Bohlen-Pierce cont.

Here is some further information on the Bohlen-Pierce scale of 9 tones
from the 13th root of 3/1 gamut. J.R. Pierce constructed it from four
3:5:7 triads as I explained in my post and mapped these notes into the
13th root of 3/1 ET gamut. The melodic pattern is 1 2 1 2 1 2 1 2 1
degrees, a mode of the MOS whose generator is 3 degrees.

The ratios generated by Pierce's derivation are one of several possible
just intonations for his set of 9 tones and by extension of the entire
set of 13 notes. Bohlen gives the following JI interpretation of the
entire 13-tone scale: 1/1 27/25 25/21 9/7 7/5 75/49 5/3 9/5 49/25 15/7
7/3 63/25 25/9 3/1.

Bohlen used a method based on the 1/1 5/3 7/3 triad and combination
tones to derive this tuning for the 13 tone scale and selected two
subscales from the entire set. His delta (from dur, major) scale
(2 1 2 1 1 2 1 2 1 ) is a mode of Pierce's tempered form and has the
ratios 1/1 25/21 9/7 75/49 5/3 9/5 15/7 7/3 25/9 3/1. The gamma (minor?)
scale is not a mode of the above scale nor even an MOS, though it has
9 tones as well. Its successive intervals are 1 2 1 2 1 2 2 1 and its
JI form is 1/1 27/25 9/7 7/5 5/3 9/5 49/25 7/3 25/9 3/1.

In addition to these 9-tone scales, Bohlen also mentions a 12-tone
scale repeating at the 3/1. It is generated from the 1/1 7/4 5/2
triad. In ratios, it is 1/1 11/10 6/5 30/23 10/7 11/7 7/4 21/11 21/10
23/10 5/2 11/4 3/1. From the 1/1 3/2 5/2, he derives a 7 tone per
octave, 11 tone per 3/1, scale: 1/1 10/9 6/5 4/3 3/2 5/3 9/5 2/1 9/4
5/2 27/10 3/1. These are obviously related to quite different just
intonations for the 13th root of 3 scale.

An 8-tone set derived from the 1/1 7/5 9/5 chord is also described
by Bohlen, but dismissed as a curiosity.It has the ratios 1/1 10/9 6/5
9/7 7/5 14/9 5/3 9/5 2/1 and repeats at the octave.

To be honest, I do not understand how these scales are
derived from the basic chords.

Bohlen also proposes a logarithmic system of 1300 "hekts" to
the 3/1 analogous to the cycle of 1200 cents to the 2/1. To
calculate hekts, multiply the log of the interval by 1300 and
divide by the log of 3. One assumes that hekt means a hundredth
of a scale step and derives from the Greek hekaton (100) and the
metric prefix hecto-.

The refs:

H. Bohlen (1978) 13 Tonstufen in der Duodezime. Acustica 39: 76-86.
(An independent and earlier discovery of 13th root of 3 scale by
Bohlen. He refers to his unpublished manuscripts dated 1972 and 1974.
R. Wille may have also described the whole gamut in 1975 as well, but
Bohlen would appear to have priority.)

Mathews, M. V., Pierce, J. R., Reeves, Alyson, and Roberts, Linda
A. (1988). Theoretical and experimental exploration of the
Bohlen-Pierce scale, J. Acoust. Soc. Am. 84, 1214-1222
(I've seen this only as a pre-print. It has some short musical
compositions in the system and a good description of how Pierce
selected 9 of the 13-tones.)

See Also Mathews, Max V. and John R. Pierce. (1990) The Bohlen-Pierce
Scale. (Chapter 13 in New Frontiers of Computer Music, Mathews and
Pierce, eds. MIT Press pp 165-173).

--John


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