Bohlen Pierce Scale
Similar to how the 12-tET scale divides an Octave (a 2/1 ratio) which "doubles" the frequency when multipled, the Bohlen-Pierce scale is tempered to the 3/1 or 'Tritave' tripling the frequency which is actually a perfectly "in-tune" fifth above the octave (since the perfect fifth is a ratio of 3/2, 3/2*2/1=6/2 or 3/1). This results in a scale which sounds tonally stretched, and at first I described the harmony originally as outside - because it's all I knew at the time. When the right timbre is used, and the strings are plucked in the center (near the octave), the octave's even partials are canceled for a square-wave like bubbley and warm timbre (allowing the perception of tritave-octave equivalence).
The intervals of the 13-tone bohlen pierce system do not include ratios above the prime 7. from the root note, the ratios of the just intonated version of the scale are:
0. 1/1
1. 27/25
2. 25/21
3. 9/7
4. 7/5
5. 75/49
6. 5/3
7. 9/5
8. 49/25
9. 15/7
10. 7/3
11. 63/25
12. 25/9
13. 3/1
The primary chord which replaces the 4:5:6 triad found in 12-tET western music is the 3:5:7 chord and it's inversion 5:7:9. However much research is still to be done in this scale, and much more music has yet to surface.
The 9-note scales which replace the typical 7-note diatonic scale contain new nominals H and J, making for a diatonic scale from C: C D E F G H J A B C. These were named Gamma and Lambda by Heinz Bohlen.
13 tone modes of Bohlen-Pierce scale:
3 3 3 3 1 BP Pentatonic
2 1 2 1 1 2 1 2 1 Delta, Moll I
1 2 1 2 1 1 2 1 2 Dur I, Delta inverse
1 2 1 2 1 1 2 2 1 Gamma
1 2 2 1 1 2 1 2 1 Gamma inverse
2 1 1 2 1 2 1 1 2 Dur II
2 1 1 2 1 2 1 2 1 Lambda
2 2 2 2 2 2 1 Krumhansl Heptatonic
1 2 1 2 1 2 1 1 2 Harmonic, Lambda inverse
1 2 1 2 1 2 1 2 1 Pierce, Moll II
3 1 2 1 3 2 1 Prooijen Major
2 1 3 1 2 3 1 Prooijen Minor
1 1 2 1 2 1 2 1 2 Walker A
1 2 1 1 2 1 2 1 2 Walker B
2 1 2 1 2 1 2 1 1 Walker I
2 1 2 1 2 1 1 2 1 Walker II
2 2 2 1 2 2 2 Sirius-7
1 3 1 3 1 3 1 Canopus-7
1 2 1 1 2 1 1 1 2 1 Canopus-10
1 4 1 1 1 4 1 Arcturus-7
1 1 3 1 1 1 3 1 1 Arcturus-10
The Bohlen-Pierce scale was discovered first in the 70's by Heinz Bohlen, and by another John Pierce shortly after during his systemic exploration of equal temperaments. Heinz Bohlen stumbled upon the scale in a much different manner. I specualte that John Pierce first stumbled onto the 13-tone 3/1 scale by looking for tonality from his octatonic scale. He had made an 8-tET scale which was criticized for lack of tones, and enough tonal stability to generate musical interest. Looking into the math and wishing to preserve the melodic effect of the size of the steps, he realized a scale with the similar sized steps could be created with 13-tones to the tritave.
Heinz Bohlen's discoveries of this scale are properly documented on his website which is an excellent resource for more theory. http://www.huygens-fokker.org/bpsite/ - Heinz's website.
The Bohlen Pierce scale has been associated with indian music due to the 146 cents interval, as well as an overall generally pscyhedelic sound which I believe is due to the large steps and stretched nature of the scale. In 2010, I presented the 9-string bohlen pierce guitar I built pictured at the top of the page, along with the theory book avilable in the books section to students at various colleges around the United States including Berklee and Northeastern Conservatory.
