencies.

It shouldn't surprise us that the results are consistent with doubling-at-14.4, since Fletcher and Rossing derived that result using equations for Q that had the same form as those of Moloney and Hatten. What this new table gives us is an absolute result: we don't have to assume that we know the optimum for any whistle in advance.

Now, my question is: are these numbers (and hence the doubling-at-14.4 rule) consistent with the experience of those who build very low or very high whistles?

[Daniel_Bingamon]

I've not tried this 14.4 rule but I have make lower than usual whistle. My Basswhistle, 146.832Hz Octave below Low-D is 1.63" diameter with 0.14" Wall thickness. It has a 1" wide windway.
The whole first register plays quite well with these dimensions.

[Tunborough]

I've not tried this 14.4 rule but I make a lower than usual whistle. My Basswhistle, 146.832Hz Octave below Low-D is 1.63" diameter with 0.14" Wall thickness. It has a 1" wide windway.
The whole first register plays quite well with these dimensions.

Thanks, Daniel. It was with you in mind that I extended the chart to low-low-C. Is that 1.63 I.D. or O.D? That would be either 41.4 mm I.D., which agrees pretty well with my chart, or 34.3 mm I.D., not so much.

How's the second register? If the model I'm using is right (which still isn't decided), 41.4 mm I.D. (1.63") would give a second register about as good as the first, and 34.3 mm I.D. (1.35") would give balanced performance over the first register, but leave the second register weak.

On another topic, how long is the window, from fipple to lip?

[Tunborough]

I went astray in the calculations for the last chart I posted. With the errors corrected, Hans and I have come up with the following for the optimum diameter for a given bottom note. These are based on the theoretical formulas in Moloney and Hatten. Their experimental results don't give consistent results, and I'm not convinced they apply to whistles anyway.

D = 2620 * f^(-5/6)
f = 12640 * D^(-1.2)

D is the inside diameter of the tube that will have the same (theoretical) Q at the top and bottom of a 2-octave range starting at f, and a higher Q in between. We're presuming that this will give the best balance of playability between the top and bottom ends of the range, without going to extremes in voicing. In practice, the constants in these formulas may not be dead on, but we're pretty confident on the scaling factor, which corresponds to doubling every 14.4 semitones.

Code: Select all

Key     f0     Opt Dia   Q
C3     130.8    45.1    73.0
D3     146.8    41.0    70.2
E3     164.8    37.2    67.5
F3     174.6    35.5    66.3
G3     196.0    32.2    63.8
A3     220.0    29.3    61.3
Bb3    233.1    27.9    60.2
C4     261.6    25.3    57.9
D4     293.7    23.0    55.7
E4     329.6    20.9    53.6
F4     349.2    19.9    52.6
G4     392.0    18.1    50.6
A4     440.0    16.4    48.7
Bb4    466.2    15.7    47.8
C5     523.3    14.2    46.0
D5     587.3    12.9    44.2
E5     659.3    11.7    42.6
F5     698.5    11.2    41.7
G5     784.0    10.1    40.2

[hans]

...and here is a graph, based on those functions.
The red curve plots the points of theoretical maximal Q for a whistle of a given bore diameter.
Right and left are zones for whistles with wider and narrower bore,
+-5% inner green zone, and +-15% outer green zone
(but there are no fixed boundaries, I just wanted to give some indication of deviation from the middle).


created using the great and open source Graph software from here: http://www.padowan.dk/graph/

[s1m0n]

Every time I see this thread title I think that it should be about an insomniac's search for bedtime reading.

[Tunborough]

[Thread revival]

Turns out, there is no optimum bore.

A whistle is balanced when performance at the top end is the same as at the bottom end, and it looks like the Q factor is a great measure of that balance. A wide bore favours the Q factor at the bottom end, and a narrow bore favours the Q factor at the top end. But the length of the window, between the windway exit and the blade, also matters. A longer window favours the Q factor at the bottom end, and a shorter window favours the Q factor at the top end. There are probably limits at the extremes, but you can have balanced whistles with wide bores and long windows, or narrow bores and short windows. Further investigation is in order of what the trade-offs are.

[Terry McGee]

Turns out, there is no optimum bore.

Is that too bold a statement just yet, Tunborough, or do you think it stands up? I'm thinking you have confirmed that there is no optimum cylindrical bore, for the reasons you give. But I'm wondering if we can yet rule out an optimum tapered bore, or whether a well chosen tapered bore would trump a well chosen cylindrical bore. Thoughts?

[stringbed]

This is a recently revived thread and it may be worth restating one of the remarks at the start of its first iteration as a question:

Optimum for what and for whom?

[hans]

[Thread revival]

Turns out, there is no optimum bore.

