Relationship between Bohlen-Pierce and octave-ful temperaments

Just as conventional music theory is associated with multiple temperaments (the rank-1 temperament 12EDO and the rank-2 temperament meantone), the Bohlen-Pierce system is based on at least two different temperaments. The first is 13EDT (13ED3), the rank-1 temperament where 3/1 is divided into 13 equal parts. Many explanations of Bohlen-Pierce simply state that this equal division is Bohlen-Pierce. But just as a description of conventional music theory wouldn't be complete if you just said it's 12 equal divisions of 2/1 and left it at that (ignoring diatonic scales, key signatures, and so on), the Bohlen-Pierce system is also strongly associated with the rank-2 temperament (of the 3.5.7 subgroup) tempering out 245/243. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament".

Relationship of 13EDT to octave-ful temperaments

Equal-tempered Bohlen-Pierce or 13EDT is a rank-1 temperament whose mapping is ⟨13 19 23]. It tempers out infinitely many commas including 245/243 and 3125/3087. What happens if we add the prime 2 back in to this system? If we add it as an independent dimension, we get a rank-2 temperament whose mapping (with the unusual basis ordering 3.5.7.2) is ⟨13 19 23 8], ⟨0 0 0 1]. So in addition to the ~146 cent step of the Bohlen-Pierce equal temperament, there is another generator of about 28 cents which is used only for intervals with the prime 2 (in other words, ratios that contain even numbers). This temperament has a name, bohpier (named after Bohlen-Pierce because of this relationship).

If we take that same abstract temperament, bohpier, and re-express it with the standard 7-limit basis ordering 2.3.5.7, we get the mapping ⟨1 0 0 0], ⟨0 13 19 23]. The generators when viewed through this lens are ~1200 cents and ~146 cents, and the unique thing you can immediately see from the mapping matrix is that 3, 5, and 7 are all multiples of the ~146 cent generator, with no octaves required for any of them. This makes sense because if you remove octaves from bohpier and use only the generator chain, it becomes equal-tempered Bohlen-Pierce again.

Bohpier has a near-equal MOS of 8 notes, and 41EDO is a particularly good equal temperament that supports it. Therefore 41EDO secretly contains a version of Bohlen-Pierce you can get by taking every fifth note. (Other EDOs are 33EDO and 49EDO but they are not so good.)

Relationship of rank-2 "Lambda" temperament to octave-ful temperaments

The rank-2 temperament under discussion here is the 3.5.7 temperament that tempers out only 245/243 (not any other commas such as 3125/3087). Its mapping matrix is ⟨1 1 2], ⟨0 2 -1] and the two generators are ~1902 cents and ~440 cents. (The generator represents 9/7 and two of them make a 5/3 because of 245/243 vanishes.) In 3/1-equivalence world, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is what's known as the BP "Lambda" scale.

If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "octarod".

What if we want a rank-2 system? In that case we must find a suitable interval in the system to represent 2/1. In theory any of the many 7-limit temperaments tempering out 245/243 (the sensamagic family) could be considered a relative of Bohlen-Pierce. But most of them divide either 3/1 or the ~440 cent generator into some number of equal parts, and so don't have the same generators/lattice as rank-2 BP itself. (In math terms the "index" is greater than 1.) For example, magic divides 3/1 into 5 equal parts. Superpyth, on the other hand, leaves 3/1 unsplit but makes 9/7 no longer the generator - instead if has complexity 6.

To find a rank-2 temperament more closely analogous to Lambda temperament but containing the prime 2, 2/1 must be mapped to some already existing note of Lambda temperament rather than a new note obtained by splitting up the 3/1 period or ~440 cent generator. There are two obvious candidates for this mapping of 2/1: +7 generators and -6 generators. (Note 7+6=13 so in 13ed3 these notes are "enharmonically equivalent" to each other.)

Sensi

Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale. Seven generators up from C is H#, so this means H# represents 2/1 above C. In equal-tempered BP H# is only 1170 cents, so in order for this to accurately represent 2/1 the generator needs to be widened by a few cents, from ~440 cents up to ~443 cents.

The result of doing this is sensi temperament.

Hedgehog

Now consider what happens if 2/1 is mapped to -6 generators instead. Six generators down from C (in the Lambda scale notation) is Jb, so now Jb represents 2/1 above C. In this case the generator must be narrowed from ~440 to ~436 cents in order for Jb to end up at a reasonable 2/1.

Furthermore, since 2/1 is being mapped to (2, -6) (up two ~1902 cent periods and down six ~436 cent generators), and both 2 and -6 are even numbers, that means 2/1 splits into two equal parts in this system. (This is in contrast to sensi in which 2/1 is intact and doesn't split into any equal parts.)

The result of doing this is hedgehog temperament.

Note that although the period and generator of hedgehog are usually given as (~600 cents, ~164 cents), an equally valid choice of generator is (~600 cents, ~436 cents), because 436 is the half-octave complement of 164. When 3/1 is taken as the period, the ~436 cent generator must be used (because the ~164 cent one doesn't generate all the notes).