The Art of Microtonal Fretting

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 that softens major thirds to 31-EDO that approximates just intonation harmonies with remarkable fidelity. 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 empowers luthiers, hobbyists, and theorists to explore exotic scales by producing accurate fret positions and corresponding frequencies for any equal division of the octave.

Equal Division of the Octave Explained

In an equal division of the octave system, the frequency ratio between adjacent steps is constant. 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 $\frac{1^{\frac{n}{m}}}{}$ of its original length, its frequency increases by $2^{\frac{n}{m}}$. Our task is to determine the distance from the nut where this shortening occurs for each fret.

Fret Position Formula

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. The expression inside the fraction represents the remaining string length after fretting, computed by dividing the scale length by the exponential ratio. Subtracting that value from the full length yields the distance along the fingerboard. This formula generalizes the well-known 12-EDO expression and ensures each microtonal step remains consistent. The calculator implements this equation using JavaScript’s power function, ensuring millimeter-level accuracy suitable for instrument design.

Frequency of Each Fret

Beyond physical placement, microtonal composers often need to know the pitch associated with each fret. If the open string vibrates at frequency $\mathrm{f\_0}$, fretting at position $n$ yields a frequency of $\mathrm{f\_n}=\mathrm{f\_0}\times 2^{\frac{n}{m}}$. These frequencies do not correspond to traditional note names; instead, they create new harmonic landscapes. Some musicians assign numeric labels, while others map microtonal steps to colors or symbols. The calculator outputs a table pairing fret distances with their frequencies, providing a foundation for naming conventions, notation, or digital tuning files. Because the frequencies are derived from mathematical ratios, they remain consistent across octaves and strings, simplifying ensemble work for multiple instruments tuned to the same system.

Table of Fret Positions

The table below will display after you submit your parameters. Each row lists the fret number, the distance from the nut in millimeters, and the frequency produced by fretting at that position. The first fret typically lies at a noticeable distance from the nut, while higher-numbered frets cluster ever closer together. Microtonal divisions with large $m$ values produce especially tiny intervals near the bridge, challenging both luthiers and performers. Instruments that support 53-EDO or higher often rely on moveable frets, fanned fretboards, or even digital pitch shifting to accommodate the compressed spacing.

Historical and Cultural Context

Microtonality is not merely a contemporary experimental fad. Ancient Greek theorists like Aristoxenus debated alternative divisions of the tetrachord, while Middle Eastern maqam and Indian raga traditions make use of intervals far subtler than the Western semitone. In the 16th century, theorist Nicola Vicentino designed the archicembalo, a keyboard with 36 keys per octave aimed at realizing just intonation. The 20th century saw pioneers such as Julián Carrillo, Alois Hába, and Harry Partch craft instruments with 16, 17, 31, or 43 pitches per octave. Partch built an entire orchestra of custom instruments to explore the tonalities of 43-EDO, including adapted guitars and marimbas. Today, microtonality thrives in genres ranging from avant-garde classical to math rock and electronic dance music. Software synthesizers now allow musicians to load custom tuning files instantly, fueling interest in building physical instruments that match these digital scales. This calculator participates in that tradition by making the geometric side of microtonal design accessible to anyone with a browser.

Practical Building Considerations

While the formulas guarantee precise placements, implementing microtonal frets on a physical instrument introduces practical challenges. Wood expands and contracts with humidity, potentially altering fret distances. To mitigate this, luthiers often cut slots slightly wider than the fret tang and glue the frets in place. Some builders experiment with metal or 3D-printed fingerboards to preserve exact measurements. Additionally, fret height and string gauge can affect intonation; thicker strings may require compensation at the bridge to stay in tune across the neck. Musicians venturing into EDOs above 24 sometimes adopt partial fretboards that cover only the most musically relevant regions, leaving upper registers blank to maintain playability. These pragmatic decisions underscore that calculations are only one part of a holistic design process.

Microtonal Harmony and Composition

Understanding fret placement opens the door to exploring harmony in new ways. In 19-EDO, for instance, the equivalent of a perfect fifth spans 11 steps, creating chords with a subtly different flavor from their 12-EDO counterparts. Composers may discover fresh consonances or dissonances that inspire unique progressions. Because microtonal systems can approximate just intonation more closely, they enable resonances that align with harmonic overtone series, producing a sense of clarity or shimmer absent in standard tuning. The calculator’s frequency output lets composers map these intervals accurately in digital audio workstations or notation software, ensuring that written scores and live performances align.

Integrating with Digital Tools

Many contemporary musicians blend acoustic microtonal instruments with digital synthesizers and samplers. Tools like MIDI Tuning Standard (MTS) or Scala files allow electronic instruments to adopt arbitrary tunings. By exporting the frequencies from this calculator into a tuning file, you can synchronize a physical microtonal guitar with virtual instruments. Some performers even use pitch-tracking software to drive microtonal synthesizers from a fretless instrument, effectively using the fingerboard as a controller. Accurate fret position data ensures that these hybrid setups maintain consistent intervals, preventing the ear from detecting discrepancies between acoustic and electronic sounds.

Beyond Equal Temperament

Though this calculator focuses on equal divisions of the octave, the microtonal landscape includes many unequal tuning systems such as just intonation, meantone, or Bohlen-Pierce. Extending the formulas to these domains requires replacing the constant ratio with specific rational fractions or logarithmic sequences. Nevertheless, mastering EDO fret spacing lays a solid foundation for tackling more complex tunings. Many luthiers prototype ideas in EDO before committing to the more labor-intensive task of cutting frets for an unequal system. Future versions of this tool may incorporate custom ratio lists, opening even wider vistas of sonic exploration.

Using the Calculator

To begin, enter your instrument’s scale length, the total number of frets you wish to install, the desired EDO value, and the open string frequency. Classical guitars often use a scale length around 650 mm, but builders may choose shorter or longer lengths for ergonomics or tonal reasons. After pressing the generate button, the calculator computes fret distances and frequencies with JavaScript’s high-precision math functions and populates the results table. A copy button appears, allowing you to transfer the data into spreadsheets, CAD software, or notation documents. Whether you’re designing a 17-EDO travel guitar or a 24-EDO experimental bass, the output serves as a blueprint for your luthier work.

Interpretation of Results

The results provide a practical map of your microtonal fretboard. Distances are measured from the nut, making them easy to mark with a ruler. Frequencies help you verify tuning during construction and can guide choice of string gauges or tension. Because the exponential spacing causes frets to crowd near the bridge, some builders choose to stop at a lower fret number than the calculator allows. Remember that physical factors like string stiffness, action height, and bridge compensation may require slight adjustments after construction. Nevertheless, starting from mathematically derived positions dramatically increases the odds that your instrument will play in tune across its range.