Plotting 432 & 440
In preparation for submitting to Fukt's upcoming open call for The Sound Issue, I decided to explore the delta between 432 Hz and 440 Hz. For context, 440 Hz is the international standard for tuning to A. It's the ISO standard codified in 1955, but its formal advocacy can be traced back to the 1930's. While there's conspiracy abound regarding potentially nefarious origins with respect to WW2, there had already been an ongoing international effort to standardize.
432 is often related to Scientific Pitch, the standard from the 19th century where middle C is set to 256 (so this would actually make A around 430.54 Hz). Pre standardization, tuning varied by region. A perfect example is Verdi famously writing his Requiem with A4 tuned to 435 Hz, and then later indicating orchestras should tune it to 432 Hz.
The significance of 432 as it relates to sacred math, geometry, and natural frequencies, is thought to be the main reason for asserting its preference as a tuning standard for A. Here are a few notable relationships:
- Highly composite number. 432 = 2⁴ × 3³, resulting in 30 divisors. This extends well to integrating with harmonic series; octaves, perfect fifths, fourths, and thirds.
- Fits within the Pythagorean Tuning structure where octaves = 2:1, perfect fifths = 3:2, and perfect fourths = 4:3.
- Closely relating with timing and the cosmos, found in everything from temple measurements, astrological timing, and geometric ratios.
- Fits naturally into base 12, base 60, and circular systems.
Below are recordings of both 432 Hz and 440 Hz. Enable both to hear the interference pattern.
Listen for 432
Listen for 440
While the auditory experience of the interference pattern between 432 and 440 Hz is immediate, translating that into visual form invites both scientific and aesthetic interpretations. To that end, this project proposes three distinct explorations:
Chladni Patterns
Ernst Chladni, the German physicist and musician is best known for his invention of a technique to showcase modes of vibration on a rigid surface. Essentially these patterns are the visual representation of standing wave modes. When sand or a particulate is poured on the surface, the patterns form along the areas or nodal lines (areas of no motion) when the plate vibrates. Different patterns correspond to different resonant frequencies, otherwise known as eigenmodes; stable shapes the particles assume on the plate at specific frequencies.
Paul Bourke formalized that we could simulate these patterns for a square plate of length L, the nodal lines can be plotted given the following equation:
For a circular plate with a radius R, the nodal lines are described by:
Thinking about how to model this in something like Processing, we need to find the easiest way to represent plate geometry, materiality, and tonal frequency applied to the plate. The above formulas give us the zero displacement contours for standing waves on plates; its limitations are spatial, and it only gives values for modes m and n.
What this means physically and visually:
- For values m and n where the result equals zero indicates all points where no vibration occurs (where particles accumulate)
- Each standing wave pattern produces a unique nodal line geometry
How might we think about plotting this for a rectangular plate?
- Set values m and n
- For each point(pixel) on the plate (within the window):
- If result = 0, plot the coordinate
- Additionally, if we accept a tolerance (acceptable distance from nodal lines), we can also plot the point
Removing plots for values where m = n, a distinct progression begins to emerge for values 0...10. These are solutions to the eigenvalue problem of a vibrating surface. The goal, however, is to bridge thoery with tonal experience. Specifically we're looking to invert the model to demonstrate how different frequencies (432 vs 440 Hz) excite given different modes on the physical shape.
This portion proved a bit more challenging as it's difficult to model real world materiality. On a square plate what do M and N mean?
Nis the nubmer of half wavelengths across the x axisMis the number of half wavelengths across the y axis Each mode (combination) represents a standing wave patterns so the higher either value, the more complex and therefore higher frequency the vibration mode is.
Animated Sweep: Frequencies from 0 to 500 Hz
Processing was used to create a dynamic sketch that calculates and visualizes resonant patterns across frequencies from 0 to 500 Hz. Each frame estimates the closest integer mode shape (N, M) for a given frequency, producing an emergent animation of how vibration modes evolve with pitch.
Zooming In: 432–440 Hz
When focusing specifically on the 432–440 Hz range, only three frequencies yield distinct nodal arrangements: 432 Hz, 435 Hz, and 439 Hz. These are the only ones that resolve cleanly into discrete standing wave shapes with clear nodal geometry.
To explore this, we extracted the SVGs for each and generated an interpolated animation that cycles through their visibility over time, creating a looping transition that mimics wave-based modulation.
Reflections
Despite how compelling some of the eigenmode plots are, Chladni figures may not be the most precise tool for communicating the interference between close frequencies like 432 and 440 Hz. The mode shapes are based on structural resonance, not audio superposition or frequency, which limits their use in this context.
Still, the resulting SVGs and generated point data remain visually and conceptually rich. These artifacts may be used in future explorations, perhaps as building blocks for a more abstract visualization system.