source file: m1454.txt
Date: Mon, 22 Jun 1998 05:49:22 -0500

Subject: Double Equal Temperament

From: "Benjamin Tubb" <brtubb@cybertron.com>

I'm including the following info for those who might be interested in trying
this tuninig system if they haven't already heard of it.

Joseph Schillinger created Double Equal Temperamnet and described/referenced it
on pages 273, 274, and 665 of "The Mathematical Basis of the Arts":

page 273 (footnote 1): See Appendix C, which presents a primary selective
system (tuning system) in music worked out by Schillinger for the execution of
intonations not possible in our present tuning system. Schillinger called it
Double Equal Temperament to distinguish it from our equal temperament system.
(Ed.)"

page 274: "The primary selective system, which dominates the music of the
western world, is known as equal temperament and consists of a series of 12
semitones. It is possible to construct a tuning system which would permit
execution of other systems of intonation, such as mean temperament, just
intonation, and the inflections of special types of intonation. The author has
devised a system of tuning, "double equal temperament," which successfully
unifies these systems of intonation. (See Appendix C.)"

page 665 (Appendix C): "Double Equal Temperament is a primary selective system
of tuning, designed in accordance with thos theory. Double Equal Temperament
successfully unifies all systems of intonation used in the western world today.
It consists of the basic intervals: the 12th root of 2, combined with
micro-intervals: 144th root of 2, which serve as deviation units and are
arranged in relation to each basic unint. The micro-units are best averages for
all differences between units of the 12th root of 2 and just intonation
(natural scale). This tuning permits one to execute with a high degree of
precision: twelve-unit equal temperament, just intonation, and the string and
vocal inflections of special types of intonation (chamber, jazz, Gypsy music,
etc.).

An electronic organ with micro-tuning and a specially designed keyboard was
built in 1932 for the author by Leon Theremin for the performance of Double
Equal Temperament."

page 664 illustrates and defines Double Equal Temperament as:

C   1
C+  2^(1/144)
C#- 2^(11/144
C#  2^(1/12)
C#+ 2^(13/144)
D-  2^(23/144)
D   2^(1/6)
D+  2^(25/144)
D#- 2^(35/144)
D#  2^(1/4)
D#+ 2^(37/144)
E-  2^(47/144)
E   2^(1/3)
E+  2^(49/144)
F-  2^(59/144)
F   2^(5/12)
F+  2^(61/144)
F#- 2^(71/144)
F#  2^(1/2)
F#+ 2^(73/144)
G-  2^(83/144)
G   2^(7/12)
G+  2^(85/144)
G#- 2^(95/144)
G#  2^(2/3)
G#+ 2^(97/144)
A-  2^(107/144)
A   2^(3/4)
A+  2^(109/144)
A#- 2^(119/144)
A#  2^(5/6)
A#+ 2^(121/144)
B-  2^(131/144)
B   2^(11/12)
B+  2^(133/144)
C-  2^(143/144)
(C) 2

-------------
Benjamin Tubb
AIM: brtubb
ICQ: 650264
brtubb@cybertron.com
http://home.cybertron.com/~brtubb/theory.html

The Music of Stephen Collins Foster (1826-1864)
http://www.geocities.com/Nashville/9958/