Part 5: PTS vs. 12TET–Plotting the Scale of Harmony

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.

Leave a Reply

Your email address will not be published. Required fields are marked *