Microtonal Fret Spacing Calculator

The Art of Microtonal Fretting

Microtonal fretting begins with a different way of dividing the guitar string. Standard Western guitars divide the octave into twelve equal parts, a system known as twelve-tone equal temperament (12-EDO). Each fret represents a semitone step, and the distances shrink exponentially as they approach the bridge. Yet musicians across cultures and experimental scenes have long desired intervals that fall between these semitones. The microtonal approach replaces the fixed dozen divisions with any number of equal steps, from 19-EDO, which offers a different placement for familiar interval shapes, to 31-EDO, which supplies much finer pitch steps. Building a fretboard that supports these alternative tunings demands precise calculations. A small error in placement can render chords sour or melodies out of tune. This calculator gives luthiers, hobbyists, and theorists fret locations and corresponding frequencies for any equal division of the octave.

Equal Division of the Octave Explained

This microtonal fret calculator uses the constant step ratio of an equal division of the octave. If an octave spans a frequency ratio of 2:1, dividing it into $m$ equal parts gives a ratio of $2^{\frac{1}{m}}$ between neighboring pitches. For traditional 12-EDO, this ratio is the twelfth root of two, approximately 1.05946. In 19-EDO the ratio becomes the nineteenth root of two, around 1.03715, creating more gradual pitch increments. Translating these ratios into physical distances on a fretboard requires applying the same exponential relationship to string length. When a string is shortened to $\frac{1}{2^{\frac{n}{m}}}$ of its original length, its frequency increases by $2^{\frac{n}{m}}$. The relevant layout question is the distance from the nut at which that shortening occurs for each fret.

Microtonal Fret Position Formula

For an EDO fretboard, the location of the $n$th fret on a scale length $L$ in a system with $m$ divisions of the octave follows a concise formula: $d=L-\frac{L}{2^{\frac{n}{m}}}$. Here, $d$ is the distance from the nut to the fret. Equivalently, the remaining vibrating string length is $\frac{L}{2^{\frac{n}{m}}}$. The expression inside the fraction is the remaining vibrating string length after fretting, found by dividing the scale length by the exponential ratio. Subtracting that length from the full scale gives the mark along the fingerboard. This is the familiar 12-EDO rule generalized to any chosen number of equal octave divisions. The calculator uses JavaScript’s power function and displays the resulting distances to two decimal places in millimeters.

Frequency of Each Microtonal Fret

The microtonal fret table also reports the pitch multiplier at every fret. If the open string vibrates at frequency $f_0$, fretting at position $n$ yields a frequency of $f_n=f_0\times 2^{\frac{n}{m}}$. These frequencies need not correspond to traditional note names; they identify the equal-step pitches generated from the open string frequency you enter. Some musicians use step numbers, colors, or symbols rather than note labels. The generated table pairs each fret distance with its frequency, which can support a personal notation scheme or comparison with a separately prepared digital tuning. Because every value comes from the same ratio, equivalent fret counts on strings with different open frequencies retain the same interval structure.

Table of Microtonal Fret Positions

The microtonal fret-position table appears after you submit the instrument parameters. Each row gives the fret number, its distance from the nut in millimeters, and the frequency produced at that location. The first fret normally sits a substantial distance from the nut, while successive frets gather toward the bridge. High EDO values make neighboring fret marks especially close together, particularly high on the neck, so builders should consider whether the physical fret wire and intended playing area can accommodate the chosen division. The table is a geometric layout reference; it does not determine bridge compensation or other setup details.

Historical and Cultural Context for Microtonal Frets

Microtonal fret design belongs to a long history of questioning the twelve-note octave. Ancient Greek theorists such as Aristoxenus debated alternative divisions of the tetrachord, while Middle Eastern maqam and Indian raga traditions use pitch practices that do not reduce neatly to the Western semitone. In the 16th century, theorist Nicola Vicentino designed the archicembalo, a keyboard with 36 keys per octave aimed at realizing alternative interval systems. The 20th century saw pioneers such as Julián Carrillo, Alois Hába, and Harry Partch build instruments with unusual pitch resources per octave. Partch developed an ensemble of custom instruments around a 43-tone just-intonation system, including adapted string instruments and marimbas. Today, microtonality appears in experimental classical music, rock, electronic music, and many other practices. Digital synthesizers make custom tunings readily available, which in turn encourages builders to make physical instruments that can share a chosen tuning plan. This calculator addresses the geometric side of an equal-division fretboard.

