BP equal temperament scales of more than 13 steps

The pages 24, indexed c through i, of Heinz Bohlen's original notes from 1972 deal with equal temperament scales that would approximate the just intonation Bohlen-Pierce scale better than this is done by the 13-step ET scale. He lists two groups of scales with 43, 56, 69, 82 (the "n x 13 + 4" group) and 75, 88, 101 steps (the "n x 13 + 10" Group) respectively, favoring the 101-step version because it does for BP in terms of approximation accuracy what 53-step ET does for the traditional JI Western scale. Including more recent results, the range of ET scales, sorted according to ascending accuracy regarding the approximation of just intonation BP, looks as follows:

Those marked bold approximate all tones better than 13-step ET does. The four highlighted ones were brought to the attention of the caretaker by a letter of Paul Erlich in October 1999. It should also be noted that both the 258-step and the 271-step version appear in Kees van Prooijen's publication of 1978 already.

271 notes per "tritave" (perfect twelfth)

The "chromatic" version of BP employs four different halftones: 27/25, 49/45, 375/343 and 625/567. As shown in "Development of the chromatic BP scale" they form a distinctive relationship (diamond) with
- the major BP diesis D (3125/3087 or 21.18 cents) being the distance between 375/343 and 27/25 as well as between 625/567 and 49/45, respectively,
- the minor BP diesis d ( 245/243 or 14.19 cents) being the distance between 49/45 and 27/25 as well as between 625/567 and 375/343, respectively,
- and consequently D - d (16875/16807 or 6.99 cents), called "a" in the following, being the distance between 375/343 and 49/45.

a = D - d is the smallest diesis between any just intonation BP enharmonics, with d being about twice its size, and D about three times. Its complementary interval is b = d - a (823543/820125 or 7.20 cents).

Each member in the chain of enharmonic BP intervals is composed of a kind of "quarks", steps of approximately 7 cents, from the small diesis (1 step) to the great semitone (24 steps). These steps are not equal, and a and b definitely are members of this group. The "quarks" appear already in the earliest notes on the scale where the semitones are shown as related like 24 : 22 : 21 : 19. The whole chain then consists of 271 steps (12 diamonds, 5 steps each, + 3 great links, 19 steps each, + 2 major links, 17 steps each, + 4 minor links, 16 steps each, + 4 small links, 14 steps each).

The special status of mini-intervals in the order of 7 cents becomes obvious, and an equal temperament scale of 271 steps of 7.018 cents each is indeed a most exact approximation of all tones (enharmonics included) of the just BP scale, as shown above.

But, as Paul Erlich points out, referring to an interesting web page of Kees van Prooijen : "...a scale of 271 notes per tritave - in just intonation - is the smallest such set that can be considered a more closed system than 13 notes - in just intonation - . The two concepts are different because, when an equal temperament is approximating just intonation, the comma at the end of the chain is averaged out over all the intervals in the chain, so each one may have a very small deviation from JI; while when JI is approximating an equal temperament, the comma is not averaged out and applies in full force..." (from a private letter of Paul Erlich, November 1999).

271 notes per "tritave" in just intonation? The following considerations give at least a flavor of their shape. We have already identified the small BP diesis a = D - d of 6.99 cents as one of the mini-intervals that the scale consists of, and its complementary interval b = d - a (7.20 cents). It is easy to find out that both a and b constitute a major part of a 271-step just intonation BP scale, as the following table shows. But there are (most probably two) other mini-intervals in the 7 cents range that still have to be found. Until their advent we need a kind of crutch. In this case the small semitone ST = 27/25 is used for that purpose; it contains a so far uncertain mix of nineteen mini-intervals, a number of them with certainty not identical with a or b. The "small link", for instance, with its 97.866 cents differs only by a fraction of a cent from 14-times the small diesis (97.865 cents), but in the sense of this investigation, this is just accidental.

(Links are the elements between the "diamonds" of the enharmonic chain of tones.)

Paul Erlich holds that it should be not too difficult to construct a lattice that reveals the complete 271-step JI scale, and he is most probably right. However, it would finally be of academic interest only, because it is highly unlikely that anybody would be able to distinguish acoustically between that scale and 271-step ET.