Relationship between Bohlen–Pierce and octave-ful temperaments: Difference between revisions

Just as conventional music theory is associated with multiple temperaments (the [[rank]]-1 temperament [[12edo|12EDO]] and the rank-2 temperament [[meantone]]), the [[Bohlen-Pierce]] system is based on at least two different temperaments. The first is [[13ed3|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 [[just intonation subgroup|subgroup]]) tempering out [[245/243]]. This temperament has been referred to by various names including "Lambda temperament" or "BP Diatonic temperament".

Equal-tempered Bohlen-Pierce or 13EDT is a rank-1 temperament whose mapping is {{val|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 {{val|13 19 23 8}}, {{val|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 {{val|1 0 0 0}}, {{val|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|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|33EDO]] and [[49edo|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 {{val|1 1 2}}, {{val|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.

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.)

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.

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.)

[[Category:Bohlen-Pierce]]

[[Category:Theory]]