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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 37z (43d) | Vol. 1Bz (23d) No. 2 | Year 1180z (1968d) ▶️



Click to view PDF

Editor: Jamison “Jux” Handy, Jr.

Associate Editor: Henry Churchman

Contents:

  • Chromatic Musical Scales and Notations | Erich Kothe (page 27z, page 45z)
    • Part One: Hearing and Sound (page 27z)
      • Figure 1: Range of frequencies and their intensities which are perceived by the human ear (page 27z)
      • Figure 2: Semidiagrammatic section of the right ear (page 28z)
      • Figure 3: Apparent shift in frequency with high sound pressure (page 29z)
      • Figure 4: Loudness level contours (page 29z)
      • Figure 5: Relationship between loudness and loudness level (page 2Az)
      • Figure 6: Curves showing sensitivity of the ear to changes in sound pressure (page 2Az)
      • Figure 7: How pitch scales varies with frequency (page 2Bz)
      • Figure 8: Curves showing sensitivity of the ear to changes in frequency (page 30z)
      • Figure 9: Functional diagram of heterodyne type of complex wave analyzer (page 45z)
      • Figure 10d: Overtones page 46z)
      • Figure 11d: Beats page 48z)
      • Figure 12d: Masking Effects page 49z)
      • Figure 13d: Degree of Consonance page 49z)
  • Counting and Tallying | B. A. M. Moon, Director, Computing Centre, Univ. of Canterbury (page 31z, page 4Az)
  • Our Miserable Mess of Measures? by Henry C. Churchman (page 33z)
  • More Evidence of Base Twelve Economy | Tom Linton (page 35z)
  • Conversion of Decimal Constants | Stan Bumpus (page 36z)
    • Pi equals, duodecimally (page 37z)
    • Euler’s constant equals, duodecimally (page 38z)
    • Khintchine’s constant equals, duodecimally (page 39z)
    • e equals, duodecimally (page 3Az)
    • 1/e equals, duodecimally (page 3Bz)
  • Number-Base Oddments | Shaun Ferguson (page 40z)
  • More on Douze Notre Dix Futur, work of Jean Essig (page 41z)
  • Exhibit of a Computer Program | Robert R. McPherson (page 42z)
  • Excerpts from Letters and Comments (page 43z)
  • Dedicated to an Unknown English Mathematician | Henry C. Churchman (page 4Bz)
  • Counting in Dozens (page 50z)
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.