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When
building a stave type ashiko, usually 12 or more staves are cut separately using
a table saw, then glued together to form a cone as illustrated at left.
Calculating all the proper angles can be difficult, and requires some
fairly extensive geometric equations. There
is a wonderful program called DrumCalc that calculates most of the angles for
you. There is one angle that
DrumCalc doesn’t give you, and this page tries to show what that angle is, why
it is important, and how to calculate it.
The
basic angle used to cut the staves is noted at left as Theta.
This angle is simply 360/N where N is the number of staves.
When the staves are cut on the table saw, the blade must be
set at an angle. Since the staves
are laid flat when cut, the angle to use is not Theta, but Theta’.
Theta’ is formed by drawing a line from the middle of the top of the
drum to the stave such that it is perpendicular to the stave, as above.
And Theta is not quite the same as Theta’.
The more angled the stave is, the more the difference between the 2
angles. The difference is small for
most ashikos, but when the same calculation is used for the bell of a djembe,
the correct angle is critical. If
you don’t make the correction, your staves will not fit together tightly.
Here
the 2 angles are shown together, on a stave that I have made short and with a
steep angle to better accentuate the differences.
Note that this stave is pretty close to what you might use for a djembe
bell, though.
Here I have labeled all the important vertices so that I
can refer to them for the calculations. I
am also going to use the following notation:
L1 = BC = stave top outside dimension
L2 = DE = stave bottom outside dimension
L3 = FG = stave width at the perpendicular
AB = R1 = top drum radius (see notes on this
below) HD = R2 = bottom drum radius
AF = R3 = radius of perpendicular
BD = V = outside length of stave
AH = DK = h = drum height
N = number of staves
For the purposes of the calculations, all angles will be in
radians.
Using simple geometry:
θ = 2π / N ;
L1 = 2 * R1 * sin (θ / 2) ;
L2 = 2 * R2 * sin (θ / 2).
Using similar logic, we can say
L3 = 2 * R3 * sin (θ’ / 2)
The triangles AFB and EKB are similar.
Therefore:
AF / AB = DK / BD , or R3 /
R1 = h / V ; therefore R3 = R1 * h /
V
where V can be calculated as:
V = Sqrt ( h2 + BK2 ) ; where BK = R1 – R2
if we define x = BF = the distance from the top to the perpendicular, we
can similarly note:
x / R1 = (R1 – R2) / V
Since the sides of the stave are all straight:
L3 = L1 – (L1 – L2) * (x / V)
We now have calculated L3 and R3 in terms that are known, and using the
formula above, we can calculate θ’ as:
θ’ = 2 * sin -1 ( ˝ * L3 / R3)
The closed form is lengthy and I don’t think it is worth trying to type
it all out here. Using the formulas
above, it is easy to set up a spreadsheet to calculate all the parameters.
The link below is to an Excel file that does just that, for those who
have access to Microsoft Excel. I am
not enough of a programmer to be able to put together a Java program or
something as neat as DrumCalc that does it all by itself.
<----- Click on this to download the Excel file.
Below is a sample calculation using some parameters from a drum I am
currently making. Note that there is about a 4.5% difference between
θ and θ'.
