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

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 {{val|1 1 2}}, {{val|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.

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.

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#<ref group="note">This page uses the standard where the nominals are assigned starting from C to the LssLsLsLs mode of the scale. See [[4L 5s (3/1-equivalent)#Notation]] for a further discussion.</ref>, 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.

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.)

[[Category:Bohlen-Pierce]]