User Guide: Harmonica and Vibrating Strings Series Explorer | Legatto Institute

Welcome to Harmonic Series Explorer, An interactive application designed to visualize and listen to the physics of vibrating strings. This tool is ideal for students of musical alchemy, sound designers, and anyone interested in understanding how frequencies and mathematical ratios give coherence and harmony to our universe.
Through this guide, you will learn how to navigate the interface, adjust fundamental frequencies, and explore the different geometric and acoustic display modes.

1. Main Controls: Frequency and Amplitude

At the top right are the "Play All" and "Stop" buttons to listen to and turn off the sound of the active harmonics at the same time.
The left side panel (or top panel on mobile devices) contains the master controls that define the behavior of the sound wave.
  • Fundamental Frequency (f₁): It is the fundamental note from which all other harmonics are built. You can use the slider to adjust the value or click directly on the number (for example, "1 Hz") to enter a specific frequency. Practical tip: Try entering alternative and therapeutic tuning frequencies such as 432 Hz or 528 Hz to observe its behavior.
  • Amplitude (Volume): Controls the overall volume (master gain) of the generated sine waves.
  • Animation Speed: It modifies the speed at which the waves move on the screen. This does not alter the pitch of the sound, but facilitates the visual study of acoustic physics, allowing the movement of the waves to be observed in slow motion.

2. View Modes: Geometric Visualization of Sound

To fully understand the nature of sound, the application offers different ways to perceive space and vibration:
  • 2D View: It displays the wave in a classic two-dimensional plane, ideal for clearly identifying the amplitude height and wavelength.
  • 3D View: It displays a three-dimensional representation using WebGL technology, allowing you to observe the depth and movement of the vibrating string in space.
  • View of Both (Both): It splits the screen to simultaneously display 2D and 3D representations, offering a comprehensive visual learning experience.

3. Harmonic Matrix: Listening to the Proportion

The "Harmonics" section presents an interactive grid of 20 cells. Each cell represents a harmonic that vibrates at an exact multiple of the fundamental frequency (n × f₁).
  • Activate/Deactivate: Click in the center of any cell to turn that specific harmonic on or off. Its color will change to indicate that it is active.
  • Button 'M' (Mute): It maintains the visualization of the harmonic on the screen (in gray), but mutes its audio output. Useful for visually isolating frequencies without saturating the sound.
  • Button 'S' (Only): It instantly silences all other harmonics and allows you to hear only the selected one, keeping the visualization intact.

4. Presets: Exploring Chords and Scales

The Harmonic Series Explorer includes quick access buttons that activate specific mathematical and musical settings, revealing the profound connection between sound and sacred geometry:
  • All: Activate all 20 harmonics simultaneously.
  • Odd / Even: Alternate between even or odd number harmonics, demonstrating how certain resonances build specific acoustic timbres (like closed or open tubes).
  • Octaves: Select only the frequencies that double their value (harmonics 1, 2, 4, 8, 16), creating a pure resonance.
  • Cousins: Activate the harmonics that correspond to prime numbers.
  • Pythagoreans: Activate the fundamental harmonics that give rise to Pythagorean tuning and perfect proportions.
  • Triads: Select the harmonics that form natural major chords, revealing the music inherent in the nature of sound.
  • Clean: Turn off all frequencies to start over.

5. Graph Interpretation: Nodes and Antinodes

In the viewing area (dark background), the moving waves allow you to study the physical behavior of sound:
  • Nodes: They appear as bright points along the string. They are the points of zero displacement where the string doesn't move. Remember: the harmonic number n will always have n+1 nodes (including the two fixed ends).
  • Antinodes: These are the points where the wave reaches its maximum amplitude (the highest and lowest point of the curve).
  • Legend and Information: At the bottom, a status bar will keep you informed at all times about the active fundamental frequency, the number of harmonics on, and what the highest frequency playing at that moment is.
Use this tool to deepen your conscious listening, visualize the actual behavior of sound pressure, and take your understanding of harmony to a transpersonal level.