For all your makers and tweakers out there.
Anyone know how to calculate tone hole sizes and locations for CONICAL bores?
I thought I’d experiment with a poor man’s version of a conical bore, modifying a pvc or cpvc tube with liquid plastic to create a pseudo conical bore.
Hi Dan,
I use 1.618 as a division tool…I first start by cutting the tube til the bell note is reached. then measure from the labium to the end of your tube. divide by 1.618 by 1.618 by 1.618…this will layout the nodes of your tube, layout your hole based on your nodes…you can see how the hole gets bigger closer to the node…Initially I choose my locations and reem the holes out until my hole size reaches the desired note…once I have the hole sizes I then choose a drill bit closest to the sizes uncovered by reeming. cover the holes you drilled with tape…turn your tube 180 degrees and drill new holes from the bottom up…and adjust new holes up or down to reach desired note…be prepared to scrap a tube or two…
I hope this helps…this is The way I do it…and it works, I don’t trust magic formula’s myself…It all comes down to customizing chaos for me 
cheers,
Greg
I hope this helps…this is The way I do it…and it works, I don’t trust magic formula’s myself…It all comes down to customizing chaos for me
It is all magic in the end though! That’s a pretty reasonable approach too, Greg.
Feadoggie
Ye gods, how I want to make a whistle!
(Sorry, somewhat off topic, I know. But reading Greg’s post, it just sounds like a lot of fun to me! 
)
I know of no online sources for calculating holes in a conical instrument. Nederveen’s book (‘acoustical aspects of woodwind instrument’) has the equations for both cylindrical and conical bores and much more, which include the first order approximations used by the spreadsheets that can be found online by, or based on Pete Kosel’s work which are I believe also are used by the flutomat calculator.
So as not to mislead you the book does not give you a packaged solution - it is not a how to book!
A couple refinements to my cylindrical whistle design and I’m going to move on to trying a conical design, and diving into the math followed by almost certainly many prototypes.
Bill
Greg - I wish I could understand your procedure, and the rational behind it. All I know is that dividing by 1.618 is the golden ratio, but i have no idea what that has to do with nodes and hole sizes 
~Hans
[u]Peter Hoekje’s Exel spreadsheet[/u] gives the possibility to set the tube diameter for each tone hole location, so you can use it for a tapered bore.
[u]Pete Kosel’s Flutomat javascript form page[/u] does not give that option and can be used only for cylindrical bores.
Both I find to be not very precise, but good for a first approximation.
Note that for a whistle design you need quite a different length correction for the whistle window than for the flute embouchure hole, i.e. the formula used for calculating the correction for the emb. hole needs adjustment.
~Hans
Greg - I wish I could understand your procedure, and the rational behind it. All I know is that dividing by 1.618 is the golden ratio, but i have no idea what that has to do with nodes and hole sizes
~Hans[/quote]
Pluck a string on a guitar, and search for the harmonics by lightly touching the string without making it touch the frets and you will find pure Fibonacci and 1.618 relationships. A guitar string is not much different then the column of air in your whistle…Hence the fret board is also layed out this way…no wonder they call the number “golden”
Hans, here is a picture of a generation d I missed the “G” ohwell 
…but thats ok..it’s just a rough diagram showing the divisions…I also use this ratio for my wind way calculations
thank you for the pic, greg. even without the G, it is pretty easy to understand. very cool. so… pardon my ignorance, but where did the 1.618 come from? when i make my flutes i use a much different way of calculating based on the 12 half tones from root to octave that make up the chromatic scale.
be well,
jim
Maybe the question is, “where did we come from” and “what are we doing here?” 
I was trained as a professional trades person, In my everyday work I always found this number. I also build violins and found every key measurement was based on 1.618 …strange? not at all!!! turns out the construction of our bodies are inherent to this ratio…and possibly what we find pleasing has to do with who we are and how we’re made!..The shape of 1.618 is a pentagram, the inner ear is also shape to this natural phenom…don’t make me draw another picture!! 
!
Check this out…http://goldennumber.net/music.htm
http://goldennumber.net/dna.htm
http://goldennumber.net/solarsys.htm
well, isn’t that just cool as hell! absolutely fascinating! thank you very much for posting the links.
the geek/nerd in me is jumping up and down like a little child in a candy store.
be well,
jim
I am just having a short break from making one of these:
http://www.ehhs.cmich.edu/~dhavlena/low-d.htm
using one inch diameter thin wall aluminium allow tubing and a bit of broom handle.
Within an hour I had a tube sounding a low D!
Making the holes will be for tomorrow, but I think using a tapered reamer is the way to go for fine adjustment of sizes.
The guy who wrote that article says there is no difference in spacing between conical and cylindrical bore.
I don’t think this is the case.
