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", although the standard name by which it is referred to on this wiki is Bohlen–Pierce–Stearns.

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 Bohlen–Pierce–Stearns temperament to octave-ful temperaments

The rank-2 temperament under discussion here is the 3.5.7 temperament known as BPS 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 (the period, 3/1) and ~440 cents (which represents a sharpened 9/7, two of which make a 5/3 because 245/243 vanishes.) In 3/1-equivalence world, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is 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 it 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 this temperament's genchain 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 (which corresponds to 125/63 in the 3.5.7 subgroup) and -6 generators (which corresponds to 49/25 in the 3.5.7 subgroup). In 13ed3 these notes are "enharmonically equivalent" to each other as 13 = 7 + 6; this corresponds to tempering of the additional comma 3125/3087.

Sensi

Consider the first possibility, that 2/1 is mapped to +7 generators in the BP lambda scale, therefore tempering out 126/125. Seven generators up from C is H#, so this means H# represents 2/1 above C. In 13edt, 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, therefore tempering out 50/49. 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 which each represents 7/5~10/7. (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).