Do numbers walk in tune with music?
🎶 A Curious Parallel: Tone Meets Toroid
After extending the Preston Toroidal Scale (PTS) to 1-, 2-, 3-, and 4-digit sets, a pattern emerged—not just numeric, but musical.
The 3PTS, in particular, displayed a smooth logarithmic curve when values were sorted and plotted—eerily similar in shape to the frequency progression of musical octaves in 12-tone equal temperament (12TET).
So we asked:
Could PTS be reflecting harmonic relationships the way 12TET approximates them?
📈 The Plot Thickens
We mapped the sorted 3PTS values to their ordinal positions (1–81), then plotted them against log-scaled positions to simulate frequency spacing.
Key observations:
- The curve was not linear
- The spacing resembled a logarithmic sweep
- The distribution closely approximated a power curve, similar to an octave doubling model
This isn’t to say the PTS is a tuning system—but it certainly rhymes with one.
🎼 Comparing Curves
In 12TET:
- Each semitone increases frequency by a factor of the 12th root of 2 (~1.05946)
- An octave doubles the base frequency
In PTS:
- No explicit base frequency is defined
- But the sorted numbers show relative gaps similar to tone intervals
Overlaying both curves reveals:
- PTS is not perfectly aligned—but has regions that mirror 12TET steps
- Inflection points in 3PTS spacing cluster near expected semitone regions
- The differences may suggest a distinct harmonic dialect
🔍 What About Roots and Resonance?
Let’s not forget:
- All 3PTS values reduce to 3, 6, or 9 under digit summation
- These same digits were idealized by thinkers like Tesla as “keys to the universe”
- In musical terms, they hint at stability, thirds, triads, resonance
And in PTS:
- These are built-in. Every number participates in that triadic structure.
🎛️ Implications
This invites speculation, sure—but also experimentation:
- Could we build a tuning system using PTS-derived numbers as base frequencies?
- Could we sonify the toroidal walk itself?
- Could this become a non-12TET microtonal alternative?
Possibly. The shapes are there. The math is repeatable. The resonance is measurable.
🌐 Coming Next
We move from musical metaphor to mathematical machinery:
A look into recursive and closed-form generation of Lo Shu rank-n squares.
From fractal walks to expanding grid logic—we’ll build bigger magic.
Until then:
Sometimes, numbers don’t just count—they hum.