Alternative Temperaments

[Paul H. Erlich]

Hi all, I'm not subscribed to this list but I've been asked to respond to
it; please forgive me if that is poor manners.

Any equal temperament that is a meantone tuning (i.e., consonant major
thirds are formed from four fifths, and consonant minor thirds are formed
from three fifths; fifths around 693-700 cents) can be notated in the
conventional manner. Simply figure out how many steps the fifth is and
construct the circle of fifths for as many notes as required. For example,
let's construct 19-tone by starting with D and going up 9 fifths and up 9
fourths, using the fact that adding a fifth will mean adding 11 mod 19, and
adding a fourth will mean adding 8 mod 19:

D  0
A  11
E  3
B  14
F# 6
C# 17
G# 9
D# 1
A# 12
E# 4

D  0
G  8
C  16
F  5
Bb 13
Eb 2
Ab 10
Db 18
Gb 7
Cb 15

If you continued further, you'd find Fb=E# and Cb=B#.

So in 19-tone, a diatonic semitone is 2 steps, and a chromatic semitone is 1
step. This notation for 19-tone is quite standard. In 31-tone, a diatonic
semitone is 3 steps and a chromatic semitone is 2 steps. Additional
accidentals are often used (e.g., by Fokker) so that 1-step alterations are
easily notated and performed.

Non-meantone tunings, such as 22-equal, cannot be notated in the
conventional manner. The usual approach for just intonation and equal
temperaments with fifths around the Pythagorean size or larger is to use new
accidentals (such as + and -) to indicate raising or lowering by a comma (or
its equivalent). That is, four fifths above C is E, but a consonant major
third above C is E-.

My paper offers an alternate solution for 22-tone. Since my pseudo-diatonic
scale has 10 notes instead of 7, an "octave" gets 5 lines and 5 spaces on
the staff. The key signatures, shown on the last page of the paper, come out
really nice. This notation is practical since the keyboard mapping that I
propose for 22-equal uses two keyboard octaves for one acoustical octave,
and the black keys are the "naturals" and the white keys are the "sharps"
and "flats".

There is a large body of literature on microtonal notation, and a great deal
of diversity. But for meantone tunings like 19-tone there is clear
consensus. After all, common-practice harmony developed in the age of
meantone tuning!