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Dozenal Society of America
Promoting base twelve and alternative base mathematics
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The Dozenal Society of America
The DSA is a voluntary, nonprofit education corporation, organized for the conduct of research and education of the public in the use of dozenal (also called duodecimal or base-twelve) in calculations, mathematics, weights and measures, and other branches of pure and applied science.

The Duodecimal Bulletin

◀️ Issue: Whole Number 65z (77d) | Vol. 33z (39d) No. 2 | Year 119Az (1990d) ▶️



Click to view PDF

Editor: Dr. Patricia Zirkel

Cover Title: Rocked Around The Clock

Contents:

  • Board of Directors Meeting
  • Rock − It | A Puzzle/li>
  • Music, Scales and Dozens, part II: Just Intonation and the Chromatic (Dozenal) Scale | Dr. John Impagliazzo
    • The Scale of Just Intonation
    • Defects in Tonal Scales
    • The Chromatic Scale, or Scales and Dozens
    • Conclusion
  • How Many Fruits on the Tree of Life? | B.A.M. Moon
  • Second Order Duodecimal Root Bracelets | Charles W. Trigg
  • Dozenal Jottings
    • Charles Ashbacher | published solution to difficult geometry problem
    • Dr. Tony Scordato (NCC)
    • Robert Broda | twelve houses of zodiac
    • Anetta Trigg | Charles W. Trigg room at Mathematical Association of America headquarters
    • Dudley & Kay George
    • Jamison & Vera Handy
    • Neela Lakshmanan
    • Ian Patten | working on two books
    • Charles & Miriam Bagley | old letter from George Terry
    • Bruce Moon (Australia)
    • Donald Hammond (DSGB)
    • Fred Newhall
    • Richard T. Trelfa
    • Robert R. McPherson
    • New member: Robert A. Broda
  • More Pi
  • Strange Digits
  • “A Basic Solution” Solution | Jean Kelly
  • Ralph Beard Memorial Award — 1990d (119Az): Peter D. Thomas (Australia)
  • Why Change?
  • Counting in Dozens
Copyright © 11B9–120Az (2013–2026d). The Dozenal Society of America. Contact Us.
Creative Commons License
This site and the documents on it are licensed under a Creative Commons Attribution 4.0 International License.

This site expresses numbers in alternative bases, particularly dozenal (also called duodecimal, or base twelve).
The DSA does not endorse any particular notation or pronunciation scheme for alternative bases.
However this site defaults to the following to ensure online readability across all platforms:
“Computerese” uppercase letters represent transdecimal digits: A = ten, B = eleven, C = twelve, etc.
Single-character subscripts annotate bases: d = decimal, z = dozenal, x = hexadecimal, etc., e.g., 2014d = 11BAz = 7DEx.

(0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...)d
(0 1 2 3 4 5 6 7 8 9  A  B 10 11 12 13 14 15 16 17 18 19 1A 1B 20 21 22 23 24 25 26 27 28 29 2A 2B 30 ...)z
(0 1 2 3 4 5 6 7 8 9  A  B  C  D  E  F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20 21 22 23 24 ...)x

PDF documents on this site reflect changing practices among dozenalists over the years:
Various symbols have been used to represent digits ten and eleven:
Dwiggins 𝒳 and Ɛ (often rendered X and E); Bell Telephone * and #; Pitman turned 2 and 3 (↊ and ↋); etc.
Italics were once used to distinguish dozenal numbers from decimal: e.g. 0.5 = 0.6.
Later, semicolon became a separate‑identity “dozenal point”, versus period as “decimal point”: e.g., 0.5 = 0;6.
However, subscript base annotations provide a more versatile solution that is least‑change: e.g., 0.5d = 0.6z = 0.8x

Dozenal offers superior divisibility over decimal, which can make everyday arithmetic easier to learn:
⅙ = 0.2z | ¼ = 0.3z | ⅓ = 0.4z | ½ = 0.6z | ⅔ = 0.8z | ¾ = 0.9z | ⅚ = 0.Az
⅙ = 0.1666…d | ¼ = 0.25d | ⅓ = 0.333…d | ½ = 0.5d | ⅔ = 0.666…d | ¾ = 0.75d | ⅚ = 0.8333…d

EACH ONE, TEACH ONE.