Fret Calculator: Single Scale and Multiscale Layout

The ideal 12-TET fraction from a nominal nut to fret n is $f_n=1-2^{-n/12}$. Mottola's Calculating Fret Positions, retrieved text page 2, gives the modern exponent rule and a 25.5 in worked example. The octave fraction is 1/2; the two-octave fraction is 3/4.

The two-dimensional construction follows the geometry described in Aaron Spike's FretFind2D documentation (attached text pages 1–2). Here x runs toward the bridge and y toward the bass side. Let $d=(W_s-W_n)/2$. For a multiscale neck, the outer strings' longitudinal projections are $A_t=\sqrt{L_t^2-d^2}$ and $A_b=\sqrt{L_b^2-d^2}$. In single-scale mode both projections equal the entered centreline scale.

At neutral fraction $f_0$, place each nominal nut at $x_N=x_0-f_0A$, with $x_0=f_0\max(A_t,A_b)$; the nominal saddle is at $x_B=x_N+A$. This makes the chosen nominal fret transverse. The x origin is the earliest outer-string nominal nut contact, not the earliest board corner; some board-edge coordinates can therefore be negative.

Interpolate the string endpoints across the nut and saddle by string order. Each fret intersection is $P_{j,n}=N_j+f_n(B_j-N_j)$. Intermediate actual scales are the Euclidean lengths $|B_j-N_j|$, not an arithmetic interpolation of outer scale lengths. Equal fractions are collinear, so straight fret slots pass through every nominal string intersection. Intersect each fret line with the two straight board edges; edge margins are transverse at the nominal nut and saddle.

Before evaluating slot endpoints, certify both side edges over the continuous range from the nominal nut through the board end. Each edge must advance toward the bridge, remain on its proper side of the outer strings, and intersect the fret family in forward order. The oriented intersection denominator is affine in fret fraction, so checking its two endpoint values excludes an intervening zero. For a positive denominator, the numerator of the intersection-progress derivative is monotone; its two endpoint values therefore certify forward progress throughout the interval. Rejected geometry blocks the results and bypasses the endpoint divisions.

The board end uses $f_{end}=f_{last}+e_{end}/[(A_t+A_b)/2]$. Its centreline coordinate is the entered extension past the last fret. Its angle follows the fret-line interpolation at this extended fraction and usually differs from the last installed fret's angle.

Signed fan is $\alpha_n=\operatorname{atan2}(x_{b,n}-x_{t,n},W_n+f_n(W_s-W_n))$, measured from the transverse direction. Positive means bass farther toward the bridge.

Known compensation offsets move contacts along each nominal string: $N'_j=N_j+c_{N,j}(B_j-N_j)/L_j$ and $B'_j=B_j+c_{S,j}(B_j-N_j)/L_j$. The open length becomes $L_j-c_{N,j}+c_{S,j}$; frets do not move. These offsets do not by themselves enforce ideal intonation after fretting. Mottola's compensation discussion distinguishes nominal fret scale from saddle setback.

The plan uses millimetres with identical x and y scales. Its 100 mm horizontal and vertical bars provide independent print calibration. The reported full-graphic width includes annotation space, at least a 50 mm coordinate border at each longitudinal end, and the renderer's additional margins. The drawing library uses 640 px across, 28 px padding on each side, and a maximum height of 460 px; the larger of the width- and height-limited physical extents determines the reported full-graphic width.

• Twelve equal semitones per octave; ideal flexible strings define the nominal grid.
• The neck centreline is straight. String spreads are centred on it and measured transversely.
• Single scale means fundamental centreline length; multiscale inputs mean actual outer-string lengths before compensation.
• String centres are equally spaced at each nominal end. Nut and saddle endpoints of intermediate strings lie on straight nominal end lines.
• The board has straight side edges with equal transverse margins at the nominal nut and saddle. Its end follows the fret-line interpolation at the extended fraction, with the entered extension along the centreline; the end angle may differ from the last fret.
• Compensation inputs are externally determined along-string distances, linearly interpolated between the two outer strings. This simple interpolation is a design assumption, not a universal string-set compensation pattern.
• Row 0 denotes the nominal nut datum. Its shifted-nut offset is negative when a positive nut shift is applied; the real open string length appears in the string table.

For planar 12-TET layouts with 2–12 strings, 1–36 installed frets and positive input dimensions within the displayed ranges. Both outer multiscale lengths must exceed their transverse endpoint change, the shifted nut must remain behind fret 1, and the board must end before the nominal saddle. Both board sides must advance toward the bridge without a fold, and fret-edge intersections must remain nonsingular and forward-ordered throughout the entire board, including between installed frets. Invalid outlines block the results. A fan beyond 40° still prompts a separate playability warning; this is a design advisory rather than an intonation limit.

This is a geometric layout, not an automatic compensation or structural-strength model. It excludes string stiffness, gauge-dependent spacing, fretting stretch, action, fretboard radius, crown/contact geometry, zero-fret hardware details, tooling kerf and non-12-TET scales. The 12th-fret result remains a reference even if fewer than 12 frets are installed.

The viewer and its Report fit scenes to the available page. They are not automatically full size. Use the reported plan canvas width or both 100 mm bars to set physical scale. Confirm both axes after any printing, scaling or tiling. Numerical precision does not imply the same manufacturing accuracy.