A whistle is balanced when performance at the top end is the same as at the bottom end, and it looks like the Q factor is a great measure of that balance. A wide bore favours the Q factor at the bottom end, and a narrow bore favours the Q factor at the top end. But the length of the window, between the windway exit and the blade, also matters. A longer window favours the Q factor at the bottom end, and a shorter window favours the Q factor at the top end. There are probably limits at the extremes, but you can have balanced whistles with wide bores and long windows, or narrow bores and short windows. Further investigation is in order of what the trade-offs are.

Nice to see this old thread revived! Happy New Year to you all! I quite agree to your observations, and thank you for adding the matter of relative window length. I think the geometry of the window is a fairly subjective matter for a whistle maker, and that explains the great variety of geometries amongst makers. It is subjective, because it is a matter of choice of tone and voicing.

I see that the image I posted a few posts up is no longer working, so I post this again with corrected links. This corresponds to a scaling factor of doubling the diameter every 14.4 semitones:

https://music.bracker.uk/Music/Searchin ... -Bore.html

Optimum for what and for whom?

My search was for the optimum bore of a whistle, for anyone. The "optimum" being searched as some objective measure.

Personally, very subjectively, I favour most of the time to play whistles with a a narrower bore than what I came to think of "optimum bore" (represented by the red line in the graph), a narrower bore lying in the lighter shaded zone to the left of the red line. And personally, very subjectively, I favour whistles with a window geometry of double the width to the length. I am not advocating this as the answer. I like playing my narrower whistles, because they play a little sweeter and a bit quieter, just great for my living room and solo playing. For ensemble playing, and outdoors, I find a whistle with a more "optimum" bore more suitable.

[pancelticpiper]

This topic has always interested me on a pragmatic level (I'm no scientist).

I know the length-to-bore ratio naturally gets narrower as whistles get lower, long known to pipe organ builders.

So staying with whistles in the Mezzo sizes down to Low D and Low C, I've found that whistles with a ratio of around 24.5 to 25 are my favourite players.

These include:
Home-made Mezzo A 24.46
Freeman Mezzo G 25
Goldie Mezzo F 24.99
Goldie Low D 24.6

However a whistle that took time to get used to, that's now a favourite, is rather narrower:
Alba Low E 25.86

And my great-playing sub-Low D whistles are narrower, as would be expected:
Goldie-Overton Low C 28.33
Alba Bass A 29.15

Whistles I don't care for, in some cases nearly unplayable, have been wide:
Susato Mezzo G 23.35
Burke Mezzo F 22.28
Burke Low D 23.3
Alba Low C 24.23 (made with the same huge tubing as the Alba Bass A)

[pancelticpiper]

As far as the definition of "optimum" being wide open to personal taste, yes, but within fairly tight limits.

I don't think any player wants

-high notes that require great force and volume of air, that are much louder than the rest of the notes, that have a harsher tone than the rest of the notes, that are difficult to sound without squawking.

-low notes that require backing off significantly on the blowing, that are much quieter than the rest of the notes, that are feeble and nearly impossible to sound without breaking into the 2nd octave.

As I've mentioned before, a chat with Chris Bleth was illuminating. https://www.imdb.com/name/nm1259350/

While I and many other players coming from the trad side want easy nimble high notes, he told me that he preferred having a clear break-point between the octaves.

On the other hand another non-trad professional musician (originally a sax player but now makes his living playing whistle) seems to like the same whistle performance I do, with sweet high notes yet full low notes.

[Tunborough]

I'm thinking you have confirmed that there is no optimum cylindrical bore, for the reasons you give. But I'm wondering if we can yet rule out an optimum tapered bore, or whether a well chosen tapered bore would trump a well chosen cylindrical bore. Thoughts?

For purposes of this thread, I'm taking "optimum" to be the best balance in ease of playing the top notes and bottom notes. Varying the bore profile certainly makes it easier to balance the tuning between the octaves, but a tapered bore would still be subject to constraints on absolute bore diameter and window length in balancing the ease of playing. I'd say the latter considerations have to wait until we know more about optimum bore profiles for tuning.

Optimum for what and for whom?

A fair caution... I'd say the goal is not to dictate the perfect bore, but to draw a line in the sand for equal balance between bottom notes and top notes. If you never play above high B, or are looking for a session cannon with a robust bottom end, you'll look for a fatter whistle. If you want a sweet, nimble top end, or want to play into the third octave, you'll look for a thinner whistle. Also, organ builders have long known that wide tubes give a "flutey" sound and narrow tubes give a "reedy" sound. If you want a flutey sound, you'll look for a whistle with a wide bore, and maybe a short window length. If you want a reedy sound, you'll look for a narrow bore, and maybe a long window length. (I don't yet know what "short" and "long" mean in this context.) I share Hans's preference for the "thinner" side of the line; knowing where that line is helps me make choices that suit my preferences.

Nice to see this old thread revived! Happy New Year to you all!

Nice to see you drop by again, Hans. Happy New Year to you, too.