Introduction: Practical Microtonal Fretboard Building Considerations

Microtonal fret positions are mathematical targets, but cutting those locations into a physical neck introduces practical constraints. Wood expands and contracts with humidity, potentially affecting measured distances. Fret-slot width, fret tang fit, and installation technique can all influence the finished result. Fret height, string gauge, action height, and string stiffness also affect intonation, and a bridge may need compensation during final setup. Players working in EDOs above 24 may choose a limited fret range when the upper-neck spacing becomes difficult to machine or play. These construction choices do not change the equal-division calculation, but they matter when turning its nut-to-fret measurements into an instrument.

Microtonal Harmony and Composition

Microtonal fret placement makes an EDO’s interval map tangible for harmony and composition. In 19-EDO, for instance, an eleven-step interval is close to a perfect fifth but differs from the 12-EDO fifth. Composers can test unfamiliar chord shapes and hear how the chosen number of equal divisions changes their character. Some EDO systems offer close approximations to selected just intervals, while others deliberately emphasize a distinct equal-tempered sound. The frequency column helps relate a physical fretboard to the pitch values needed in notation or audio software. It does not assign harmonic function by itself; that interpretation depends on the musical context and the step relationships a player chooses.

Integrating Microtonal Fret Data with Digital Tools

A microtonal guitar or bass can be coordinated with digital instruments when both use the same EDO relationship. MIDI Tuning Standard (MTS) and Scala-based workflows are common ways to configure compatible electronic instruments for non-12-EDO pitch sets. The table’s displayed frequencies can be copied for reference when preparing a separate tuning file, spreadsheet, CAD drawing, or notation project. Accurate nut-to-fret dimensions help keep the acoustic instrument’s intended intervals aligned with the selected digital tuning. The calculator supplies the equal-division positions and frequencies only; any file formatting or device-specific tuning setup is handled in the software or hardware you use.

Beyond Equal Temperament

This fret-spacing calculator is specifically for equal divisions of the octave, not every microtonal tuning method. Unequal systems such as just intonation, meantone, or other ratio-based layouts need individual fret locations derived from their own interval ratios rather than one repeated EDO step. Learning the EDO geometry is still useful: it shows how a frequency ratio translates into a remaining string length and then a nut-to-fret measurement. A builder considering an unequal system should calculate each intended ratio independently instead of treating this table as a substitute for that design work.

How to use: Generating an EDO Fret Table

To generate a microtonal fret table, enter the scale length in millimeters, the number of frets to list, the desired EDO value, and the open-string frequency in hertz. Classical guitars often use a scale length around 650 mm, but builders may choose shorter or longer lengths for ergonomics or tonal reasons. After you press the generate button, the calculator calculates each nut-to-fret distance and frequency, then fills the results table. The Copy Table button lets you copy the displayed rows for use in a spreadsheet, CAD workflow, or personal notes. Whether the project is a 17-EDO travel guitar or a 24-EDO bass, verify the selected scale length and EDO before transferring marks to the fingerboard.

Interpreting Microtonal Fret Spacing Results

The microtonal fret results are a marking map for the specified scale and equal division. Distances are measured from the nut, so each value is a direct reference for laying out a fret slot. Frequencies show the pitch produced from the entered open-string frequency at the idealized fret position. Since exponential spacing compresses marks toward the bridge, inspect the smallest gaps in the requested fret range before deciding how many frets to install. Physical setup factors such as string stiffness, action, and bridge compensation can require final adjustment, but mathematically derived positions provide the correct equal-division starting layout.

Worked example: comparing EDO choices on one fretboard

For a meaningful microtonal fret-spacing comparison, hold the scale length, number of frets, and open-string frequency constant, then generate one table for each EDO under consideration. A higher EDO places the early frets closer to the nut than a lower EDO because each step represents a smaller fraction of an octave. Compare the resulting fret gaps, especially in the upper register, and confirm that the spacing is practical for the material, fret wire, and playing technique planned for the instrument.

Limitations and assumptions for EDO Fret Spacing

This calculator models ideal equal-division fret locations on a string of the scale length you enter. It assumes each fret produces the exact mathematical shortening ratio and does not model string stiffness, action, compensation, neck movement, fret installation tolerances, or an unequal temperament. Use consistent millimeter and hertz inputs, and treat the displayed measurements as a layout starting point to be checked during instrument construction and setup.