I discovered this scale in the seventies during my research in generic equal temperaments. Apparently in the same timeframe, it was discovered by Heinz Bohlen and later also by John Pierce. They have done such a wonderful job in studying and documenting this scale that it is only just that it is known by their names: the Bohlen-Pierce scale.
My first fascination with this scale was through the harmonic equivalence with the well known just intonation scale in normal 12-tone orientation in the octave. This scale is constructed from the major triads on the tonic, dominant and sub-dominant and is best depicted in a grid where the dimensions are determined by the 3rd and 5th harmonics.
12/2:
5/3
5/4
15/8
4/3
1
3/2
9/8
a:9
e:4
b:11
f:5
c:0
g:7
d:2
In exact analogical ways we can do the same thing for 13 tones in the 3rd:
13/3:
7/5
7/3
35/27
9/5
1
5/3
25/9
e:4
a:10
d:3
g:7
c:0
f:6
b:12
This gives a lovely asymmetrical scale:
|
c |
d |
e |
f |
g |
a |
b |
c |
It's funny to notice that this harmonically equivalent construction results in a melodically reversed scale.
When we modulate to the dominant the result is:
a:10
d:3
g#:9
c:0
f:6
b:12
e+:5
So like in 12/2 just intonation we have to sharpen two tones of the scale. Only in 12/2 we rarely notice this effect because one of these 'sharpenings' is enharmonically non-significant. In the 13/3 case the 'normal' sharp constitutes two steps, and the e+ equals one step. Here is my suggestion for a notation, depicting two successive dominant and two sub-dominant modulation scales:
It is now just as easy to construct equivalent minor scales:
b-:11
e:4
a:10
d:3
g:7
c:0
f:6
|
e |
f |
g |
a |
b- |
c |
d |
e |
My piece 'Odd Piano' is strictly in 'C Major' whereas 'Variations on a theme by Anton Webern' takes the big step to 13 tone serialism. Serial notation can use accidentals like this:
reference:
Prooijen, Kees van. "A Theory of Equal-Tempered Scales", Interface vol. 7 no. 1, June 1978, pp. 45-56.