On a guitar the second harmonic is the fifth above the octave (first harmonic) and it occurs when dividing the string by three. Lightly touching the string one third way, so two thirds of the string remain, will sound the first harmonic (octave) of the fifth, i.e the same note which is fingerd a fifth above octave with a third of the string length. Two thirds is 0.666…, not 0.618 (golden proportion; 1/1.618=0.618).
Pressure nodes and anti-nodes in a flute or whistle are located similar, with the slight difference that the sounding air column is a little longer than the tube.
See flute acoustics. The pressure node in a flute for the first harmonic (octave) is located half way down the tube length, the pressure anti-nodes are located ca. one quarter and three quarters along the tube. The Golden Proportion has no bearing.
I have studied Fibonacci numbers and the Golden Proportion in nature, and find these to be fascinating aspects of naturally evolved efficient design, but when it comes to determine acoustically efficient location for flute tone holes I rather follow the maths of logarithmic divisions. To locate tone holes in places we organically evolved beings can reach comfortably with our fingers I then need to alter hole diameters (shift hole up: make it smaller; shift hole down: make it bigger) in a precise mathematical fashion which takes account of all the holes, or simply empirically by trial and error.
The resulting spread of holes has again nothing to do with the Golden Proportion. I aim to have somewhat equal distances between holes 1, 2 and 3, as between holes 4, 5 and 6. This is of course a big “somewhat”, and usually a compromise between equal spacing and acoustically efficient spacing is best.
Great fun, all of this!
~Hans
thanks hans…if it works it works, your much smarter then me… 
carry on,
Greg
if i may be allowed… having played guitar for about 35 years, i can assure you that Hans’ info is correct. additionally, if you lightly touch the vibrating string of a guitar at the 12th fret (equal to one half the length of the string), the tone produced will be one octave higher than that produced by plucking an unfingered string.
it was this info that helped me map out the fingerhole placements on the flutes i make. it seems to work, after making adjustments for human “limitations,” all of my final flutes have excellent intonation. like any cylindrical flute/whistle, the upper registers need to be “blown” into tune, but they are very close before being “blown” in. using a wedge similar to doug tipple’s tipple-fijardo wedge helps a lot with upper register tuning/intonation.
and i agree, hans… man, this stuff is fun!
be well,
jim
Hans, I am just a simple Carpenter from out around the bay and can’t argue with anyone with more then grade five…I divide by 1.618, drill my holes…and play a tune.. 
as in the diagram above drawn by me…(no camera tricks) shows accurate placement. and by the way seem to work just fine…however it sounds you have put alot of study into this and frankly the thought of it all mesmorizes me again if you divide by 1.618 you will find the hole layout…I’ve done it about 500 times…I feel confident that the next whistle will work out just fine again 
great fun
well, i’m intrigued enough by the 1.618 that i’m going to give it a go on my next flute. so, i thank you for posting the pic, greg.
be well,
jim
Boy, we’re having some fun now. Messing with all these variables at once makes my head spin. But getting them all to happy values is the the point, so they all need to be messed with.
Greg, interesting application of Phi here. I’m not quite able to follow your drawing and comments. You start with a good bell note and divide that length by Phi to find what? The spot where an octave D would go? I assume that’s your “D” in the drawing that doesn’t match a hole. Then you divide that by Phi to find … (and I’m missing something here.)
Using the notation of L1 for left hand top finger etc. what would your procedure be?
Bell note length / Phi = What?
What / Phi = L1 or is it R1?
R1 / Phi = R2?
R2 / Phi = R3?
I’d like to explore this approach some, but I want to start with your method rather than err and make up something of my own. (But I suppose that’s part of the tradition. It’s how new tunes are born!)
Cheers,
Carey
The method I used was percentages which is probably the same thing
Once you have the tube of your low D copper whistle cut to
the proper length to sound a low D note, carefully measure
the distance from the mouthpiece “lip” to the right hand end
(open-end) of the instrument.The center of the 1st hole (the hole nearest the mouthpiece)
should be located 44.74% of this overall “lip” to open-end
measurement.The center of the 2nd hole should be located 52.47% of this
overall “lip” to open-end measurement.The center of the 3rd hole should be located 60.38% of this
overall “lip” to open-end measurement.The center of the 4th hole should be located 68.82% of this
overall “lip” to open-end measurement.The center of the 5th hole should be located 74.93% of this
overall “lip” to open-end measurement.The center of the 6th hole (the hole nearest the open end of
this whistle) should be located 84.10% of the overall “lip”
to open-end measurement.How to convert these percentages into actual measurements:
- First move the percentage’s decimal point two
units to the left (ie: 44.74 becomes .4474)- Then simply multiply this number by the
“lip” to open-end distance