Part 2: Walking the Grid – How the Preston Toroidal Scale Works

From keypad wanderings to harmonic structure


🧭 The Setup: A Simple 3×3 Grid

Imagine a standard phone keypad:

[7 8 9]
[4 5 6]
[1 2 3]

Each of the 9 positions becomes a starting point. We define directional steps using coordinate pairs:

  • N = (0, -1)
  • NE = (1, -1)
  • E = (1, 0)
  • SE = (1, 1)
  • S = (0, 1)
  • SW = (-1, 1)
  • W = (-1, 0)
  • NW = (-1, -1)
  • O = (0, 0) (stand still)

We take 3 steps in any of these directions from each cell, wrapping around the edges.

For example, starting at 7 and moving East:

  • Step 1: 7 → 8 → 9 → wrap to 7 → …

The digits visited form a 3-digit number. This is repeated across all positions and directions.


🔢 Result: 81 Unique Three-Digit Numbers

Every combination generated follows a toroidal path and uses only digits 1–9.

Surprising outcomes:

  • Every result is divisible by 3
  • All values reduce to 3, 6, or 9
  • Includes classic angel numbers (e.g., 111, 222, …)
  • Captures the Solfeggio group all together in one round (396, 417, etc.)

Sorted, the PTS looks like:

[111, 123, 132, 147, 159, 168, 174, 186, 195,  
 213, 222, 231, 249, ... up to 999]

🧠 Code Demo Tabs (Placeholder)

Python Tab:

# Define the 3x3 grid
# Define all directional vectors
# Walk from each position in each direction
# Collect and sort unique 3-digit results

JavaScript Tab:

// Create grid array
// Define directions as [dx, dy]
// Use modular arithmetic to wrap
// Build three-digit numbers

Live Demo (Placeholder): [ GridWalker.js demo with interactive starting points + path lines ]


🎼 Why It Matters

The structure that emerged wasn’t random. It hinted at something underlying—not just a pattern, but a harmonic container.

From this, further patterns emerged:

  • 9×9 magic square substitutions with preserved sums
  • Lower and Higher-order PTS extensions (2PTS, 4PTS)
  • A numeric analog to tone systems like 12TET

The numbers didn’t just align—they sang, subtly but with seeming intention.

Next up: We map the PTS into the framework of the 9×9 Lo Shu-inspired magic square—and discover why 4995 might just be the most harmonious number you’ve never heard of.

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