Unison:
650.0 mm
Octave:
325.0 mm
Perfect Fifth:
433.3 mm
Perfect Fourth:
487.5 mm
Major Third:
520.0 mm
Minor Third:
541.7 mm
Major Sixth:
390.0 mm
🎯 A Simple Example: Building a One-String Zither
You're building a simple one-stringed zither (monochord) for a science project to demonstrate how harmony works. You've cut a piece of wood that allows for a "vibrating string length" (the distance between the two fixed ends) of exactly 1000mm. Where should you place your markings to play a "Perfect Fifth"?
Just do this:
1️⃣ Set "Total String Length" to 1000.
2️⃣ Ensure "Measurement Unit" is set to Millimeters.
3️⃣ Look at the result for "Perfect Fifth": it says 666.7 mm.
4️⃣ Measure exactly 666.7mm from one end of the string and place a mark or a movable bridge there.
5️⃣ Pluck the string—the note produced when pressing the string at that mark will be a pure, harmonic fifth above the open string!
Pro tip: Always measure from the same end (the "Nut" or "Zero" point). If you measure some notes from the left and others from the right, your intervals won't align correctly!
Potential risks:
Data Source: Harmonics (Claudius Ptolemy / 19th C. Revivals) • Public domain • Solo-developed with AI
The monochord is perhaps the most significant scientific instrument in the history of music. Used by Pythagoras in the 6th century BCE and later standardized by Ptolemy, it served as the physical proof that music—something seemingly ephemeral and emotional—was rooted in the immutable laws of mathematics. By dividing a single string into simple integer ratios, the ancients discovered the intervals that form the basis of Western harmony.
In the 19th century, during the revival of classical education and the rise of experimental acoustics (notably by Hermann von Helmholtz), the monochord returned to the laboratory. It was used to demonstrate Just Intonation, a system of tuning where frequencies are related by whole-number ratios, as opposed to the "Equal Temperament" used on modern pianos.
The physics of a vibrating string is straightforward: the frequency of vibration is inversely proportional to its length. If you halve the length of the string (1/2), you double the frequency, creating the Octave. If you take two-thirds of the string (2/3), you produce the Perfect Fifth.
This tool calculates the physical "node" position from the nut (zero point). To build your own, you simply need a rigid board, two fixed bridges (the ends), and a movable bridge (the nut) that can be slid along the string to these calculated positions. When the movable bridge is placed at the major third position (4/5 of the total length), the resulting tone will be a pure, beating-free harmonic Third.
Why calculate these positions today? In our digital world of perfectly tuned synthesizers, we have lost the "organic" resonance of pure ratios. Modern luthiers, acoustic engineers, and experimental musicians use the monochord to explore Microtonality and the psychoacoustic effects of Just Intonation.
Using this tool, a maker can build a "Slide Guitar" or a "Diddley Bow" calibrated to historical harmonics. It allows for the construction of instruments that resonate with the natural harmonic series, providing a richness of tone that standard Western tuning often filters out. It is also an essential tool for physics students to visualize the relationship between tension, mass, and length in wave mechanics.
For the best results, use a high-tension steel or brass wire. A total length of 600mm to 1000mm is ideal for tabletop experiments. Ensure your soundbox—even a simple hollow wooden box—is reinforced where the tuning pins are inserted, as the cumulative tension of even a single string can exceed 20 lbs of force.
🐱 From the Lab Cat's Desk:
The humans spent three thousand years arguing about "Perfect Fifths" and "Major Thirds." I, however, solved music theory in a single afternoon. If the string makes a high-pitched 'twang' that startles me from my nap, it is out of tune. If it vibrates at a frequency that mimics the purr of a well-fed feline, it is a masterpiece. Also, please do not use the movable bridge as a chew toy. It is surprisingly hard on the gums.