Following the harmonic breadcrumbs from 45 to 49995
🧱 The Natural Extensions of a Toroidal Walk
After discovering the 3-digit Preston Toroidal Scale (3PTS), a natural curiosity arises:
Can this pattern extend?
What happens when we apply the same walking procedure—but adjust either the length of the result (number of digits) or the movement rules?
That curiosity led to three important expansions:
- 1PTS: The basic set of single digits [1–9]
- 2PTS: A reduction of 3PTS members by chopping off the third digit
- 4PTS: A wrapped extension, cycling the first digit to the end
These aren’t arbitrary—they emerge directly from the structure of the toroidal process.
🔢 One-Digit: The Foundation (1PTS)
Simple, linear: just the digits 1 through 9.
Their sum? 45.
That number becomes surprisingly important.
🔁 Two-Digit: Trimming the Walk (2PTS)
By taking each 3-digit value from 3PTS and removing the third digit, we derive 81 two-digit numbers. This isn’t just random chopping—it preserves directional lineage while projecting it into a simpler form.
Their magic constant (when arranged in a magic square-style layout)? 495
Already, a pattern emerges:
1PTS sum = 45
2PTS magic = 495
3PTS magic = 4995🔄 Four-Digit: Wrapping the Circuit (4PTS)
Rather than reduce the digits, what if we extend them?
Take each 3PTS value, and append its first digit to the end—forming a closed-loop that reflects the toroidal nature of the walk.
Example:
- 147 → 1471
- 321 → 3213
- 693 → 6936
This yields 81 four-digit values.
When these are organized into a 9×9 grid similar to previous experiments, the row sums suggest a new magic constant:
49995
It fits perfectly:
1PTS: 45
2PTS: 495
3PTS: 4995
4PTS: 49995You may notice a pattern forming: a recursive structure around the digits 4, 5, and 9.
Or as you put it:
4(5+4)5, 4(5+4)(5+4)5, 4(5+4)^n5…
🧮 Digit Root Behavior
Another surprising feature? Only 3PTS maintains the strict reduction to digit roots of 3, 6, or 9.
- 1PTS has full spread: 1–9
- 2PTS mixes evenly
- 3PTS: 100% reduce to 3, 6, or 9
- 4PTS: scattered, despite being direct derivatives of 3PTS
This makes 3PTS unusually “resonant”—a sweet spot of numerical coherence.
🎻 Harmonics in the Structure
Plotted as distributions, these four scales create a fascinating shape:
- 1PTS: linear and finite
- 2PTS: slightly curved
- 3PTS: logarithmic-like rise
- 4PTS: warped echo of 3PTS, repeating patterns
The stair-step of harmonic density lines up with the musical metaphor:
- The 3PTS curve resembles an equal-temperament octave sweep
- The 4PTS resembles a reverberation
This opens a doorway to our next post: analyzing how the PTS system maps against musical scales like 12TET.
➿ Summary
The PTS structure doesn’t just extend—it amplifies.
Each level seems to scale in harmony with the last, reinforcing the same core digits, factors, and summations with uncanny consistency.
And all of it echoes from a little walk on a keypad grid.
Coming next: frequency plots, tone curves, and why the PTS might be humming a tune that’s been with us all along.