Relationship between Bohlen–Pierce and octave-ful temperaments: Difference between revisions
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m "octarod" is a really old name and this isn't referring to the 11-limit extension |
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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 the world of tritave equivalency, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the Bohlen–Pierce "Lambda" scale. | 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 the world of tritave equivalency, its MOS sequence goes 4, 5, 9, 13..., and the 9-note MOS is known as the Bohlen–Pierce "Lambda" scale. | ||
If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[ | If we add the prime 2 back into this system, we get a rank-3 system that has been given the name "[[sensamagic]]". | ||
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. | 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. | ||