Critical Dimensions for the Tagelharpa Instrument

The Tagelharpa’s design is based on a set of proportional relationships that scale the instrument’s dimensions according to the chosen scale length and number of strings. Below is an explanation of each critical dimension using traditional mathematical formulas.

A. String Scale (Peg-to-Bridge Distance)

The distance from the nut (or peg) to the bridge is defined as the string scale.
A = scaleMm, where scaleMm = scaleCm × 10.

B. Overall Instrument Length

The overall length is calculated piecewise from the scale length (in cm):
For scaleCm ≤ 40:
B = scaleCm × 1.66 × 10
For scaleCm ≥ 56:
B = scaleCm × 1.43 × 10
For 40 < scaleCm < 56, let f = (scaleCm – 40) / 16, then:
B = scaleCm × [1.66 – 0.23 × f] × 10

C. Window Length

The window (soundboard opening) is designed for a full octave:
C = 0.5 × scaleMm + 20

D. Window Width

The window width is based on the number of strings and a fixed peg spacing:
D = pegSpacing × numStrings, where pegSpacing = 36 mm.

E. Peg Spacing

The spacing between pegs is fixed:
E = 36 mm

F. Headstock Width

The headstock width is determined by the window width plus an additional 50 mm:
F = D + 50

G. Bridge Width

The bridge width is defined piecewise:
If scaleCm ≤ 30: G = 60 mm;
If scaleCm ≥ 56: G = 70 mm;
For 30 < scaleCm < 56, let f = (scaleCm – 30) / 26, then:
G = 60 + 10 × f

H. Bridge Spacing

After subtracting 12 mm from each side of the bridge width, the usable span is:
Usable Span = G – 24
If more than one string is used, the gap between string centers is:
H = Usable Span / (numStrings – 1)

I. Body Minimum Width

The body’s minimum width is determined from empirical data. For example:
- For scaleCm ≤ 32: I = 160 mm
- For 32 < scaleCm ≤ 37: I is interpolated from 160 to 190 mm.
- For 37 < scaleCm < 56: I is interpolated from 190 to 210 mm.
- For scaleCm ≥ 56: I = 210 mm

J. Body Minimum Depth

The body’s depth is given by:
J = clamp(scaleMm × 0.12, 45, 70), meaning the calculated value is limited between 45 mm and 70 mm.

K. Tail Top Width

The tailpiece’s top width is defined as:
K = H × numStrings
(where H is the gap between strings.)

L. Tail Bottom Width

The tailpiece’s bottom width is a fraction of the top width:
L = K × 0.7

M. Tail Length

The tailpiece length is given by:
M = 0.4 × (B – scaleMm), where B is the overall instrument length.

N. Headstock Thickness

The headstock thickness is fixed:
N = 35 mm

O. Soundhole Centre

The centre of the soundhole is calculated by:
O = ((scaleMm – C – (cutOutTop/2)) / 1.85) + C + cutOutTop
where C = windowLength and cutOutTop is the vertical offset for the window.

P. Neck Start

The neck start position is determined by the distance between the bottom of the window and the top of the soundhole. With:
bottomOfWindow = windowLength + cutOutTop
the neck start is:
P = ((O – bottomOfWindow) / 3) + bottomOfWindow
This sets the vertical point where the neck begins.

Q. Neck Thickness

Neck thickness is set based on scale; for instance:
Q = { 35 mm if scaleMm > 450; otherwise 25 mm }
This parameter accounts for the need for a thicker neck in larger instruments.

R. Width at Neck Level

The width of the instrument at the neck level is found by linear interpolation between the top width and the bottom width of the instrument. Let:
r = P / B, where P is the Neck Start and B is the Overall Length.
Then the width at the neck is:
Width_neck = F + (I – F) × r
Here, F is the headstock width (top width) and I is the body minimum width.