The Bohlen-Pierce Site
Continued Fractions and BP

Last modified: June 1, 2009

The relationship between the fundamental just BP intervals and the division of the tritave (3:1) into 13 equal steps is a natural one. That can easily be shown using the mathematical tool of Continued Fractions.
(One of several good websites to learn about Continued Fractions is this one here).

If for any fundamental BP interval with the ratio n:d we calculate the continued-fractions convergents of the following expression:

log (n:d) / log 3

then the following table evolves (only those convergents are listed that approach the just interval values with an error of less than 1 %, and at the same time have denominators < 100):

 Ratio Convergents 9:7 3/13 8/35 - - 7:5 4/13 15/49 - - 5:3 6/13 7/15 13/28 20/43 9:5 7/13 8/15 15/28 23/43 15:7 7/10 9/13 34/49 - 7:3 7/9 10/13 27/35 -

This means nothing else than that these fundamental just intervals claim the places 3, 4, 6, 7, 9 or 10, respectively, in a scale that divides the ratio of 3:1 into 13 equal steps. Not surprisingly, two of the non-fundamental just intervals fit the pattern, too: 49:25 wants to be 8 out of 13, and its complimentary interval 75:49 opts for place no. 5.

That this relationship is something special can easily be demonstrated by a comparison with the division of the octave into 12 steps, using the same tool. In this case we investigate the convergents of the expression

log (n:d) / log 2

n:d meaning here the ratios of the fundamental just intervals of the Western scale:

 Ratio Convergents 6:5 5/19 - - 5:4 9/28 19/59 - 4:3 5/12 17/41 22/53 3:2 7/12 24/41 31/53 8:5 19/28 40/59 - 5:3 14/19 - -

There is no clear vote for any division. Only 3:2 and 4:3, as its complimentary interval, favor the division of 2:1 into 12 steps. Small wonder that this regime is under siege.