The Twelfth Root

analytical · neon · twelve laws · under keeper eris

neon

{{ act || react || change }} · the sculpture

analytical

Twelve equal semitones multiply to one octave, because the twelfth root of two, taken twelve times, is two.

LIT Let r = 2^(1/12). The product r·r·…·r, twelve times, is 2, which is one octave, 1200 cents. The page multiplies the factors one by one rather than calling a single power for the product.

AMBER Twelve-tone equal temperament was calculated in the 16th century, notably by Zhu Zaiyu (1584). The citation is the history; the product is the check.

MODEL A stair of twelve rises. The top step sits one octave above the first and on the same angle, because twelve equal turns make a full circle and the frequency has doubled.

Sources AMBER

  1. Equal temperament (reference)

LIT on load: the identity is recomputed (it throws on mismatch), the training target scores 1, 200 rounds improve the shared sketch, the workspace loop logs 5 entries, and the OBJ and SVG exports match the sketch.

Gallery of kept sketches:

Workspace improvement loop MODEL

  1. Measure: every 40 rounds, record the acceptance rate, the current score and how many rounds since the last improvement.
  2. Try next: if the score has been flat for 40 rounds, widen the mutation step. If acceptance is under 15%, shrink it; over 45%, widen it. Otherwise keep it.
  3. Log back: append {entry, round, measured, tried, step} to the workspace log (it is in the export JSON and the list below).
  4. Repeat: re-check the identity before claiming it. The seal stays write-if-missing.

Live workspace log:

    Shared board MODEL

    localStorage key mayhem-twelve-laws-board. Each page writes its best shared reward.

      Attempt board MODEL

        ← The Fifths Ring · Twelve Laws · Kepler's Cubes → · ERIS

        Background wash: AMBER MODEL a field seeded from this page's seal. The viewport is a deep colour, not black, over a vivid page.

        proving…

        Seal sha256(name|slug|blurb): c6a0b12ab7623dc56b1e79f0d6abd0ab34c9b1d66ee6f3243556cc0ff504290e