Relationship between Bohlen–Pierce and octave-ful temperaments: Difference between revisions
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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. | 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 | Bohpier has a near-equal MOS of 8 notes, and [[41edo|41EDO]] is a particularly good equal temperament that [[support]]s 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 == | == Relationship of rank-2 "Lambda" temperament to octave-ful temperaments == | ||